Direct Current

This section introduces you to the foundational concepts of direct current (DC) circuits. It begins with an exploration of key electrical terms—current, voltage (potential difference), resistance, and electromotive force (EMF)—and their interrelationships using Ohm’s Law. You will be guided through the identification and behaviour of ohmic and non-ohmic conductors, and how to construct and analyse circuits using resistors in series and parallel. The section further examines the structure and function of fixed and variable resistors, including potentiometers and rheostats, and their applications. It culminates in an in-depth study of Kirchhoff’s Current and Voltage Laws and how these are used to determine unknown quantities in complex networks. You will also explore how galvanometers are adapted into ammeters and voltmeters. Prior knowledge from earlier years on charge, electric fields, and magnetic effects of currents is linked to deeper understanding.

KEY IDEAS

· Circuit Analysis Using Kirchhoff’s Laws: Application of Kirchhoff’s Current and Voltage Laws to determine currents and potential differences in complex electrical networks.

· Direct Current (DC) Circuit Fundamentals: Exploration of current, voltage, EMF, resistance, and their relationships using Ohm’s Law.

· Galvanometers and Their Adaptations: Understanding how galvanometers are converted into ammeters and voltmeters using shunt and multiplier resistors.

· Ohmic and Non-Ohmic Conductors: Identification of materials that obey Ohm’s Law versus those whose resistance varies with voltage or temperature.

· Resistor Types and Applications: Description of fixed and variable resistors, their classification, and their practical uses in controlling current and voltage.

· Resistors in Series and Parallel: Analysis of how different resistor arrangements affect total resistance, current distribution, and voltage drops in circuits.

Understanding Electricity: The Basics

Welcome to the exciting world of electricity! You encounter electricity every day – it powers our homes, phones, and countless devices that make modern life possible. But what exactly is it, and how does it work? At its core, electricity is all about the movement of tiny particles called electrons. Imagine them like incredibly small, energetic dancers moving through a wire. When these electrons have a net movement, we have an electric current.

Electric Circuits

An electric circuit is a complete path that electric charges move through. An

example is the wiring/connections in your room that makes it possible for your light and fan to work. For the charges to flow continuously, the path must be closed, meaning there are no breaks in it. A simple circuit is the most basic setup that allows electricity to flow. It consists of three important components:

1. A Power Source (e.g., Battery): This provides the Electromotive Force (EMF) or the “push” (voltage) to get the electrons moving.

2. Conductors (e.g., Wires): These are materials that allow electrons to flow easily, providing the path.

3. A Load (e.g., Light Bulb, Resistor): This is the device that uses the electrical energy and offers resistance to the flow.

Figure 3.1: Simple Circuit

Key Terminologies of Circuitry

To understand how electricity works in our devices, we need to learn some important terms related to circuits.

1. Current (I) Electric current is the time rate of flow of electric charge past a point. Think of it as how many electrons are passing a certain spot in a wire every second.

I = q/t Where q is electric charge in Coulombs (C), t is time in seconds (s).

Current is also measured in Amperes (A)

2. Potential difference It is defined as the work done to move a unit charge between two points in a circuit. Commonly known as voltage, is the electrical “push” that makes electrons move from one point to another, just like water pressure that drives water through pipes.

V = W/q Where W is the work done in Joules (J), q is the charge in Coulombs (C) and V is the potential difference in Volts (V).

Potential difference is what makes current flow through circuit elements such as resistors or bulbs, causing energy to be used up or transformed. Its unit is volt (V).

3. Electromotive Force (EMF)

It is the total energy supplied by a source (like a battery or generator) per unit of charge to drive the charge around a complete circuit. It’s the cause of the potential difference that drives the current. While closely related to voltage, EMF specifically refers to the energy provided by the source before any energy is lost within the source itself.

EMF is the maximum potential difference the source can provide when no current flows (open circuit). It corresponds to the source’s capability to convert other forms of energy (chemical, mechanical) into electrical energy.

When current flows in a circuit, the voltage or potential difference measured across the terminals of the source (called the terminal voltage or terminal potential difference) is usually less than its EMF. This is due to internal resistance of the source (energy lost inside the source).

Hence, E = V + Ir where E = EMF, V= terminal potential difference, I= current, r= internal resistance of the source.

Ir is known as the ‘lost voltage’ If there is no current (I=0), then V=E Difference Between Electromotive Force and Potential Difference Electromotive Force (EMF) Potential Difference (p.d.)

Energy supplied per unit charge by the power source Energy transferred per unit charge between two points Across terminals of the power source when no current flows A c r o s s a n y t w o p o i n t s / components in the circuit Includes energy lost inside the source E xc l u d e s i n t e r n a l e n e rg y losses, shows energy used by components Generally constant for a given source Varies depending on load and circuit conditions Related to energy provided by sources (battery, generator) Related to energy used by c o m p o n e n t s ( e . g . , b u l b s , resistors) Drives the current by supplying energy Causes energy to be transformed within the circuit

4. Resistance (R) Electrical resistance is the opposition to the flow of electric current. Imagine it as something that tries to slow down those flowing electrons. Its unit is ohms (Ω).

Figure 3.2: Water Flow Analogy for understanding key terminologies of circuitry We can easily understand the concepts of electricity by imagining a system of water flowing through pipes. In this simple analogy, the tiny electrons that make up electric current are like the water molecules, and the wires they travel through are the pipes. The Electromotive Force (EMF) from a power source, such as a battery, acts as the water pump that initiates the flow. This initial push creates the electrical pressure, or voltage, that drives the current. The opposition to this flow is called resistance, which is analogous to a valve that restricts the water’s path and causes a drop in pressure (voltage) across it. Finally, the useful work performed by the system, like a water jet spraying from a nozzle, represents electrical power, or wattage, which is the energy delivered by the electrical current.

Ohm’s Law

Now that we understand what voltage, current, and resistance are, let us talk about Ohm’s Law. This is a fundamental rule in electricity that tells us exactly how these three things are related to each other. It’s named after German physicist Georg Simon Ohm, who published his findings in 1827. This law is crucial for understanding how electrical components interact and for designing and analysing virtually any electronic circuit.

Ohm’s Law states that the current passing through a metallic conductor or wire is directly proportional to the potential difference across its ends provided the temperature and other physical conditions remain constant.

This fundamental relationship can be expressed by the following mathematical formula:

V ∝ I V = IR Where:

V = Voltage (measured in Volts, V) I = Current (measured in Amperes, A) R = Resistance (measured in Ohms, Ω) Understanding the Proportionalities

1. Direct Proportionality between Voltage and Current (for fixed Resistance)

a. If you increase the voltage across a bulb, the current flowing through it will increase proportionally, assuming the resistance remains constant.

b. Example: If a 10 Ohm bulb has 5 Volts across it, it draws 0.5 Amps. If the voltage is doubled to 10 Volts (and resistance stays 10 Ohms), the current also doubles to 1 Amp.

2. Inverse Proportionality between Current and Resistance (for fixed Voltage)

a. If you increase the resistance in a circuit while keeping the voltage constant, the current flowing through the circuit will decrease. Conversely, if you decrease the resistance, the current will increase. Where there is less resistance, more electrons move.

b. Example: With a constant 12 Volt battery, a 6 Ohm resistor will draw 2 Amps. If you replace it with a 12 Ohm resistor, the current will halve to 1 Amp.

It is important to note that Ohm’s Law applies directly to ohmic materials (like most metals and resistors), where resistance remains constant regardless of the applied voltage. However, for non-ohmic materials (like diodes or transistors), the resistance can change with voltage or current, and Ohm’s Law is still used, but the resistance value is considered dynamic.

Ohmic Conductors

An ohmic conductor is a material or component that strictly obeys Ohm’s Law.

This means that its electrical resistance (R) remains constant over a wide range of applied voltages and corresponding currents, provided that physical conditions like temperature do not change significantly.

Characteristics

1. Linear V-I Characteristic

When a graph of voltage (V) versus current (I) is plotted for an ohmic conductor, it results in a straight line passing through the origin. The slope of this V-I graph is constant and represents the resistance (R) of the conductor (R = ∆ V___ ∆ I = slope). Note that conventionally these axes are drawn the other way around, meaning that the gradient of the graph actually represents the inverse of the resistance.

Figure 3.3: A graph illustrating Ohm’s law

2. Constant Resistance

The resistance value of an ohmic conductor does not change with variations in the applied voltage or the current flowing through it.

3. Temperature Dependence (Implicit)

While ohmic conductors maintain constant resistance under constant physical conditions, their resistance can still change if the temperature changes. For most metals, resistance increases with increasing temperature. However, within a limited operating temperature range, they are considered ohmic.

Examples of Ohmic Conductors

1. Most metallic conductors: Copper wires, aluminium wires, silver, gold.

2. Fixed Resistors: Standard carbon-composition resistors or wire-wound resistors, designed to have a specific, constant resistance value.

3. Heating elements (at constant temperature): The resistance of the filament in a toaster or kettle can be considered ohmic if its temperature is kept constant.

However, in typical operation, as they heat up, their resistance changes, making them behave non-ohmically over a wide range of operation. For introductory purposes, they are often used as simple ohmic examples within a narrow temperature range.

Non-Ohmic Conductors

A non-ohmic conductor is a material or component that does not obey Ohm’s Law. For these materials, the electrical resistance changes with variations in the applied voltage, the current flowing through them, or other external factors like temperature or light intensity.

Characteristics

1. Non-linear V-I Characteristic

When a graph of voltage (V) versus current (I) is plotted for a non-ohmic conductor, it results in a curve, not a straight line. This indicates that the ratio V/I (which is effectively the resistance) is not constant and changes depending on the operating point.

2. Variable Resistance

The “resistance” of a non-ohmic conductor is not a fixed value but rather a dynamic quantity that varies with the operating conditions. This varying resistance is sometimes referred to as dynamic or differential resistance.

3. Sensitivity to External Factors

Many non-ohmic devices are specifically designed to have their resistance change based on external stimuli (like temperature, light, pressure or applied voltage), which makes them useful for various electronic applications.

Examples of Non-Ohmic Conductors

1. Semiconductor Devices

a. Diodes: These devices allow current to flow easily in one direction (forward bias) after a certain “threshold” voltage is reached, and barely at all in the opposite direction (reverse bias). Their V-I graph is highly non-linear.

b. Transistors: Fundamental components for amplification and switching, their resistance is controlled by an input signal, leading to highly non- linear behaviour.

2. Incandescent Light Bulb Filaments

As current flows through the tungsten filament, it heats up significantly.

The resistance of tungsten increases dramatically with temperature, so as the voltage increases, the filament gets hotter, its resistance increases, and the current does not increase proportionally (it increases less rapidly than predicted by Ohm’s Law with a fixed resistance). This makes them non- ohmic over their full operating range.

3. Thermistors These are resistors whose resistance changes significantly with temperature.

They are used as temperature sensors.

4. Varistors (Voltage-Dependent Resistors, VDRs)

Their resistance changes with the applied voltage, typically decreasing as voltage increases. They are used for surge protection.

5. Gas Discharge Lamps

(E.g., neon lamps, fluorescent lamps) require a high voltage to ionise the gas, after which their resistance drops sharply, leading to a non-linear V-I characteristic.

Figure 3.4: Current-voltage characteristics of Ohmic conductor (left) and an example of a non-ohmic (right) conductor. The component characteristics shown in this particular example (left) are for a diode.

Electrical Symbols for Circuit Diagrams

Circuit diagrams are like blueprints for electrical and electronic circuits. Instead of using words or pictures, they use a universal set of standardised symbols to represent different components and their connections. Learning these symbols is essential for anyone who wants to understand, design, or troubleshoot electrical circuits. Think of electrical symbols as the “alphabet” or “vocabulary” of circuit diagrams. Just like you need to know letters to read a book, you need to know these symbols to “read” a circuit diagram.

Below are some of the most fundamental and frequently encountered electrical symbols you’ll see in circuit diagrams. Have a go at drawing a circuit diagram for a battery powered torch. Compare your drawing with your neighbour.

Figure 3.5: Common electrical circuit symbols and their components

Activity 3.1 Researching and Connecting Key Electrical Terms

Objective: To enhance understanding of key electrical terms by engaging in collaborative research, peer teaching, and visual mapping.

What you need

1. Access to internet for online research

2. Notebook or digital document for notes

3. Pen or pencil

4. Large paper for drawing a concept map, or a computer/digital whiteboard app What to do

1. Choose one or more friends to work with. Form a small group (ideally 2–4 learners).

2. As a group, list these electrical terms:

a. Current

b. Resistance

c. Potential Difference (Voltage)

d. Electromotive Force (EMF)

e. Power

f. Energy

3. Divide the terms so each person researches at least one term.

4. For your assigned term, research and find out:

a. What does it mean? (Definition)

b. What is its scientific symbol?

c. What is its standard unit (and symbol)?

d. Where is it used? Give a real example in an electrical circuit or daily device.

5. Together, organise your notes clearly. Get ready to explain your term(s) to others briefly and clearly. Practice if you want.

6. Each group presents their findings to one other group. Listen carefully to other groups and take notes.

7. After all presentations, join with everyone to create a concept map showing how the terms connect.

a. Write each term on a large sheet or digital board.

b. Draw arrows to link related terms.

c. Add short notes or formulas on the arrows to explain the relationships (e.g., “Power = Voltage × Current”).

d. Include examples to make connections clearer.

Activity 3.2 Match Electrical Terms with their Symbols, Units & Definitions Objective: Test your understanding of voltage, current, resistance, and power by matching each term with its correct definition, symbol, unit, and exploring how they are related.

What you need

1. Pen or pencil

2. Notebook or blank paper

3. (Optional) Cut out printed tables as cards for matching or use sticky notes What to do

1. Review the Four Tables Below

Table 3.1: Electrical Terms

1. Voltage

2. Current

3. Resistance

4. Power

Table 3.2: Symbols

A. I B. R C. P D. V

Table 3.3: Units

a. Watt (W)

b. Ampere (A)

c. Volt (V)

d. Ohm (Ω)

Table 3.4: Definitions and Relationships

i. The flow of electric charge through a circuit; affected by voltage and resistance.

ii. The potential energy difference per unit charge between two points in a circuit.

iii. The property that opposes or limits the flow of electric current in a circuit.

iv. The rate at which electrical energy is converted into other forms like heat or light.

v. Increasing resistance decreases current if voltage stays constant.

vi. Equals voltage multiplied by current (= V × I).

2. Create a 4-column table in your notebook with these headers:

| Term | Symbol | Unit | Definition/Relationship |

3. Fill in your answers by matching the items from Tables 1-4. Use the numbered, lettered, and letter/roman-numeral codes to help keep track.

4. Under your completed table, write short sentences describing how these terms relate, for example: Increasing resistance (iii) causes current (i) to decrease if voltage stays constant (v).

5. Use your matching table to quiz yourself or a friend. Try to explain in your own words how the electrical terms connect and what happens in a circuit if one changes.

Activity 3.3 Using the PhET Circuit Construction Kit to Explore Voltage, Resistance, and Current Objective: Use the PhET Circuit Construction Kit (DC) simulation to build circuits where you manipulate voltage and resistance independently and together. Observe and explain how current changes using the formula I = V/R .

What you need

1. Access to the simulation here: PhET Circuit Construction Kit: DC

2. Notebook or digital document for recording your data and observations

3. Pen or keyboard What to do

1. Go to the PhET simulation link above. Click “Intro” or “Start” to open the workspace where you can build circuits.

2. Set Up Your First Circuit: Constant Resistance, Varying Voltage

a. Drag a battery onto the workspace and set a starting voltage (e.g., 3 V).

b. Place a resistor with a fixed resistance value (e.g., 10 Ω).

c. Connect a wire to complete a simple circuit including an ammeter to measure current.

d. Record the current reading (you can choose any point on the wire/ components to measure the current).

e. Gradually increase voltage (e.g., 3 V, 5 V, 7 V, 9 V) by clicking the battery and adjusting the voltage slider.

f. For each voltage, record the current displayed by the noncontact ammeter.

g. Note how current changes with voltage when resistance is constant.

3. Set Up Your Second Circuit: Constant Voltage, Varying Resistance

a. Fix the battery voltage (e.g., 9 V).

b. Change the resistor’s resistance to different values (e.g., 5 Ω, 10 Ω, 20 Ω, 40 Ω) by clicking on the resistor and adjusting it.

c. Record the current for each resistance value.

d. Observe how current changes as resistance increases with voltage constant.

4. Set Up Your Third Circuit: Vary Both Voltage and Resistance

a. Try different combinations by changing voltage and resistance together.

b. For example, increase voltage and decrease resistance, or vice versa.

c. Record current readings for at least 4 different voltage-resistance pairs.

d. Look for patterns in how current changes with both variables changing.

5. Create a table in your notebook with columns: Voltage (V), Resistance (Ω), Current (A), and your observations.

6. Analyse Your Results

Using the formula, explain

a. How does current change when you increase voltage but keep resistance the same?

b. How does current change when you increase resistance but keep voltage the same?

c. What happens to current when you change both voltage and resistance?

7. Write a short paragraph describing how voltage and resistance affect current in a circuit based on your observations from the simulation.

Activity 3.4 Testing Electronic Components and Plotting V–I Graphs to Identify Ohmic and Non-Ohmic Behaviour Objective: Use a circuit to measure the voltage across and the current through different components (such as resistors, bulbs, and diodes). Plot voltage-current (V–I) graphs to determine which components obey Ohm’s Law (ohmic) and which do not (non-ohmic) and explain the differences in their behaviour.

What you need

1. Power supply (variable voltage)

2. Resistors of different values

3. Filament bulbs

4. Diodes

5. Connecting wires and switch

6. Ammeter (to measure current)

7. Voltmeter (to measure voltage)

8. Rheostat or variable resistor (to change circuit voltage or current)

9. Breadboard or circuit board (optional)

10. Graph paper or software for plotting graphs

11. Notebook and pen for recording data What to do

1. Connect the power supply, ammeter, variable resistor/rheostat and the component to be tested (resistor, bulb, or diode) in series as shown in the typical circuit diagram for V–I characteristics. Ensure that a voltmeter is connected in parallel across the component.

2. Measure Voltage and Current

a. Close the switch and start with the power supply at a low voltage.

b. Record the voltage across the component from the voltmeter and the current through it from the ammeter.

c. Gradually increase the voltage in small steps (e.g., 0.5 V increments), recording multiple pairs of voltage and current readings (aim for 5–6 readings).

d. Take multiple readings at each step and average the current to improve accuracy.

e. For diodes, keep the supply voltage very small and increase by 0.1V at a time if possible. Also, reverse the polarity of the power supply to observe the behaviour in reverse bias.

3. Repeat for Each Component

a. Replace the resistor with the filament bulb and repeat the measurements.

b. Then test the diode similarly (with care to record forward and reverse bias behaviour).

4. Plot V–I Graphs

a. For each component, plot a graph of voltage (V) on the x-axis and current (I) on the y-axis using your recorded data.

b. Draw smooth curves or straight lines depending on the shape of your data.

5. Analyse Your Graphs

a. Identify which graph is a straight line passing through the origin – this indicates ohmic behaviour (component obeys Ohm’s Law with constant resistance).

b. Identify which graphs are curved or non-linear – these indicate non- ohmic behaviour, where resistance varies with voltage or current.

6. Safety Reminder

a. Work carefully with electrical equipment.

b. Do not exceed voltage ratings of components to avoid damage.

c. Switch off power supply when changing components or wiring.

Summary Table for Recording (one per component) Voltage (V) Current (A) Repeat 1 Current (A) Repeat 2 Current (A) Repeat 3 Current (A) Average (Add rows as needed)

Activity 3.5 - Choosing Efficient Lighting for a Solar-Powered Community Library Objective: Solve a real-world problem by comparing different bulbs to find lighting that uses the 12V solar battery efficiently. Calculate current for each bulb using Ohm’s Law, test bulb brightness in a circuit (physical or simulated), and justify your choice using both calculations and observations.

What you need

1. Calculator

2. Notebook or paper for recording data and calculations

3. Pen or pencil

4. Access to a circuit simulation tool (e.g., PhET Circuit Construction Kit) or physical components:

a. 12V power supply (or 12V battery/simulation)

b. Several bulbs or LEDs with known voltage and resistance ratings

c. Connecting wires and switch

d. Ammeter and voltmeter (for physical circuit) or built-in meters in simulation What to do

1. Understand the Problem

Imagine your community library runs on a 12V solar battery. The lighting should provide enough brightness but also preserve battery life by not drawing excessive current.

2. Collect Bulb Data

For each bulb, note the following (from product specs or simulation settings):

a. Rated voltage (should be near 12V)

b. Resistance (if given) or power rating (Watts)

3. Calculate Current Using Ohm’s Law

For each bulb, calculate the current drawn when connected to the 12V battery. Use the formula: I = V/R where V=12V and R is bulb resistance.

4. Set Up the Circuit or Simulation

a. Using a physical circuit or a circuit simulator, build a simple circuit with the 12V source and one bulb.

b. Connect the bulb and measure the current flowing through it (or observe the simulated current).

c. Observe the brightness of the bulb.

5. Test All Bulbs

a. Repeat the circuit or simulation setup for each bulb.

b. For each, record:

i. The measured/calculated current

ii. Your observation about bulb brightness

6. Compare Results

a. Make a table listing bulbs, their calculated currents, observed brightness, and your notes.

b. Consider which bulbs draw less current but still provide adequate brightness.

7. Choose the Best Bulb

a. Based on your calculations and observations, decide which bulb best balances brightness and battery life.

b. Write a short explanation justifying your choice using numbers (current values) and your brightness observations.

c. Answer the question: What affects the brightness of a bulb in a circuit?

Example Table to Fill

Bulb Type/

Model Resistance

(Ω) Calculated/ Measured

Current (A) Observed

Brightness Notes/

Recommendation Bulb A

Bulb B Bulb C

Activity 3.6 Applying Ohm’s Law to Calculate Current and Resistance Study the worked example below carefully before attempting the example questions that follow.

Worked Example 1

An electric heater is connected to a 240 V power supply. If the heating element has a resistance of 20 Ω, how much current flows through the heater?

Solution

Step 1 - Identify the knowns and unknowns Voltage (V) = 240 V Resistance (R) = 20 Ω Current (I) =? (unknown)

Step 2 - Introduce the appropriate formula I = V/R

Step 3 - Substitute the values and calculate I = 240/20 I = 12 A Practice Problems Now, using the worked example as a guide, solve the following problems individually or in groups.

1. An electric fan is operating on a 220 V supply. If the motor has a resistance of 110 Ω, how much current does the fan draw?

2. A smartphone charger outputs 5 V and provides a current of 2 A to your phone. What is the equivalent resistance of your phone’s charging circuit at that moment?

Resistors What is a Resistor?

In any electrical circuit, all components, including the wires, offer some resistance to the flow of electrons. This inherent resistance is often undesirable, as it can lead to energy loss. However, there are many cases where we need a specific, controlled amount of resistance in a circuit to limit the current. This is where a resistor comes in.

A resistor is a dedicated component engineered to provide a precise amount of resistance, ensuring that the current is reduced to a desired level for a particular part of the circuit.

In simpler terms, it is a passive two-terminal electrical component device that opposes the flow of electric current.

Figure 3.6: A resistor (right) and its circuit symbol (left) Why Do We Use Resistors?

Resistors cannot be ignored in electronics and are used for a variety of crucial functions:

1. Current Limiting: One of the most common uses is to protect sensitive components (like LEDs or integrated circuits) from excessive current that could otherwise damage them. By adding a resistor in series, you can control the current flowing through that part of the circuit.

2. Voltage Division: Resistors can be arranged to create specific voltage levels from a higher supply voltage. This is achieved by placing two or more resistors in series, where the voltage drops across each resistor are proportional to their resistance values.

3. Biasing Components: In circuits with transistors or other active components, resistors are used to set the correct operating conditions (bias points) for optimal performance.

4. Pull-up/Pull-down Resistors: In digital circuits, these resistors are used to ensure that a digital input has a defined state (either high or low) when no other signal is applied, preventing “floating” inputs that can lead to unpredictable behaviour.

5. Signal Conditioning: Resistors are used in filters, timing circuits, and other signal processing applications to shape or modify electrical signals.

Types of Resistors

Resistors come in many forms, each suited for different applications based on their construction, precision, and power handling capabilities:

Resistors are mainly classified into two broad types: linear resistors and non- linear resistors.

Linear Resistors

These resistors have a resistance value that largely remains constant or changes predictably with applied voltage and temperature. They generate a voltage drop proportional to the current passing through them. Linear resistors are further divided into:

1. Fixed Resistors: These have a set resistance value that does not change.

2. Variable Resistors: These resistors allow their resistance value to be adjusted.

i. Potentiometers: These are three-terminal devices that function as adjustable voltage dividers. They are commonly used for volume control in audio equipment, brightness control for lights, or as input sensors for microcontrollers where a variable voltage is needed.

ii. Rheostats: Typically, two-terminal devices (or potentiometers used with only two terminals), rheostats are primarily used to control current in a circuit. They are often found in higher power applications.

Non-Linear Resistors

Their resistance varies non-linearly with voltage, temperature, or other factors.

Key types include:

1. Thermistors: Their resistance changes significantly with temperature. They are widely used as temperature sensors (NTC thermistors decrease resistance with increasing temperature, PTC thermistors increase).

2. Photoresistors (LDRs - Light Dependent Resistors): Their resistance decreases as the intensity of light falling on them increases. They are used in light-sensing applications, like automatic streetlights or light meters.

3. Varistors (Voltage Dependent Resistors - VDRs): Their resistance changes with the applied voltage, typically decreasing sharply at a certain voltage.

They are primarily used for surge protection, diverting excess voltage away from sensitive components.

Resistor Colour Code

The resistor colour code is a system that uses coloured bands painted on the resistor body to indicate its resistance value, tolerance, and sometimes reliability or temperature coefficient. This system allows you to quickly determine resistor specifications without reading numerical labels.

Here is how it works

1. Each colour represents a specific digit, multiplier, or tolerance value based on an international standard (IEC 60062).

2. The colour bands are read from one end, typically starting closest to a band that is spaced farther apart or a tolerance band.

3. Commonly, resistors have 4 or 5 colour bands:

a. In a 4-band resistor, the first two bands represent the significant digits, the third band is the multiplier (power of ten), and the fourth band indicates tolerance (how much the actual resistance may vary).

b. In a 5-band resistor, the first three bands are significant digits, the fourth is the multiplier, and the fifth is the tolerance, allowing more precise values.

Figure 3.8: Resistor colour code chart To Read a Resistor Colour Code

1. Identify the tolerance band (gold, silver, or none), usually spaced apart or at one end.

2. Read bands from the opposite end towards the tolerance band.

3. Convert the first two or three bands into digits.

4. Apply the multiplier from the next band.

5. Note the tolerance from the last band.

Resistors in series and parallel In electrical circuits, how resistors are connected drastically affects the total resistance and how current flows through the circuit. Two fundamental ways to connect resistors are in series and in parallel, each influencing voltage, current, and resistance differently.

Resistors in Series

Figure 3.9: Two resistors in series When resistors are connected end-to-end, one after the other, they form a series circuit. This simply means the electrons making up the current will enter the resistors one after the other (serially), from first to the last. In this arrangement, the electric current has only one path to follow. Because of this, the same current flows through each resistor without changing. This is an important point: current is constant through resistors connected in series. However, the voltage across each resistor in the series varies depending on its resistance. The total voltage supplied by the power source is divided among the resistors. The voltage drop across each resistor can be different, depending on how much each resistor resists the current.

To find the total resistance of resistors in series, simply add their resistance values:

Rₜₒₜₐₗ = R₁ + R₂ + R₃ + …Rₙ This means that the total resistance increases as more resistors are added in series Proof

Figure 3.10: Circuit diagram of three resistors connected in series Let us consider three resistors, R₁, R₂and R₃, in series with a voltage source V.

Current I flows through both.

From Ohm’s law, V = I Rₜₒₜₐₗ Voltage across each resistor Voltage across R₁ V₁= IR₁ Voltage across R₂ V₂ = I R₂ Voltage across R₃ V₃ = I R₃ Total voltage supply V = V₁ + V₂ + V₃ By substitution I Rₜₒₜₐₗ = I R₁+ I R₂+ I R₃ I Rₜₒₜₐₗ = I (R₁+ R₂+ R₃) Divide both sides by I Rₜₒₜₐₗ = R₁ + R₂ + R₃ This can be extended to any number of resistors in series:

Rₜₒₜₐₗ = R₁ + R₂ + R₃ + …Rₙ The total resistance of a series connection is greater than the single greatest resistance in the connection.

Resistors in Parallel

Figure 3.11: Three resistors in parallel Resistors are connected in parallel when they are connected across the same two points, giving the current multiple paths to flow. Again, this simply means the current will divide and enter the resistors at the same time (parallel or concurrent entry).

Unlike series circuits, the voltage across each resistor in parallel is the same, and equal to the total voltage supplied by the source.

Because the current can split and flow through each resistor separately, the total current flowing into the parallel network is the sum of currents through each resistor.

The key characteristic of resistors in parallel is that the total or equivalent resistance is always less than the smallest individual resistor in the group. This happens because adding more paths allows more current to flow, reducing overall resistance.

The formula for total resistance in a parallel circuit is 1_ Rₜₒₜₐₗ = 1/R₁ + 1/R₂ + 1/R₃ + … + 1/Rₙ Proof

Figure 3.12: Circuit diagram of three resistors connected in parallel Consider three resistors, R₁, R₂and R₃, in parallel across a voltage source V.

The voltage across each resistor is the same (V) because their ends are directly connected.

By Ohm’s law, the current through each resistor is:

I₁ = V/R₁ I₂ = V/R₂ I₃ = V/R₃ Also, total current Iₜₒₜₐₗ = V/Rₜₒₜₐₗ The total current Iₜₒₜₐₗ supplied from the source is the sum of currents through each resistor by Iₜₒₜₐₗ = I₁ + I₂ + I₃ Substitute:

V/Rₜₒₜₐₗ = V/R₁ + V/R₂ + V/R₃ V/Rₜₒₜₐₗ = V( 1/R₁ + 1/R₂ + 1/R₃ ) Divide both sides by V 1_ Rₜₒₜₐₗ = 1/R₁ + 1/R₂ + 1/R₃ This formula also generalises to n resistors in parallel:

1_ Rₜₒₜₐₗ = 1/R₁ + 1/R₂ + 1/R₃ + … + 1/Rₙ Many practical circuits are not just simple series or parallel but are a combination of both. In these cases, you can simplify complex circuits step-by-step.

First, identify groups of resistors that are clearly in parallel or series. Calculate their equivalent resistances individually, then replace those groups with their equivalent resistor in the circuit. Repeat this process until you find the total resistance between two points.

Figure 3.13: Series-parallel connections of resistors Looking at the circuit above; how would you expect the supply voltage to be shared between each of the resistors in the circuit? How would the current through each resistor compare?

Resistivity Resistivity is an intrinsic property of a material that quantifies how strongly that material opposes the flow of electric current. It is a fundamental characteristic of the material itself, regardless of its shape or size.

Resistivity depends only on the type of material (e.g., copper, silver, wood, glass) and its temperature. The unit of resistivity is the Ohmmeter (Ω⋅m) The resistance R of a uniform conductor depends on its resistivity ρ, its length l, and the cross-sectional area A, described by the formula:

R = ρl/A where R = resistance (ohms, Ω) ρ = resistivity of the material (Ω·m) l = length of the resistor (m) A = cross-sectional area (m²) This formula tells us

1. Resistance increases with greater length (l) because electrons encounter more obstacles as they travel farther.

2. Resistance decreases with a larger cross-sectional area (A) because a thicker conductor allows more paths for electrons to flow.

3. Different materials have different resistivity values ρ, affecting their resistance even if they have the same size and shape.

Variable Resistors

In many electronic circuits, we need to be able to adjust the resistance to control the current or voltage. Think of a volume knob on a radio, a dimmer switch for a light, or a speed control for a fan. These devices all rely on a variable resistor.

A variable resistor is an electrical component whose resistance can be changed manually. The two most common types of variable resistors are the rheostat and the potentiometer. While they share a similar internal construction, they are used for very different purposes in a circuit.

The Rheostat: Controlling Current

A rheostat is a two-terminal variable resistor used to control the current in a circuit. It works by varying the total resistance of the circuit. By changing the resistance, it changes the total current flowing through the circuit, according to Ohm’s Law (I=V/R).

Figure 3.14: Rheostat

The Structure

A rheostat typically consists of a resistive wire wound around an insulating core.

A sliding contact, or “wiper,” moves along the wire. The two terminals are:

1. One end of the resistive element.

2. The sliding contact.

When you move the wiper, you change the length of the resistive wire that the current must travel through. A longer path means more resistance, and a shorter path means less resistance.

Common Applications

1. Dimmer Switches: For lights, a rheostat controls the amount of current flowing to the bulb, changing its brightness.

2. Motor Speed Control: Rheostats can be used to control the speed of a DC motor by varying the current flowing through it.

3. Some Heaters: By changing the resistance of a heating element, a rheostat can control the amount of heat produced.

The Potentiometer: Controlling Voltage

A potentiometer (often called a “pot”) is a three-terminal variable resistor used to control the voltage in a circuit. It acts as a voltage divider, providing an output voltage that is a fraction of the input voltage.

Figure 3.15: A typical single turn Potentiometer The Structure A potentiometer has three terminals:

1. Terminal A: One end of the resistive element.

2. Terminal B: The other end of the resistive element.

3. Terminal C (Wiper): A sliding contact that moves along the resistive element.

When an input voltage is applied across the two ends (A and B), the wiper (C) can be moved to “tap” off a fraction of that voltage. The output voltage is measured between one end terminal (e.g., B) and the wiper (C).

The Voltage Division Principle

The total resistance between terminals A and B is constant (Rₜₒₜₐₗ). The resistance between the wiper C and terminal B (Rₒᵤₜ) varies as the wiper moves. The output voltage (Vₒᵤₜ) is a fraction of the input voltage (Vᵢₙ) given by the formula:

Vₒᵤₜ = Vᵢₙ Rₒᵤₜ_ Rₜₒₜₐₗ By moving the wiper, you change Rₒᵤₜ, which in turn changes Vₒᵤₜ.

Figure 3.16: Voltage divider circuit using two resistors Vₒᵤₜ = Vᵢₙ R₂_ R₁ + R₂ Common Applications

1. Volume Control: The volume knob on an audio system is a potentiometer.

It adjusts the voltage signal sent to the amplifier, which controls the output volume.

2. Sensor Input: Potentiometers can be used as simple position sensors. For

example, a joystick or a rotary sensor can be a potentiometer, where the position of the wiper sends a different voltage signal to a microcontroller.

3. Dimmer Switches: While a rheostat can be used, many modern dimmers use a potentiometer to control the voltage supplied to a more complex power control circuit.

While many potentiometers are small, rotary components used for volume controls and everyday electronics, the sliding wire potentiometer is a specialised, high- precision instrument. It is a fundamental tool in physics laboratories and electrical engineering for accurate measurement of electromotive force (EMF) and voltage without drawing any current from the circuit under test. This type of potentiometer is not used for continuously variable output in everyday devices, but rather as a precise measuring instrument based on the principle of null deflection.

The sliding wire potentiometer operates on the principle that the potential drop across a uniform resistance wire is directly proportional to its length, provided a constant current flow through it. The core idea is to find a “balance point” where the voltage drop across a section of the wire is exactly equal to an unknown EMF. At this point, no current flows through the measuring circuit, which means the potentiometer measures the true, open-circuit EMF of the source. This is a significant advantage over a voltmeter, which always draws some current and therefore measures a slightly lower terminal voltage.

Figure 3.17: Sliding wire potentiometer The Structure A sliding wire potentiometer typically consists of:

1. A long, uniform resistance wire: This is the “sliding wire” itself. It is often made of a material like manganin or constantan, which has a high resistivity and a very low temperature coefficient to ensure the resistance is uniform and stable.

2. A driving or “driver” cell: A battery that provides a constant current through the entire length of the sliding wire. A rheostat is usually connected in series with this cell to adjust and standardise the current.

3. A jockey (sliding contact): A metal contact that can be moved along the length of the wire to tap off a specific voltage drop.

4. A standard cell: A cell with a precisely known EMF (e.g., a Weston cell) used to “standardise” or calibrate the potentiometer before making a measurement.

5. A galvanometer: A sensitive current-detecting device used to indicate when no current is flowing in the measuring circuit (the null point).

Figure 3.18 Potentiometer circuit How it Works (The Null Method) The potentiometer uses a null method to accurately measure the EMF of an unknown cell (E2) without drawing any current from it. A driving cell (E₁) sets up a constant potential drop (voltage) across the entire length of a uniform resistance wire A to B (let’s call this length L₁). The cell to be measured (E₂) is connected in a separate circuit with a galvanometer and a sliding contact called a jockey. The jockey is moved along the wire until the galvanometer shows zero deflection.

This is the null point. At this null point, the voltage drop across the length of the wire from A to the jockey X (let’s call this length L₂) is exactly equal to the EMF of the unknown cell (E₂). Since no current flows from the unknown cell at this point, the measurement is of its true, open-circuit EMF, which is the most accurate possible.

E ∝ L E₁_ L₁ = E₂_ L₂ E₂ = L₂_ L₁ × L₁

Activity 3.7 Determining and Verifying Resistor Values Using Colour Codes and a Multimeter Objective: To develop practical skills in identifying and verifying resistor values by applying knowledge of the resistor colour code and using a digital multimeter.

What you need

1. A variety of resistors (at least 4–6, each with different bands of colours)

2. Resistor colour code chart (printed or digital)

3. Digital multimeter (set to measure resistance)

4. Pen and notebook or worksheet for recording values

5. Calculator (optional) What To do

1. Identify a friend or friends to form a small learner group.

2. Observe the example below Suppose you have a resistor with the colour bands: Red, Violet, Yellow, Gold

Figure 3.19: Resistor with colours red, violet, yellow, gold

a. Use the colour code chart

i. Red = 2 (first digit)

ii. Violet = 7 (second digit)

iii. Yellow = ×10,000 (multiplier)

iv. Gold = ±5% (tolerance)

b. Calculate the value Value = (2 7) × 10,000 = 270,000Ω or 270kΩ Tolerance = ±5% So, this resistor’s value is 270kΩ ±5%

3. Take a set of resistors from your kit or group’s supply. Assign each group member one or more resistors.

4. Use the colour code chart to decode the bands on each resistor. Write down

a. The colours

b. Each digit and the multiplier value

c. The calculated resistance value and tolerance

5. Use a table to organise your findings Resistor Number Colours Calculated Value (Ω) Tolerance 1 Red, Violet, Yellow, Gold 270,000 ±5% 2 ... ... ...

3 ... ... ...

6. Measure with a Multimeter

a. Set the multimeter to the resistance (Ω) setting.

b. Hold the resistor leads and measure the resistance.

c. Record the measured value in your table.

Resistor Number Calculated Value (Ω) Measured Value (Ω) Difference 1 270,000 268,500 -1,500 2 ... ... ...

7. Compare and Discuss

a. Compare each calculated value (from colour codes) with the measured value. Does it fall within the tolerance indicated on the colour bands?

b. Write down any differences or discrepancies.

c. Discuss possible reasons for differences (e.g., tolerance limits, accuracy of multimeter or resistor, small manufacturing variations).

8. Group Discussion

a. As a group, share your findings and explanations.

b. Reflect on why resistors have tolerances and the importance of accurately knowing resistance in circuits.

Activity 3.8 Identifying Appliances That Use Resistors and Understanding Their Role Objective: To enhance understanding of the practical applications of resistors by exploring their roles in everyday electrical appliances.

What you need

1. Notebook or digital device for writing

2. Internet access for research

3. Pen or keyboard What to do

1. Make a list of common electrical appliances you know or find that use resistors. Examples include heaters, toasters, electric stoves, chargers, LED lights, and fan speed controllers.

2. Think about and write down: Why do you think these appliances need resistors? Consider what resistors do in electrical circuits, such as controlling current, dividing voltage, or converting electricity to heat.

3. Pick one appliance from your list that interests you most.

4. Use reliable online resources to find out

a. How resistors are used in this device.

b. Why the resistors are important for the device’s operation or for protecting other components.

c. Any specific types of resistors used in the appliance (e.g., variable resistors, heating elements, ballast resistors).

5. Summarise what you learned in your notebook or digital document.

Explain the role of resistors clearly and briefly in your chosen appliance, using your own words.

6. If possible, share your findings with a friend or group. Discuss how resistors help make appliances work safely and efficiently.

Activity 3.9 Building and Investigating Series and Parallel Resistor Circuits Objective: Build resistor circuits in series and parallel using a breadboard, bulbs, resistors, and batteries. Observe bulb brightness, measure current, voltage, and equivalent resistance using a multimeter, and compare your measurements with calculated values.

What you need

1. Breadboard

2. Resistors (e.g., 3 resistors of known values, such as 1 kΩ each)

3. Bulb (with holder)

4. Batteries or a DC power supply (e.g., 5V)

5. Connecting wires

6. Multimeter (to measure voltage, current, and resistance)

7. Switch (optional)

8. Calculator

9. Notebook and pen for recording data What to do

1. Set Up the Series Circuit

a. Connect three resistors in series on the breadboard.

b. Connect the bulb, also in series with the resistors, to observe brightness.

c. Connect the series circuit to the battery or power supply.

d. Connect the multimeter as follows

i. Use the ammeter mode and place it in series to measure the current flowing through the circuit.

ii. Use the voltmeter mode and connect it in parallel across the entire resistor chain or individual resistors to measure voltage drops.

2. Record Measurements for Series Circuit

a. Measure the current through the circuit and write it down.

b. Measure the voltage across each resistor and across the entire series combination, noting each reading.

c. Measure the resistance of each resistor individually using the ohmmeter function.

d. Calculate the total resistance using the formula:

Rₜₒₜₐₗ = R₁ + R₂ + R₃

e. Use Ohm’s Law (V=IR) with your measured current and voltage to calculate the total resistance and compare with the sum of individual resistors and measured total resistance.

3. Set Up the Parallel Circuit

a. Rearrange the three resistors so they are connected in parallel on the breadboard.

b. Connect the bulb adjacent to these; in series with the parallel resistor combination.

c. Connect this parallel circuit to the battery or power supply.

d. Measure current using the multimeter placed in series with the power supply.

e. Measure voltage across each parallel resistor (should be the same).

f. Measure the equivalent resistance using the formula 1_ Rₜₒₜₐₗ = 1 __ R₁ + 1 __ R₂ + 1 __ R₃

g. Use your measurements of current and voltage to calculate total resistance via Ohm’s Law and compare with your formula result and direct resistance measurement via the multimeter.

4. Observe and Record

a. Note differences in bulb brightness between series and parallel setups.

b. Record all measurements in your notebook with clear labels.

5. Compare and Explain

a. Compare measured voltage, current, and resistance values with calculated values for both circuits.

b. Explain any differences and relate how series and parallel configurations affect current flow, voltage distribution, total resistance, and bulb brightness.

Activity 3.10 How Changing Resistance Affects Current Using a Variable Resistor Objective: Understand how adjusting resistance in a circuit changes current flow by observing how it affects the brightness of a bulb or the sound of a buzzer. Relate your observations to Ohm’s Law and discuss real-world uses of variable resistors.

What you need

1. Power source (e.g., battery or DC power supply)

2. Bulb (with holder) or buzzer

3. Variable resistor (potentiometer or rheostat)

4. Connecting wires

5. Switch (optional)

6. Notebook and pen What to do

1. Connect the power source, variable resistor, and bulb or buzzer in series using the wires. If you have a switch, include it to control the circuit easily.

2. Close the switch (or complete the circuit) so current can flow.

3. Slowly turn the knob of the variable resistor, or slide the contact, to increase and decrease the resistance.

4. Observe how the bulb’s brightness or buzzer’s sound changes as you adjust the knob.

5. Note what happens to the brightness or sound as resistance increases or decreases. Write down your observations clearly in your notebook.

6. Remember the formula I = V/R Discuss in your notes how increasing resistance lowers current (dimmer bulb or quieter buzzer) and decreasing resistance raises current (brighter bulb or louder buzzer).

7. Write about some examples where variable resistors are used, like volume controls on radios, light dimmers, or motor speed controls.

8. Explain how changing resistance in these devices adjusts their output or performance.

Activity 3.11 How Wire Length, Thickness, and Material Affect Resistance Objective: Explore how different physical properties of wires—length, thickness, and material—affect their electrical resistance by measuring with a multimeter and analysing the results.

What you need

1. Wires of different lengths (e.g., 10 cm, 20 cm, 30 cm)

2. Wires of different thicknesses/gauges (e.g., thin and thick wires)

3. Wires made of different materials if available (e.g., copper, nichrome)

4. Crocodile clips or alligator clips

5. Digital multimeter (for measuring resistance)

6. Ruler or tape measure

7. Notebook and pen What to do

1. Gather various wire samples that differ in length, thickness, and material.

Label each wire to keep track of which wire you are testing.

2. Use a ruler or tape measure to record the exact length of each wire. Note the thickness or gauge of each wire if known.

3. Measure Resistance

a. Set the multimeter to resistance mode.

b. Attach the multimeter probes to each end of a single wire and record the resistance reading.

c. Repeat for every wire sample, measuring one wire at a time.

4. Record your data. Use the table below to record measurements and observations for each wire separately.

Wire ID

Material Length

(cm) Thickness/ Gauge

Measured Resistance (Ω)

Observations/ Notes

1 2 3 4

5. Analyse Your Results

a. Look for how resistance changes with wire length.

b. Compare resistance readings between wires of different thicknesses but equal lengths.

c. Compare resistance between wires made from different materials with similar sizes.

6. Draw Conclusions

a. Explain how the length, thickness, and material of the wire influence its resistance.

b. Relate your findings to the formula R = ρL/A where R is resistance, L is length, A is cross-sectional area (thickness), and ρ is resistivity (material).

Activity 3.12 Resistance in Electrical Circuits: Series, and Parallel Study the worked example below carefully before attempting the sample questions that follow.

Worked Example 1

A string of 10 identical Christmas lights is connected in series. Each light bulb has a resistance of 15 Ω. What is the total resistance of the string of lights?

Solution

Step 1: Identify the individual resistance Each light bulb (R1, R2 , ..., R10 ) has a resistance of 15 Ω.

Step 2: Introduce the series resistance formula Rₜₒₜₐₗ=R₁+R₂+...+R₁₀

Step 3: Substitute and calculate the total resistance Rₜₒₜₐₗ= 15 Ω + 15 Ω+ 15 Ω+ 15 Ω+ 15 Ω+ 15 Ω+ 15 Ω+ 15 Ω+ 15 Ω+ 15 Ω Rₜₒₜₐₗ= 150 Ω

Worked Example 2

In a kitchen, a toaster of resistance 24 Ω and a coffee maker of resistance 18 Ω are plugged into the same parallel circuit. What is the total resistance of these two appliances when they are both operating.

Solution

Step 1: Identify the individual resistance Rₜₒₐₛₜₑᵣ=24 Ω Rcoffee maker=18 Ω

Step 2: Introduce the parallel resistance formula 1_ R = 1/Rₜₒₐₛₜₑᵣ + 1_______ R_(coffee) ₘₐₖₑᵣ

Step 3: Substitute and calculate the total resistance 1_ R = 1/24 + 1/18 1_ R = 7/72 R = 72/7 R = 10.29 Ω

Worked Example 3

A house is wired with copper wire having a resistivity of 1.68×10⁻⁸Ω⋅m. If a particular circuit requires a 15-metre length of wire with a circular cross- sectional radius of 1.0 mm, what is the resistance of this wire?

Solution

Step 1: Identify the given values ρ =1.68×10⁻⁸Ωm l =15 m Radius (r) = 1.0 mm = 1.0 × 10⁻³m

Step 2 - Calculate the cross-sectional area A = π r²A = π (1.0 × 10⁻³m)²A = 3.14 × 10⁻⁶m²Step 2 - Introduce the resistance formula R = ρL/A

Step 3 - Substitute and calculate the total resistance R = 1.68 × 10⁻⁸× 15/3.14 × 10⁻⁶R = 8.025 × 10⁻²Ω Practice Problems Now, using the worked examples as a guide, solve the following problems individually or in groups.

1. A car’s taillight system has two bulbs.

Scenario A: If the bulbs are wired in series, and each bulb has a resistance of 6 Ω, what is the total resistance?

Scenario B: If the bulbs are wired in parallel, and each bulb has a resistance of 6 Ω, what is the total resistance?

Which car’s lights would you expect to shine with a greater brightness, and why?

2. You are designing a heating element for a small portable heater using Nichrome wire, which has a resistivity of 1.1×10−6 Ωm. You need the heating element to have a resistance of 50 Ω. If you use Nichrome wire with a diameter of 0.5 mm, what length of wire will you need?

Kirchhoff’s Laws

Imagine you are trying to figure out the traffic flow at a busy intersection. You know cars are coming in from three different streets and leaving on a fourth. You cannot just guess how many cars are going to each street, can you? You need a rule! The rule is simple: the number of cars entering the intersection must equal the number of cars leaving. This is the basic idea behind Kirchhoff’s Laws, but for electric current and voltage in a circuit.

Ohm’s Law is a powerful tool, but it only works for simple, single-loop circuits or for individual components. What about complex circuits with multiple loops and junctions (points where wires split)? That’s where Kirchhoff’s Laws come in.

They provide a set of rules to analyse these more complicated circuits.

Kirchhoff’s First Law- The Current Law (KCL)

The principle Kirchhoff’s Current Law, or KCL, is based on the principle of conservation of charge. Just like cars at an intersection, electric charge cannot be created or destroyed. It must flow somewhere.

Kirchhoff’s first law states that at any junction of an electrical network, the sum of all currents entering the junction is equal to the sum of all currents leaving it.

OR It states that the algebraic sum of the electric currents at any junction or node of an electrical network is zero.

Figure 3.20: An illustration of Kirchhoff’s Current Law at a circuit junction Practical Example Let’s say a circuit junction has three wires. We know the current in the first wire is 5A flowing in, and the current in the second wire is 2A flowing out. What is the current in the third wire?

Using KCL: Iᵢₙ=Iₒᵤₜ 5A = 2A+I_(third) I_(third)= 5A−2A = 3A So, the current in the third wire is 3A, and it must be flowing out of the junction.

Kirchhoff’s Second Law - The Voltage Law (KVL) The principle Kirchhoff’s Voltage Law, or KVL, is based on the principle of conservation of energy. Think of a rollercoaster. It starts at a high point, goes down, and then gets lifted back up to the starting point. The net change in its height (potential energy) after one complete loop is zero. Similarly, in an electrical circuit, the total change in potential (voltage) around any closed loop must be zero. The voltage supplied by the sources (batteries) is consumed by the components (resistors, etc.) in the loop.

Kirchhoff’s second law states that in any closed loop or circuit of an electrical network, the algebraic sum of the e.m.fs is equal to the algebraic sum of the p.ds around the same loop.

OR It states that the algebraic sum of the p.ds around any closed loop or circuit in an electrical network is zero.

Figure 3.21: simple series circuit to demonstrate Kirchhoff’s Voltage Law (KVL) for a single loop.

Vₛ + ( − I R₁) + ( − I R₂) = 0 Vₛ = I R₁ + I R₂ Important Sign Convention To use KVL correctly, we need a sign convention. This is a rule you must follow consistently.

1. For a voltage source (battery): If you trace the loop from the negative terminal to the positive terminal, the voltage is considered positive (+V). If you trace from positive to negative, it’s negative (-V).

2. For a resistor: If you trace the loop in the same direction as the assumed current flow, the voltage drop across the resistor (V=IR) is negative (-IR). If you trace against the current flow, it’s positive (+IR).

Figure 3.22: Each of these resistors and voltage sources is traversed from a to b. (a) When moving across a resistor in the same direction as the current flow, subtract the potential drop. (b) When moving across a resistor in the opposite direction as the current flow, add the potential drop. (c) When moving across a voltage source from the negative terminal to the positive terminal, add the potential drop. (d) When moving across a voltage source from the positive terminal to the negative terminal, subtract the potential drop.

Summary and How to Apply Kirchhoff’s Laws

Steps to Solve a Complex Circuit

1. Label Currents: Assign a unique current variable (I1,I2 ,I3 ,...) to each branch of the circuit. Arbitrarily choose a direction for each current. Do not worry if you guess wrong; the final answer will just be negative, indicating the current flows in the opposite direction.

2. Apply KCL: Write a KCL equation for each junction in the circuit. You do not need to do this for every single junction; once you’ve used a certain number, the rest will be redundant.

3. Apply KVL: Identify the independent loops in the circuit. Write a KVL equation for each loop, carefully following the sign convention. You will need as many independent KVL equations as there are unknown currents.

4. Solve the System of Equations: You will now have a system of linear equations (from your KCL and KVL steps). Use substitution, elimination, or matrix methods to solve for the unknown currents.

Understanding Kirchhoff’s Laws is not just an academic exercise—it is the fundamental principle that governs almost every electrical device and system you interact with daily. The concepts of KCL and KVL are so deeply embedded in modern technology that you benefit from them constantly, even if you do not realise it.

An example of how Kirchhoff’s Laws are essential to our everyday lives include household wiring and electrical safety. Your home’s electrical system is a complex network of wires, outlets, and appliances. It is a classic example of a multi-loop circuit.

1. Kirchhoff’s Current Law (KCL) in action Every outlet in your home is a “junction” where current can split off. KCL ensures that the current entering your home from the power grid equals the sum of all the currents being used by your lights, TV, refrigerator, and other appliances. This balance is crucial. If a fault causes more current to flow out than what is being used, a safety device like a circuit breaker or fuse will trip, preventing an overload that could cause a fire.

2. Kirchhoff’s Voltage Law (KVL) in action KVL explains why all your appliances receive the same voltage (e.g., 220V or 120V). The voltage from the power company is distributed across a closed loop that includes all the parallel branches in your home’s wiring. KVL guarantees that the total voltage supplied by the source is fully consumed by the appliances in each loop. This ensures consistent power and prevents appliances from receiving an unsafe voltage.

Have a discussion with the person next to you about other real-world applications of KCL and KVL and why it is important to analyse the design of these.

Galvanometer and its Adaptations What is a Galvanometer?

A galvanometer is a sensitive instrument used to detect and measure very small electric currents. It works on the principle that a current-carrying coil placed in a magnetic field experiences a torque, causing it to rotate. The amount of rotation is proportional to the current flowing through the coil. A pointer attached to the coil moves across a calibrated scale, indicating the current. However, a standard galvanometer has some limitations:

1. It is very sensitive and can only measure tiny currents (typically in the microampere range). A large current would damage the coil.

2. It has an internal resistance, which would affect the total resistance and therefore the current in the circuit that it is placed in.

3. It measures current, but we often need to measure voltage as well.

Because of these limitations, a galvanometer is almost never used on its own for practical measurements. Instead, it is the core component of two more versatile and widely-used instruments: the ammeter and the voltmeter.

Figure 3.23: Front view of a moving coil Galvanometer The Ammeter: Measuring Current An ammeter is a device used to measure the current flowing through a component or a branch of a circuit. To measure current, the ammeter must be connected in series with the component so that all the current you want to measure flows directly through it.

Why a Galvanometer Alone Will Not Work

A standard galvanometer has a certain internal resistance (R_(g)). If we place it in a circuit, its resistance will add to the total resistance, thereby changing the very current we are trying to measure (according to Ohm’s Law, I = V/Rₜₒₜₐₗ ). A good ammeter should ideally have zero resistance, so it does not affect the circuit.

Furthermore, a galvanometer’s coil cannot handle large currents.

The Adaptation: Adding a Shunt Resistor

To adapt a galvanometer into a practical ammeter, we connect a very small resistance, called a shunt resistor (Rₛ), in parallel with the galvanometer.

Figure 3.24: A galvanometer is converted into an ammeter by connecting a low-resistance shunt resistor in parallel with it. The shunt allows most of the current to bypass the sensitive galvanometer coil.

How it Works The shunt resistor is designed to have a very low resistance. Because it’s in parallel with the galvanometer, most of the total current (I) will choose the path of least resistance and flow through the shunt resistor (Iₛ). Only a very small, safe amount of current (Ig) flows through the sensitive galvanometer coil. Since the shunt and galvanometer are in parallel, the voltage across both is the same. We can use this principle to calculate the required shunt resistance for a specific range:

Vₛ = V_(g) Iₛ Rₛ = I_(g) R_(g) Since I = I_(g) + Iₛ, we have Iₛ = I − I_(g) (I − I_(g)) Rₛ = I_(g) R_(g) Rₛ = I_(g) R_(g)_ I − I_(g) This equation allows us to select a shunt resistor that extends the galvanometer’s measurement range to a desired maximum current (I) without damaging the galvanometer. The scale of the galvanometer is then re-calibrated to indicate the total current (I).

The Voltmeter: Measuring Voltage

A voltmeter is a device used to measure the potential difference (voltage) between two points in a circuit. To measure voltage, the voltmeter must be connected in parallel across the component you want to measure.

Why a Galvanometer Alone Will Not Work

If we simply place a galvanometer in parallel with a component, the galvanometer’s low resistance would provide a new, low-resistance path for the current. This would significantly change the current distribution in the circuit and alter the very voltage we are trying to measure. A good voltmeter should ideally have infinite resistance so that it draws no current and does not affect the circuit.

The Adaptation: Adding a Multiplier Resistor

To adapt a galvanometer into a practical voltmeter, we connect a very large resistance, called a multiplier resistor (Rm), in series with the galvanometer.

Figure 3.25: A galvanometer is converted into a voltmeter by connecting a high- resistance multiplier resistor in series with it. The multiplier resistor ensures that the voltmeter draws a very small current and does not affect the circuit’s voltage.

How it Works The multiplier resistor is designed to have a very high resistance. Since it is in series with the galvanometer, the total resistance of the voltmeter is very high (Rₜₒₜₐₗ=R_(g)+Rₘ). According to Ohm’s Law (I=V/R), this high resistance ensures that only a tiny, negligible amount of current flows through the voltmeter.

The voltage being measured (V) is the sum of the voltage drops across the galvanometer and the multiplier resistor. We can use this to calculate the required multiplier resistance for a specific range:

V = Vₘ + V_(g) V = I_(g) Rₘ + I_(g) R_(g) V = I_(g) (Rₘ + R_(g)) Rₘ = V____ I_(g) − R_(g) This equation allows us to select a multiplier resistor that extends the galvanometer’s measurement range to a desired maximum voltage (V). The scale of the galvanometer is then re-calibrated to indicate this new, higher voltage range.

Activity 3.13 A Two-Loop Resistor Circuit to Verify Kirchhoff’s Laws Objective: Build a two-loop circuit with resistors and batteries, use multimeter to measure currents at junctions and voltages around loops, verify Kirchhoff’s Current Law (KCL) and Kirchhoff’s Voltage Law (KVL), compare measurements with theoretical calculations, and discuss any discrepancies.

What you need

1. Breadboard or circuit board

2. Resistors (e.g., three resistors with known values, e.g., 10 Ω, 15 Ω, 20 Ω)

3. Batteries or DC power supply (e.g., two batteries of known voltage, or one battery and a voltage divider)

4. Connecting wires

5. Digital multimeter (at least one, ideally two for current and voltage measurements)

6. Switch (optional)

7. Calculator

8. Notebook and pen for recording data What to do

1. Build the Circuit

a. Arrange the resistors and batteries to form a two-loop circuit with a junction where current splits.

b. Example

i. Battery connected to resistor R1 (10 Ω) leading to junction A.

ii. From junction A, one branch contains resistor R2 (15 Ω), the other branch resistor R3 (20 Ω).

iii. Both branches rejoin at junction B, then back to battery.

c. This forms two loops sharing resistor R1 and nodes A and B.

2. Identify Measurement Points

a. Identify junction points (e.g., A and B) where current splits or combines.

b. Identify loops (Loop 1: battery − R₁− R₂− back to battery; Loop 2: battery − R₁− R₃− back to battery).

3. Measure Currents to Verify KCL (Current Law)

a. Set the multimeter in ammeter mode.

b. Measure the current entering junction A by placing the meter in series before R1.

c. Measure the current leaving junction A by placing the meter in each branch after the junction (through R₂and through R₃).

d. Record the currents.

e. According to KCL, current entering junction = sum of currents leaving junction.

4. Measure Voltages to Verify KVL (Voltage Law)

a. Set the multimeter in voltmeter mode.

b. For Loop 1 (battery − R₁− R₂): Measure voltage across each resistor (use voltmeter in parallel with each resistor).

c. Measure voltage of battery terminals.

d. Add voltage drops across resistors and verify if total equals battery voltage (around closed loop voltage sum zero).

e. Repeat similarly for Loop 2 (battery − R₁− R₃).

5. Calculate Expected Values

a. Use Ohm’s Law V= IR and series/parallel rules to calculate theoretical currents and voltages.

b. For the junction, calculate how current splits based on resistor values in parallel branches.

c. Calculate voltage drops around each loop.

6. Compare and Discuss

a. Compare your measured currents and voltages to the calculated theoretical values.

b. Discuss any differences, considering

i. Measurement errors

ii. Internal resistance of batteries or wires

iii. Accuracy of resistors (tolerance)

iv. Contact resistance in breadboard connections

Activity 3.14 Exploring Kirchhoff’s Laws Using an Online Circuit Simulator Objective: To deepen understanding of Kirchhoff’s Current Law (KCL) and Kirchhoff’s Voltage Law (KVL) by constructing and analysing circuits using an online simulator.

What you need

1. Access to Online circuit simulator (PhET Circuit Construction Kit: DC) https://phet.colorado.edu/sims/html/circuit-construction-kit-dc-virtual- lab/latest/circuit-construction-kit-dc-virtual-lab_en.html

2. Notebook or digital document for recording results

3. Calculator (optional)

4. Pen or keyboard What to do

1. Go to the PhET Circuit Construction Kit (DC) online simulator or similar.

Select “Lab” or “Intro” mode to access the workspace.

2. Build a Circuit with Multiple Loops and Junctions

a. Use batteries (voltage sources), resistors, and wires to create a circuit that has at least two loops and at least one junction where current can split.

b. For example, make a circuit where two branches come off a common node and reconnect, creating two closed loops. In each of the loops, ensure that there is at least one resistor or other component.

3. Add Measuring Tools

a. Place ammeters in each branch to measure current.

b. Add voltmeters to measure voltage across resistors and batteries.

4. Set and Adjust Circuit Values

a. Assign different resistance values to each resistor and set the battery voltage(s).

b. After taking a set of readings, change some resistor values and/or battery voltage and repeat your measurements.

5. Record Measurements

a. For each configuration, record · The current in every branch (especially at junctions) · The voltage across each resistor and battery

b. Organise your data in a table, for example:

Branch or Loop Resistance (Ω) Voltage (V) Current (A) Branch 1 Branch 2 Loop 1 ...

6. At every junction, add the current entering and compare to the sum leaving.

They should be equal if KCL holds.

9. For each closed loop, sum all the voltage gains (batteries) and drops (resistors) around the loop. The total should be zero (account for direction/ sign).

10. Vary Circuit Settings

a. Change one or more resistor values and observe how KCL and KVL still apply under new conditions.

b. Repeat measurements and checks for at least two different setups.

11. In your notebook, think about, discuss with a friend and write down:

a. How did the simulator make it easier or harder to measure and verify KCL and KVL?

b. How might results be different in a real circuit (e.g., due to wiring resistance, contact issues, or human error)?

c. What are the practical strengths and weaknesses of simulations versus hands-on circuits?

Activity 3.15 Calculating Currents and Potential Differences

Study the worked example below carefully before attempting the example questions that follow.

Worked Example 1

In the circuit diagram below, calculate the current I if the internal resistance of the cell is negligible

Solution

Let I be the current entering the junction G, I₁be the current leaving the junction G into GC and I₂the current leaving the junction G into GF.

By Kirchoff’s first law, I = I₁+ I₂…………… (1) Taking the closed loop BCGHB in anticlockwise direction and applying Kirchoff’s second law, 6 = 5I + 4I₁ I₁ = 6 − 5I/4 ………………. (2) Taking the closed loop CDFGC in the clockwise direction and applying Kirchoff’s second law, 0 = 4I₁– 6I₂ I₂ = 4 I₁___ 6 ……………. (3) Putting equations 2 and 3 into 1 give I = 6 − 5I/4 + 4I₁___ 6 Substituting equation 3 I = 6 − 5I/4 + 4(6 − 5I)_______ 24 Multiplying through by 24 gives 24I = 36 – 30I + 24 – 20I 24I = 60 – 50I 24I + 50I = 60 74I = 60 I = 60/74 I = 0.8108A. Hence the current is given as 0.81A

Worked Example 2

Study the electrical circuit below carefully.

a. Calculate current flowing through the 5Ω resistor.

b. Find the potential difference across BE.

Solution

a. Applying Kirchhoff’s first law at the junction B, I₁+ I₂= I ………….. (1) Taking the closed loop ABEFA in a clockwise direction and applying Kirchhoff’s second law, 2 = 1 × I₁+ I × 5 2 = I₁+ 5I I₁= 2 – 5I …………. (2) Taking the closed loop BCDEB in the anticlockwise direction and applying Kirchoff’s second law, 4 = 3I₂+ 5I I₂ = 4 − 5I/6 …………… (3) Putting equations 2 and 3 into equation 1 gives 2 – 5I + (4 – 5I)/3 = I 6 – 15I +4 – 5I = 3I 10 = 3I + 20I 10 = 23I I = 10/23 I = 0.435 A

b. The p.d across BE is given as VBE V_(BE)= 5I V_(BE)= 5 × 0.434782 V_(BE)= 2.174V Practice Problems Now, using the worked example as a guide, solve the following problems individually or in groups. Before answering the questions, you should identify all junctions and loops in the circuit.

1.

i. Calculate the current that enters the junction X.

ii. Find the p.d across the 3Ω resistor.

iii. Calculate the p.d across AB.

2. The diagram below shows an electrical network. Calculate the potential difference across the 12Ω resistor.

Activity 3.16 Converting a Galvanometer into an Ammeter and a Voltmeter Objective: Learn how to convert a galvanometer into an ammeter by connecting a low-resistance shunt resistor in parallel and into a voltmeter by connecting a high-resistance multiplier resistor in series. Build both configurations and test them in simple circuits to observe and understand their function.

What you need

1. Galvanometer (moving coil type)

2. Shunt resistors (very low resistance values, e.g., fractions of an ohm)

3. High-resistance multipliers (resistors with large resistance values, e.g., kilo-ohms to mega-ohms)

4. DC power supply or batteries (low voltage, e.g., 3–12 V)

5. Known resistors (for building simple test circuits)

6. Connecting wires

7. Switch or key (optional)

8. Multimeter (optional, for verification)

9. Breadboard or circuit board (optional)

10. Notebook and pen to record observations What to do Part 1: Converting Galvanometer to an Ammeter (Using Shunt Resistor)

1. Understand the Idea

The galvanometer can only safely measure very small currents directly. To measure larger currents, you connect a low-value resistor (shunt resistor) in parallel with it. The shunt bypasses most of the current, protecting the galvanometer while allowing the total current to be measured.

2. Connect the Shunt Resistor

a. Connect the shunt resistor in parallel with the galvanometer terminals.

b. Ensure good electrical contacts and secure the connections.

3. Complete the Ammeter Circuit

a. Connect this combination in series with a battery, a known resistor (load), and a switch to form a simple circuit.

b. The galvanometer with shunt now acts as an ammeter.

4. Test and Observe

a. Close the switch to allow current flow.

b. Observe the deflection of the galvanometer needle; it corresponds to the total current flowing through the circuit.

c. Record the deflection for different circuit currents by varying the load resistor or power supply.

5. Measure the actual current with a digital multimeter connected in series to compare with galvanometer readings.

Part 2: Converting Galvanometer to a Voltmeter (Using Multiplier Resistor)

6. Understand the Idea

The galvanometer measures current, so to measure voltage, you connect a high-value resistor in series with it. This limits the current flowing through the galvanometer so it can measure voltage without damage.

7. Connect the Multiplier Resistor

a. Connect a high-value resistor in series with the galvanometer.

b. Secure all connections.

8. Complete the Voltmeter Circuit

a. Connect this series combination across a voltage source or circuit element whose voltage you want to measure.

9. Test and Observe

a. Connect the voltmeter across a power supply or resistor, noting the galvanometer deflection as the voltage changes.

b. Record the deflections corresponding to known voltages (you can use a digital voltmeter for reference).

Part 3: Comparing and Discussing

10. Record Observations

a. Note how the galvanometer deflection changes with current in the ammeter setup and voltage in the voltmeter setup.

b. Discuss how the shunt resistor allows higher current measurement by bypassing excess current, while the multiplier limits current to allow voltage measurement.

11. Analyse and Reflect

In your own words, describe and explain how galvanometers are adapted to act as ammeters and voltmeters. Write your answer down.

Summary Table for Recording

Configuration Resistor Type

Resistor Value

(Ω) Circuit Description

Galvanometer Deflection

Observations Ammeter

(Shunt) Shunt resistor (low) Series with battery + load Voltmeter (Series) Multiplier resistor (high) Connected across voltage source

Activity 3.17 Calculating Shunt and Series Resistance for Converting a Galvanometer into an Ammeter or Voltmeter Study the worked example below carefully before attempting the example questions that follow.

Worked Example 1

A galvanometer has a resistance 25 Ω and gives full-scale deflection at 5 mA.

How can it be converted into an ammeter that reads up to 2 A?

Solution

Step 1: Identify known values Galvanometer resistance, R_(g)= 25 Ω Full-scale current, I_(g)=5 mA = 0.005 A Desired full-scale ammeter current, I=2 A

Step 2: Use the formula for shunt resistance Rs Rₛ = I_(g) R_(g)_ I − I_(g)

Step 3: Substitute values into the formula Rₛ = 0 .005 × 25_ 2 − 0.005

Step 4: Perform the calculation and write the final answer Rₛ = 0.0627 Ω To convert the galvanometer to a 2 A ammeter, connect a shunt resistance of approximately 0.0627 ohms in parallel.

Worked Example 2

A galvanometer has a resistance of 50 Ω and requires a current of 1 mA for full-scale deflection. What resistance must be connected in series to convert it to a voltmeter that can read up to 10 V?

Solution

Step 1: Identify known values Galvanometer resistance, R_(g)=50 Ω Full-scale current, I_(g)=1 mA=0.001 A Desired voltage range, V=10 V

Step 2: Use the formula for multiplier resistance Rm Rₘ = V_ I_(g) − R_(g)

Step 3: Substitute values and calculate Rₘ = 10/0.001 − 50 Rₘ = 9950 Ω Practice problem Now, using the worked example as a guide, solve the following problems individually or in groups.

1. A galvanometer has resistance 20 Ω and current of 2 mA. Find the value of the resistance enable it to convert it to read up to 1 A.

2. A galvanometer has 40 Ω, and current of 1.5 mA. Find the series resistance to make it read up to 15 V.

Activity 3.18 Recap for Electricity and Circuits

Objective: Review key concepts from Years 1 to 3 focused on electricity and circuits.

What you need

1. Notebook or paper

2. Pen or pencil

3. Coloured pencils or markers (optional, for drawing diagrams) What to do

1. Write short answers to the following questions

a. What is electric charge, and what are its types?

b. Explain charge conservation and how charges distribute on conductors.

c. Describe mobile charge carriers and their role in conducting electricity.

d. Define electric field strength and potential difference in your own words.

e. How does a magnetic field form around a current-carrying conductor?

f. State Ohm’s Law and explain what makes a conductor ohmic or non-ohmic.

g. Describe the roles of resistors in circuits and the difference between fixed and variable resistors.

h. What happens to current and voltage in series and parallel resistor arrangements?

i. State Kirchhoff’s Current Law and Voltage Law with simple explanations.

j. How can a galvanometer be converted into an ammeter or a voltmeter? Explain why these modifications are necessary.

2. Draw and label the following diagrams from memory

a. A simple gold leaf electroscope showing charge influence.

b. Electric field lines between two charged plates.

c. A circuit with resistors arranged in series and parallel.

d. A two-loop circuit showing junctions and loops for Kirchhoff’s laws.

e. The symbols for resistor, battery, switch, ammeter, voltmeter, and galvanometer.

3. Create a concept map linking these terms Charge, Electric field, Potential difference, Current, Resistance, Ohm’s Law, Resistor types, Series circuit, Parallel circuit, Kirchhoff’s Laws, Galvanometer, Ammeter, Voltmeter.

Show relationships with arrows and brief notes.

4. True or False Statements

Write “True” or “False” and briefly justify your choice for each

a. The resistance of a wire decreases as its length increases.

b. A voltmeter is connected in parallel to measure voltage.

c. Kirchhoff’s Current Law applies to any junction in a circuit.

d. Increasing resistance always increases current in a circuit.

e. In a series circuit, the current is the same through all components.

5. In your own words, summarise how Ohm’s Law and Kirchhoff’s Laws help analyse electrical circuits. Include why resistors are important and how different resistor arrangements affect current and voltage.

6. In a pair, one person should act as a galvanometer designer and the other as a technician converting it for field use. The galvanometer designer should explain the basic working principle of a galvanometer to the technician.

The technician should explain to the galvanometer designer how and why they will make certain adaptations to produce an ammeter and a voltmeter, and what would happen if the wrong resistor was used.

7. Self-Quiz or Group Discussion

a. Quiz yourself or with peers by asking and answering the above questions.

b. Explain answers aloud.

c. Discuss common misunderstandings and clarify them together.

Review Question 3.1

1. A student notices that when they plug in a kettle (2000W, 240V) at home, the lights dim temporarily. The home has 5 light bulbs of 60W each, and the household wiring has an internal resistance of 0.4Ω.

a. Using Ohm’s law and circuit principles, analyse why this phenomenon occurs.

b. Evaluate the potential risks this might indicate about the home’s electrical system.

c. Design and justify three different solutions to prevent this problem, considering cost, safety, and effectiveness.

Review Question 3.2

1. A learner needs to create a 15Ω resistance using only 10Ω and 30Ω resistors. Design two different circuit arrangements and justify which is more practical.

2. Compare the suitability of a potentiometer versus rheostat for controlling the speed of a 5A motor, considering their structural limitations.

Review Question3.3

1. Evaluate the advantages and limitations of galvanometer-based meters versus digital multimeter, then propose which type would be better for a rural school’s physics laboratory with limited resources.