Waves

Welcome to this section where you shall use diagrammatic skills and mathematics to find the nature of images produced in plane mirrors. Make sure that you are careful with the construction of your diagrams, and that you seek the help of a teacher or peer if you aren’t sure how to complete your sketch or use a protractor accurately!

You shall explore the phenomenon of the reflection of light in two different types of spherical mirrors, the convex mirror and the concave mirror. You will focus on ray diagrams and their formation of fascinating images. You will gain hands- on experience in understanding the properties of these mirrors, the important relationship between focal length and radius of curvature for spherical mirrors, and the formation of ray diagrams.

You will further explore light interactions with mirrors and different media, using the mirror formula, to predict image formation and the magnification formula to determine image size, whether they are enlarged, diminished, upright, or inverted. You will also delve into the laws of refraction, including Snell’s Law, to understand light’s direction and motion. By experimenting with angles and media, you will see light change direction and understand the science behind it.

At the end of this section, you should be able to;

· Deduce the laws of reflection.

· Describe the processes involved in image formation in plane mirrors and their characteristics.

· Determine the number of images formed by inclined mirrors.

· Explain the terminologies associated with spherical mirrors.

· Describe the processes involved in image formation in spherical mirrors and their characteristics using ray tracing.

· Determine the position and characteristics of images formed by spherical mirrors with mirror formula and magnification formula.

· Explain refraction and state the laws of refraction.

Key ideas:

· Reflection is the phenomenon of light rays bouncing back when they encounter a smooth, highly polished, or shiny surface. Very reflective surfaces are often called mirrors.

· When two mirrors are placed at an angle (θ) with respect to each other, multiple images can be formed due to the repeated reflection of light between the mirrors.

· Spherical mirrors are mirrors with surfaces that are part of a sphere.

These mirrors can either be concave (inward-curved) or convex (outward- curved). They reflect light in specific ways due to their curved surfaces, making them useful in various applications. Their images can either be real (formed on a screen) or virtual (formed ‘inside’ and behind the mirror), upright or inverted, magnified or diminished.

· The mirror formula calculates the position of images formed by spherical mirrors by relating focal length, object distance, and image distance.

· The magnification formula determines the size and orientation of an image.

· The laws of refraction explain how light changes speed and direction between different media.

Reflection is a phenomenon of the ‘bouncing back’ of a wave in the same medium.

One example of this is when a light ray encounters a surface; some proportion of the light will be reflected, whilst some is absorbed. Highly polished or shiny surfaces (mirrors) are some of the most efficient reflectors of light, leaving only a small proportion absorbed.

Laws of Reflection

LAW 1: The incident ray, the reflected ray and the normal at the point of incidence all lie on the same plane.

LAW 2: The angle of incidence is equal to the angle of reflection (i = r) The angle of incidence: The angle of incidence is the angle between an incident ray and the normal (a line perpendicular to the surface) of a surface at the point where the ray strikes the surface. It is measured in relation to the normal and is usually denoted by the symbol “i”.

The angle of reflection: The angle of reflection is the angle between a reflected ray and the normal to a surface at the point where the incident ray strikes the surface.

It is also measured in relation to the normal and is usually denoted by the symbol “r”.

Activity 4.1: Verifying the Laws of Reflection

Investigate the relationship between the angle of incidence and the angle of reflection when light reflects off a plane mirror.

Materials needed:

· Plane mirror · Protractor · Paper and pencil · Ray box or laser pointer Procedure:

1. Set up a plane mirror on top of a piece of plain paper, so that the mirror’s surface is at 90 degrees to the paper.

2. Draw a straight line along the surface of the mirror, leaving it in place.

3. Direct a narrow beam of light towards the surface of the mirror, such that it can be seen travelling on the surface of the paper.

4. Mark two crosses on the ‘incident’ (incoming) beam of light and two crosses on the ‘reflected’ beam of light.

5. Remove the mirror and connect the crosses, forming a complete picture of the path of the light (see image below).

Fig. 4.1: Reflection of light on a plane mirror

6. Using a protractor, measure and record the angle of incidence (the angle between the incident ray and the normal, a line perpendicular to the mirror’s surface at the point where the light meets it) and the angle of reflection (the angle between the reflected ray and the normal).

7. Repeat steps 3-5 with the beam of light directed towards the mirror at a range of angles.

8. Analyse your data and look for patterns in the relationship between the angle of incidence and the angle of reflection.

Note: In the absence of practical equipment, the PHeT simulation could be used; follow QR code below.

Activity 4.2 Calculating angles of incidence, reflection, and deviation using the laws of reflection.

Solve the questions below

1. Find the angle of reflection, r, when a light ray incident at a glancing angle ,30 °

2. What is the angle of deviation when angle of incidence of light is 30° ?

3. A light ray strikes a plane mirror making an angle of 25° with the mirror.

Calculate the angle between the incident and reflected ray.

Formation of image in a plane mirror An image is formed when two or more rays meet or appear to meet at a point.

A ray diagram can be constructed by taking a beam of light from the image and reflecting it into the top of the eye, then by taking another beam and reflecting it into the bottom of the eye (being careful to obey the laws of reflection). The rays of light can then be extrapolated back to where they ‘appear’ to originate. See images below.

Fig. 4.2: Ray diagram for the formation of a mirror image

Activity 4.3 Identifying characteristics of images formed by a plane mirror Select the correct description of an image formed in a plane mirror, using the images above for guidance.

· Laterally inverted (left becomes right and vice versa or not inverted.

· Inverted (upside down) or erect (upright) · Real (actual light rays meet) or virtual (light rays appear to meet) · The same size as the object or smaller than the object.

· Object-mirror distance is equal to image-mirror distance or image is further from mirror.

Note: see Annex 4.2 for definitions of some of the above terms.

Activity 4.4 Images formed when two mirrors are inclined at 90⁰ Materials needed:

· 2 plane mirrors · Protractor · Paper and pencil Procedure:

1. Join a mixed-ability group of 3-4 learners.

2. Set the two mirrors at an initial angle of 30 degrees to one another, as measured using the protractor.

3. Observe the number of images you see in the mirrors and record your observations.

4. Repeat steps 3-4, adjusting the angle between the mirrors to 45 degrees and then 60 degrees.

5. Once the observations are complete, discuss the following questions within your groups:

a. How does the number of images change as the angle between the mirrors is increased?

b. Can you identify any patterns or relationships between the angle of inclination and the number of images?

c. Can you explain why the number of images changes as the angle is adjusted?

When two mirrors are placed at an angle (θ) with respect to each other, multiple images can be formed due to the repeated reflection of light between the mirrors.

The number of images (N) formed is inversely proportional to the angle of inclination and can be calculated using the formula:

N = (360___ θ ) − 1.

As the angle of inclination (θ) decreases, the number of images increases. This is because the light undergoes more reflections between the two mirrors, creating a larger number of virtual images. The principles governing the formation of images in inclined mirrors build upon the fundamental laws of reflection.

Fig. 4.3: Reflection of light between two mirrors at an angle θ

Activity 4.5 Calculating the number of images and angle of inclination between two mirrors

1. Find the number of images formed in two plane mirrors inclined at

a. 60°b. 30°c. 90°2. Find the angle of inclination of the mirrors in the case that there are this many images:

a. 5

b. 11

c. 23

3. A light ray strikes a vertical mirror at incident angle of 30°which is inclined at 90°with another horizontal mirror. Sketch this arrangement, and find

a. the angle of reflection in the second mirror and

b. the glancing angle in the second mirror.

Concave mirror: It is an optical mirror, which is part of a sphere with reflecting inner surface.

Convex mirror: It is an optical mirror, which is part of a sphere with reflecting outer surface.

Activity 4.6 Labelling the key features of spherical mirrors.

Fig 4.4: Diagram of convex mirror and concave mirror The figure above shows the two types of spherical mirror. Note that the dashed lines indicate the non-reflective side. Re-draw these diagrams, labelling on and defining the following features (you should use the internet or other research resources to help you):

1. Pole P:

2. Principal axis

3. Centre of Curvature C.

4. Radius of curvature

5. Principal focus F

6. Focal length of a mirror f

(a) Concave mirror (b) Convex mirror Fig. 4.5: Images of concave and convex mirror Watch the video linked here: https://youtu.be/oDNqfxRYQY0. In the case that you do not have access to the video, find details on its message in Annex 4.4 Concave mirror - From the video, it can be concluded that concave mirrors give both inverted and erect images, but the erect image is bigger than the size of the object.

Convex mirror - Also, in the case of the convex mirror, the image is always diminished and upright.

In mirrors, images that are inverted are called real images. These are images formed as a result of the actual intersection of rays. They can be formed on a screen. Images that are erect or upright are formed as a result of the apparent intersection of rays. That means the image can be formed inside the mirror and cannot be formed on a screen.

Activity 4.7 Exploring convex and concave mirrors with flexible reflective material Materials needed:

· A piece of flexible reflective material (e.g., a shiny, bendable plastic or foil) · A light source that emits narrow beams (e.g., a laser pointer or a flashlight with a narrow beam attachment) · A ruler or measuring tape · A flat surface to work on · A stand or something to hold the reflective material in place

Note: in the absence of practical equipment the interactive PhET simulation could be used; follow QR code below.

Procedure:

1. Forming the mirrors:

a. Convex Mirror: Hold the reflective material so that it bulges outward, like the back of a spoon.

Fig. 4.6: Parallel rays of light reflected by convex mirror

b. Concave Mirror: Hold the reflective material so that it curves inward, like the inside of a bowl.

Fig. 4.7: Parallel rays of light reflected by concave mirror

2. Shining Light and Measuring Focal Length:

a. When the ‘mirror’ is in the convex position, shine the narrow beam of light parallel to the principal axis of the mirror. Trace the path of the light on to the paper.

b. Repeat for a range of distances between the light ray and the principal axis.

c. Observe where the light rays appear to diverge from behind the mirror. Trace these rays backward to find the virtual focus point.

Measure the distance from the mirror’s surface to this virtual focus point to find the focal length.

d. Repeat the procedure with the mirror in the concave position, but with an Equal radius of curvature to before (you can confirm this by ensuring that the ends of the mirror are the same direct distance apart). Observe where the light rays converge in front of the mirror. This point is the real focus. Measure the distance from the mirror’s surface to this real focus point to find the focal length.

3. Comparing Focal Lengths:

a. Discuss and compare the focal lengths for both the convex and concave mirrors.

Relationship between Focal Length and Radius of Curvature:

a. The focal length (f) of a spherical mirror is related to the radius of curvature (R) by the equation f = R/2

b. This means that the magnitude of the focal length is half the radius of curvature for both types of mirrors.

Comparison of Convex and Concave Mirrors:

a. Although the magnitude of the focal length is the same, the type of focal point differs. In a concave mirror, the focus is real and located in front of the mirror. In a convex mirror, the focus is virtual and appears behind the mirror.

b. Example:

i. If the radius of curvature (R) is 10 cm, then the focal length (f) for both mirrors will be 5 cm.

ii. For the concave mirror, f = +5 cm (real focus).

iii. For the convex mirror, f = −5 cm (virtual focus).

Activity 4.8 Quiz Game: Spherical mirrors terminologies Game Setup:

1. Participants: Divide yourselves into teams of at most four or compete individually.

2. Materials needed:

· Quiz question cards (or a digital quiz platform) · A scoreboard · Buzzers or any system to determine who answers first (optional)

3. Rules:

· Assign one person to be quiz master or take a turn each.

Alternatively, your teacher may run this quiz.

· Each correct answer earns a point.

· The team or individual with the most points at the end wins.

Quiz Questions

1. What is the point on the surface of the mirror that lies on the principal axis?

2. What is the name of the imaginary line that passes through the pole and the centre of curvature of the mirror?

3. What is the point called where parallel rays of light either converge (for concave mirrors) or appear to diverge (for convex mirrors) after reflecting from the mirror?

4. What term describes the distance between the pole and the focus of a spherical mirror?

5. What is the term for the centre of the sphere from which a spherical mirror segment is taken?

6. What is the distance from the pole to the centre of curvature called?

7. True or False: For a concave mirror, the focal length is positive.

8. True or False: The focal length of a convex mirror is negative.

9. In which type of mirror does the image always appear virtual, smaller, and upright?

10. In which type of mirror can the image be real or virtual, magnified or reduced, and upright or inverted?

11. What is the term for the imaginary plane that is perpendicular to the principal axis and passes through the focal point?

12. Which mirror is used to converge light rays to a focal point?

13. Which mirror is used in vehicles to provide a wider field of view for the driver?

14. True or False: The image formed by a concave mirror can be projected onto a screen.

15. What happens to light rays that are parallel to the principal axis when they reflect off a convex mirror?

16. In a concave mirror, what is the nature of the image when the object is placed between the focal point and the mirror?

17. What kind of mirror is used in solar cookers to focus sunlight to a single point?

18. What is the term for the type of image that cannot be projected onto a screen and appears to be located inside the mirror?

19. Which mirror would you use to get a diminished image of a large area, like in a store or an intersection?

20. In what type of mirror does the reflected image appear to be the same size as the object and upright when the object is placed at the centre of curvature?

Characteristics of image formation in spherical mirrors using ray diagram The position, nature, and size of the image formed depend on the object’s location relative to the mirror. Concave mirrors can form real or virtual images depending on the object’s position. Convex mirrors always produce virtual images that are upright, diminished and located behind the mirror.

In locating the image formed by a spherical mirror, three specific rays are commonly used. They help in understanding whether the image is real or virtual, upright or inverted, and magnified or diminished, depending on the mirror’s type and object position relative to it:

1. Paraxial Ray (Parallel Ray)

2. Principal Ray (Focal Ray)

3. Centre Ray (Through the Centre of Curvature)

Activity 4.9 Drawing a ray diagram for a concave mirror Scenario:

You have a concave mirror with a radius of curvature R=20 cm. An object is placed 15 cm away from the mirror along the principal axis.

Materials needed:

· Blank diagram with the principal axis, centre of curvature (C), focal point (F), pole (P), mirror, and object position marked.

Procedure:

1. Identify Key Points on the Diagram:

a. Principal Axis: The horizontal line on which all key points are located.

b. Center of Curvature (C): Marked at 20 cm from the pole (P).

c. Focal Point (F): Located at R/2=10 cm from the pole (P).

d. Pole (P): The point on the mirror’s surface.

2. Draw the Object:

a. Draw the object as a vertical arrow (O) at the given distance (15 cm) from the pole (P).

3. Draw the Rays:

a. Paraxial Ray (Parallel Ray):

i. Draw a ray parallel to the principal axis from the top of the object (O) towards the mirror.

ii. After hitting the mirror, draw the reflected ray passing through the focal point (F).

b. Principal Ray (Focal Ray):

i. Draw a ray from the top of the object (O) through the focal point (F).

ii. After hitting the mirror, draw the reflected ray parallel to the principal axis.

4. Using the diagram, describe the characteristics of the image formed.

Choose the correct terms to describe the image:

a. Magnified or Diminished: Is the image larger or smaller than the object?

b. Inverted or Upright: Is the image upside down or right side up compared to the object?

c. Real or Virtual: Can the image be projected onto a screen (real) or does it appear to be inside the mirror (virtual)?

5. Repeat the procedure and describe the image with object placed

a. 5 cm and

b. 25 cm from the concave mirror

6. Repeat the procedure using convex mirror with the same radius of curvature.

Deduce whether the position of an object in relation to a curved mirror affects the nature of the image produced, alongside instructions for how to draw the three rays. Construct the entire diagram, including drawing a mirror and a principal axis etc.

Activity 4.10 Virtual lab simulations for spherical Mirrors Objective: To understand the principles of image formation by concave and convex mirrors using virtual lab simulations.

Materials needed:

· Computer or tablet with internet access · Access to a virtual lab simulation tool (e.g., PhET Interactive Simulations, available at PhET or Sunflower) Procedure:

1. Setup:

a. Go to the PhET Interactive Simulations website.

b. Search for the “Geometric Optics” simulation or a similar tool that allows manipulation of mirrors and lenses.

2. Exploring Concave Mirrors:

a. Launch the simulation and select a concave mirror.

b. Place an object (such as an arrow) in front of the concave mirror.

c. Adjust the position of the object to various distances (e.g. beyond the centre of curvature (C), at C, between C and F, and between F and the mirror).

d. Observe the changes in the image formed: note the position, size, orientation (upright or inverted), and type (real or virtual) of the image.

e. Record the observations for each position of the object in a table as in the table below:

3. Exploring Convex Mirrors:

a. Select a convex mirror in the simulation.

b. Place an object in front of the convex mirror.

c. Move the object to different positions and observe how the image characteristics change.

d. Note the position, size, orientation, and type of image formed for each object position.

e. Record the observations in a table.

Table 4.1: Table to record observations made throughout Activity 4.10 Mirror Type Object Position Image Position Image Size Image Orientation Image Type Concave Beyond C Between F and C Diminished Inverted Real Concave At C Concave Between C and F Concave Between F and Mirror

Mirrors are fundamental optical devices used to reflect light and form images.

In this lesson, you will explore how to determine the characteristics of images formed by concave and convex mirrors using the mirror formula and magnification formula. By conducting hands-on experiments and engaging in critical thinking, you will develop a deeper understanding of optical principles.

The mirror formula relates to the object distance (u), the image distance (v) and the focal length (f) of a mirror. It is expressed as, 1_ f = 1/u + 1/v The magnification formula relates the height of the image (h i) to the height of the object (h o). It is given by, m = hᵢ_ hₒ = v/u Sign convention for spherical mirrors Understanding and correctly applying the sign convention in spherical mirrors is essential for accurate and consistent results in optical calculations and for understanding the nature and characteristics of images formed by these mirrors.

It avoids confusion and errors, ensuring clarity and precision in both academic and practical applications. The information below shows the signs of the various quantities in the spherical mirrors.

a. For object distance (u) and image distance (v):

i. Distances measured from the same side of the mirror as the reflective surface are negative.

ii. Distances measured from the opposite side of the mirror are positive.

b. For focal length (f):

i. For a concave mirror, f is negative.

ii. For a convex mirror, f is positive.

Fig 4.8: Sign convention for spherical mirror Sign convention for Spherical Mirrors - Class 10 - Teachoo

Activity 4.11 Investigating magnification Objective: Explore how the height of the image relates to the height of the object using the magnification formula.

Materials needed:

· Concave mirror · Convex mirror · Small object with a known height (e.g., an optical pin, a ruler or a printed scale) · Paper and pen Procedure:

Fig 4.9: schematic of the experimental setup

1. Place the concave mirror on a flat surface.

2. Position the object at a known distance (u) from the mirror.

3. Measure and record the height of the object (ho).

4. Observe and measure the height of the image (hi) formed by the mirror.

5. Use the magnification formula m = hᵢ__ hₒ = − v/u to calculate the magnification of the image.

6. Repeat the process with the convex mirror.

7. Compare the calculated magnification values with your observations.

8. Discuss how the magnification changes with different object distances (u).

Questions (see Annex 4.5 for solutions):

· How does the magnification (M) of the image change as the object distance (u) changes for concave and convex mirrors?

· What do your results tell you about the size and orientation of images formed by concave versus convex mirrors?

Activity 4.12 Exploring the Mirror Formula

Objective: Determine the characteristics of images formed by concave and convex mirrors using the mirror formula.

Materials needed:

· Concave mirror · Convex mirror · Meter ruler · Small object (e.g., a toy or a candle) · Paper and pen Procedure:

1. Place the concave mirror on a flat surface.

2. Position the small object at various distances (u) from the mirror.

3. Measure and record the object distance (u) and the corresponding image distance (v) for each position.

4. Use the meter ruler to ensure accurate measurements.

5. Use the mirror formula 1/f = 1/u + 1/v to calculate the focal length (f) of the concave mirror.

6. Repeat the process with the convex mirror.

7. Compare the calculated focal lengths with the known focal lengths (if provided).

8. Reflect on any discrepancies and consider possible sources of error.

Questions (see Annex 4.5 for solutions):

· How does the image distance (v) change as the object distance (u) changes for concave and convex mirrors?

· What do your results tell you about the nature of images formed by concave versus convex mirrors?

Activity 4.13 Calculating image properties using mirror and magnification Formulas A concave mirror with an object beyond centre of curvature

1. An object is placed 30 cm in front of a concave mirror with a focal length of 10 cm. Determine the position and nature of the image formed.

The object between focal Point and concave mirror

2. An object is placed 15 cm in front of a concave mirror with a focal length of 10 cm. Determine the position and nature of the image formed.

Object at the focal point of a concave mirror

3. An object is placed at the focal point of a concave mirror with a focal length of 10 cm. Determine the position and characteristics of the image formed.

Object in front of a convex mirror

4. An object is placed 20 cm in front of a convex mirror with a focal length of -10 cm. Determine the position and characteristics of the image formed. Sketch the ray diagram for this scenario, showing the position and nature of the image formed by the convex mirror.

Activity 4.14 Critical thinking and real-world applications Objective: Develop a deeper understanding of optical principles by applying them to real-world scenarios.

Materials needed:

· Internet access for research (optional) · Paper and pen Procedure:

1. Research real-world applications of concave and convex mirrors (e.g., telescopes, car rearview mirrors, make-up mirrors).

2. Write a summary of one application, explaining how the mirror formula and magnification formula are used.

3. Design a simple optical device using concave and/or convex mirrors (e.g., a basic periscope or a magnifying mirror).

4. Draw a diagram of your device, labelling the important parts and explaining how it works using the mirror formula and magnification formula.

5. Create a presentation or a report summarizing your research and your designed optical device.

6. Include explanations of how the mirror formula and magnification formula apply to your examples.

Questions:

i. How are concave and convex mirrors used in everyday life?

ii. How does understanding the mirror formula and magnification formula help in designing optical devices?

Activity 4.15 Verifying the mirror formula experimentally Experimental exploration – if equipment is available (otherwise, you can replicate this using the PhET simulation linked previously) Materials needed:

· Concave and convex mirror · Optical bench · Mounted object (e.g., a pin or small object) · Screen · Metre ruler or measuring tape Procedure:

` Fig 4.10: schematic of the experimental setup

1. Mount the concave mirror on the optical bench.

2. Place the object at a fixed distance (e.g., 30 cm) from the mirror

3. Move the screen along the optical bench to find the position where a sharp image is formed.

4. Measure and record the distance between the mirror and the screen (v).

5. Measure and record the distance between the object and the mirror (u).

6. Repeat steps 2 and 3 for different object distances (e.g., 25 cm, 20 cm, 15 cm).

7. Use the recorded values of u and v to calculate 1/u, 1/v and 1/f (using the mirror formula)

8. Compare the calculated focal length with the known focal length of the mirror to confirm the validity of the equation.

9. Plot a graph of 1/v (x axis) against 1/u (y axis). Add a line of best fit. Both the x and y intercepts of the graph should give the value of 1/f .

10. Repeat the experiment using a convex mirror.

Data Table:

Table 4.2: Table to record observations made throughout Activity 4.15 Object distance(u) Image distance (v) 1_ u 1_ v 1_ f = 1_ v + 1_ u 30 25 20 15 Analysis:

· Why is it important to ensure the object and screen are precisely aligned on the optical bench?

· What is the significance of the negative focal length for convex mirrors in the mirror formula?

Activity 4.16 Verifying the magnification formula experimentally Using the same experimental set up as in Activity 4.14, follow this alternative procedure to verify the magnification formula:

Procedure:

1. Mount the concave mirror on the optical bench.

2. Place the object at a fixed distance (e.g., 20 cm) from the mirror.

3. Move the screen along the optical bench to find the position where a sharp image is formed.

4. Measure and record the distance between the mirror and the screen (v).

5. Measure the height of the object (ho).

6. Measure the height of the image (hi) formed on the screen.

7. Repeat steps 2 and 3 for different object distances (e.g., 25 cm, 15 cm).

Data Table:

Table 4.3: Table to record observations made throughout Activity 4.16 Object Distance (u) Image Distance (v) Object Height (hₒ) Image Height (hᵢ) Magnification m = ʰⁱ_ ₕₒ Magnification m = −v/u

-20 cm

-25 cm

-15 cm Analysis:

• Compare the calculated magnification values from m = hᵢ__ hₒ and m = −v/u to check for consistency.

• Why does the image height vary when you change the object distance?

Refraction is a phenomenon where there is a change in the direction and velocity of light when the light travelling in a transparent medium enters another transparent medium of different optical densities.

Light Bending as it passes from Air to Water Fig 4.11: Refraction of light at the air-water interface The laws governing the phenomenon of refraction are:

1. The incident ray, the refracted ray and the normal at the point of incidence all lie in the same plane.

2. Snell’s law: The ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant for a given pair of media.

sini_ sinr = n₂_ n₁ n₁ sini = n₂ sinr

Activity 4.17 Investigate refraction at home or in the lab Materials needed:

· A transparent glass or cup · Water · A pencil or straw Procedure:

1. Fill the glass with water.

2. Place the pencil or straw in the glass.

3. Look at the pencil or straw from the side of the glass.

Fig 4.12: Apparent bending of a pencil due to refraction in water Observation:

Why did the pencil or straw appear bent? Relate your observation to the concepts and laws of refraction you have learned and share with a friend.

Activity 4.18 Research and answer the questions below about refraction in everyday life.

1. What happens to the appearance of a pencil when it is placed in a glass of water?

2. Why does this happen?

3. How do lenses in glasses and cameras help correct vision and capture clear images?

4. How is a rainbow formed in the sky?

5. Why do objects underwater appear closer to the surface than they are?

6. What causes the appearance of a mirage in deserts or on hot roads?

Activity 4.19 - Investigating refraction using water and oil Interface Objective: understand how waves refract when travelling between two different mediums other than air, specifically water and oil.

Materials needed:

· A clear rectangular container · Water · Vegetable oil · Laser pointer · Protractor · Ruler · Graph paper · Safety goggles Procedure:

Fig 4.13: Laser beam refraction

1. Join a group of 2-4 people.

2. Fill the clear container halfway with water.

3. Carefully add a layer of vegetable oil on top of the water. The oil will float on the water, creating a clear interface between the two mediums.

4. Shine the laser pointer at an angle through the oil-water interface.

Observe the path of the laser beam as it travels from the oil into the water.

5. Using the protractor, measure the angle of incidence (the angle at which the laser beam enters the oil-water interface).

6. Measure the angle of refraction (the angle at which the laser beam bends as it enters the water).

7. Record your measurements of the angle of incidence and the angle of refraction in a table. Repeat the experiment for different angles of incidence.

Analysis:

Using Snell’s Law, calculate the ratio of the refractive indices of the oil and water.

Discussion:

Compare your experimental results with the theoretical values of refractive indices for water and oil.

Activity 4.20 Investigating Snell’s law Objective: to measure the incident and refracted angles of light as it passes through a glass block and to observe how these angles change with varying incident angles.

Materials needed:

· Laser pointer · Glass block · Protractor · Ruler · Sheet of paper · Pencil Procedure:

1. Place a piece of paper on a table.

2. Position the glass block in the centre of the paper.

3. Draw a straight line along one edge of the glass block to mark its position.

Label this line as the boundary between air and glass.

4. Using the protractor, draw a line perpendicular (90 degrees) to the boundary line at the point where the laser light will enter the glass block.

This line is called the normal line.

5. Secure the laser pointer in place so it shines a beam of light at the boundary line. Ensure the laser pointer is stable and will not move during the experiment.

6. Trace the path of the light onto the paper.

7. Mark the point where the light exits the glass block on the paper.

8. Draw a line connecting the point where the light enters the glass block to the point where it exits. This is the refracted ray.

9. Use the protractor to measure the angle between the laser beam (incident ray) and the normal line. Record this angle as the incident angle (i).

10. Use the protractor to measure the angle between the refracted ray and the normal line on the exit side of the glass block. Record this angle as the refracted angle (r).

11. Change the angle at which the laser beam hits the glass block by rotating the laser pointer.

12. Repeat steps 1 to 3 for different incident angles (e.g., 10 degrees, 20 degrees, 30 degrees, etc.).

13. Record the incident and refracted angles for each trial.

Data Collection:

Table 4.4: Table to record observations made throughout Activity 4.20 Trial Incident Angle (i) Refracted Angle (r) sin (i) sin (r) 1 2 3 4 5 Analysis:

· Create a graph with sin(i) on the y-axis and sin(r) on the x-axis.

· Find the slope of the graph to give a value for n₂__ n₁ .

· n₁= 1 for air, therefore, find n₂(the refractive index of the glass).

Conclusion:

· Summarise your findings on how the incident angle affects the refracted angle.

· Discuss any patterns observed in your data.

ANNEXES Annex 4.1 – Solutions to some activities

Activity 4.2

1.

[ glancing angle ] + [ angle of incidence ] = 90 °30 °+ i = 90°i = 90°- 30 °= 60 °r = i , r = 60 °2. The angle of deviation is 180° - 60° = 120° 3.

Ans; 130°

Activity 4.3

Laterally inverted (left becomes right and right becomes left) and vice versa.

Erect Virtual (cannot be formed on the screen) Image is of the same size as object, h₁= hₒ Image distance is equal to object distance, v= u

Activity 4.5

1.

a) n = (360___ θ ) − 1 → n = (360/60 ) − 1 → n = 6 – 1 = 5 images

b) n = (360___ θ ) − 1 → n = (360/30 ) − 1 → n = 12 – 1 = 11 images

c) n = (360___ θ ) − 1 → n = (360/90 ) − 1 → n = 4 – 1 = 3 images 2.

a) θ = ( 360/n + 1) , θ = ( 360/5 + 1) = (360/6 ) = 60°b) θ = ( 360/n + 1) = θ = ( 360/11 + 1) = (360/12 ) = 30°c) θ = ( 360/n + 1) = θ = ( 360/23 + 1) = (360/24 ) =15 °3.

Illustration From the diagram above

a. 60°b. 30° Annex 4.2 – Further Information on Reflection in a Plane Mirror Laws of reflection Relationship between the angle of incidence, reflection and glancing angle:

Consider the diagram below Fig 4.14: Reflection showing angle of incidence and glancing angle glancing angle + angle of incidence = 90 °Lateral inversion is a phenomenon where a lateral side or part of a body or an object becomes the opposite side.

Fig. 4.15: Image showing lateral inversion Real and virtual images Images are formed either at the point where rays of light meet intersect or at the point where they appear to originate. An image formed can be two kinds. These are real image and virtual image.

A real image is formed by the actual intersection of two or more rays from an object (real light rays). Real images are formed on a screen. e.g. the image in a pinhole camera.

A virtual image is formed by the apparent intersection of two or more rays from an object. A virtual image is formed from two divergent rays of light that never meet but are ‘tracked back’ to the point that they appear to have originated from.

In simple terms, a plane mirror creates an image of an object you cannot touch.

All mirrors create virtual images in this manner, but plane mirrors reflect light differently than concave or convex mirrors do.

Virtual images are not formed on a screen such as image formed on a plane mirror.

Virtual images or rays are represented by dotted lines or rays.

Images formed when two mirrors are inclined at 90⁰Fig 4.16: Ray diagram of images formed when two mirrors are inclined at 90⁰Whenever two mirrors are inclined at an angle 90°to each other and an object is placed in front of them, it revealed by observation that three images are formed on the mirrors by multiple reflection the images Iₐand I_(b)are formed by simple reflection. The additional image I_(c)is produced as a result of reflection of two main images that is reflection of image ‘Iₐ’ in mirror (‘I_(b)’) and reflection image (I_(b)) in mirror (A) it is the super imposition of two images.

The relationship between the angle of inclination and the number of images is:

n = (360___ θ ) − 1 , → θ = ( 360/n + 1)

NOTE 1

Consider two mirrors inclined at an angle of 180°.

M1 M2 The number of images formed is given as:

N = 360/180 – 1 = 1 image It takes a single mirror to form one image of an object. This means that, this arrangement of the two mirrors, inclined 180°acts as a single mirror

NOTE 2

Consider the mirrors inclined at an angle 0°in which case the two mirrors are parallel to each other.

The number of images formed is given as N = (360/0 ) – 1 = ∞ images (this means multiple or uncountable images) Annex 4.3 – Solutions to some activities

Activity 4.6

Fig. 4.17: Diagram illustrating a concave mirror with accompanying terminologies

1. Pole P: It is the central point of a concave or convex mirror through which the principal axis passes.

2. Principal axis is an imaginary line joining the centre of curvature through the principal focus to the pole.

3. Centre of Curvature C is the centre of a sphere of which the mirror is part.

4. Radius of curvature is the distance from this centre of curvature to the pole of the mirror.

5. Principal focus F is a point on the principal axis, where all rays parallel and close to the principal axis either converge or appear to diverge after reflection from the curved mirror.

6. Focal length of a mirror f is the distance between the pole of the mirror and the principal focus.

Activity 4.7

Conclusion: For mirrors with the same radius of curvature (R), the magnitude of the focal length (f) should be the same, but the nature of the focal point is different:

· For a concave mirror, the focal length is positive (real focus).

· For a convex mirror, the focal length is negative (virtual focus).

Activity 4.8

Quiz Answers:

1. Pole (P)

2. Principal Axis

3. Focus (F)

4. Focal Length (f)

5. Center of Curvature (C)

6. Radius of Curvature (R)

7. True

8. True

9. Convex Mirror

10. Concave Mirror

11. Focal Plane

12. Concave Mirror

13. Convex Mirror

14. True (when the image is real)

15. They diverge as if they are coming from the virtual focus behind the mirror.

16. The image is virtual, magnified, and upright.

17. Concave Mirror

18. Virtual Image

19. Convex Mirror

20. Concave Mirror

Activity 4.9

5 a.

· Magnified: The image appears larger than the object.

· Upright: The image is right side up compared to the object.

· Virtual: The image appears to be inside the mirror and cannot be projected onto a screen.

5b · Diminished: The image appears smaller than the object.

· Inverted: The image is upside down compared to the object.

· Real: The image can be projected onto a screen.

Annex 4.4 – Further Information on Spherical Mirrors Welcome to this section. In the case where the video mentioned previously cannot be watched at the place where you are currently, join the dialogue below. I would like you to follow the picture of a man watching himself in a spherical mirror and read the conversation between Laila, Kotey and their teacher to understand what is going on.

Scenario 1 – Concave (stood far away) Scenario 2- Concave mirror (stood closer) Fig. 4.12: Image of a man watching himself in a spherical mirror (Courtesy of Manocha Academy) Teacher: Today, we are going to watch a man standing at a distance in front of a spherical mirror. What do you see and what can you say about his image?

Laila: He is turned upside down.

Teacher: Good, we say his image is inverted. Can you say anything about him again?

Kotey: Yes, his image size is smaller than himself.

Teacher: Fantastic. You are right. We say his image is diminished. Now, watch the second scenario as he moves closer to the mirror. What do you now observe?

Kotey: Well, I think his image size now looks bigger than himself.

Teacher: Good! We say his image is magnified or enlarged. Is that all?

Laila: No Sir. His image this time is not inverted. He is standing straight.

Teacher: Awesome! We say his image is upright or erect.

Scenario 3- Convex mirror Fig. 4.13: Image of a man watching himself in a convex mirror Now let us consider scenario 3 as he moves to stand in front of the next spherical mirror B.

Teacher: What do you observe?

Kotey: The man is standing upright Teacher: Precisely.

Laila: Yes! He is also diminished in size.

Teacher: Good! Your observations are right.

Rules in drawing Ray diagrams

Table 4.5: rules in drawing ray diagrams Rule Representation Paraxial ray - Ray parallel to principal axis will pass through the focus after reflection or appear to come from focus, F.

Principle ray - Ray passing through the focus will become parallel to the principal axis after reflection or a ray is in line with the focus after striking the mirror will move parallel to the principal axis.

Rule Representation

Centre ray – A ray of light that passes through the centre of curvature or in line with it after striking the mirror, is reflected back along the same path.

Ray 4 - A ray of light that strikes the pole of the mirror at an angle of incidence is reflected back at the same angle that the angle of incidence equals the angle of reflection.

NOTE: achieving equal angles is difficult in practice.

https://www.teachoo.com/10824/3118/Rules-for-drawing-Ray-Diagram-in- Mirrors/category/Concepts/ Formation of ray diagrams

Table 4.6: Table containing ray diagrams for concave and convex mirrors Ray diagrams of concave mirror 1 2 Image forming at infinity 3 4 5 6 Scenario A If you draw 2 on a light plane paper and you turn the back of the paper, you will get this diagram.

Ray diagrams in convex mirrors 1 2 Object moves from O to O’ and image moves from I to I’ closer to mirror.

3 Scenario A Two rays from infinity are parallel: one in line with C and the other in line with F Uses of spherical mirrors Convex Some examples of where spherical mirrors are used include for convex mirrors;

Car side mirrors, Security mirrors in stores, near ATM machines and Industrial and Workplace Safety Concave Makeup mirrors, Telescopes (concave), Shaving mirrors, Reflectors in flashlights, Solar cookers, Dentist’s mirrors, Streetlight reflectors and Magnifying glasses.

Annex 4.5 – Solutions to some activities

Activity 4.11

· For a concave mirror, the magnification (M) can be greater than 1 (image is larger) or less than 1 (image is smaller) and can be positive (upright) or negative (inverted) depending on the object’s distance (u).

· For a convex mirror, the magnification (M) is always less than 1 (the image is smaller) and positive (upright).

· As the object distance (u) changes, the magnification (M) changes inversely. For closer objects, the image appears larger in a concave mirror and smaller in a convex mirror.

Activity 4.12

· For a concave mirror, as the object distance (u) decreases (object moves closer to the mirror), the image distance (v) increases (image moves farther from the mirror) and can become real and inverted if within the focal length.

· For a convex mirror, the image distance (v) is always negative (virtual image) and located behind the mirror, getting closer to the mirror as the object distance (u) decreases.

· Any discrepancies could be due to measurement errors, improper alignment, or imperfections in the mirrors.

Activity 4.13

1. Given:

· u = − 30 cm · f = 10 cm Using the mirror formula:

1_ f = 1/u + 1/v 1_ 10 = 1/30 + 1/v 1_ v = 1/10 + 1/30 1_ v = 3 + 1/30 1_ v = 4/30 1_ v = 2/15 v = 15/2 v = 7.5 cm Determine the Magnification m = − v/m m = − 7.5____ − 30 m = 7.5/30 m = 0.25 Characteristics:

· Nature: Real (since v is positive) · Orientation: Inverted (since m is negative) · Size: Diminished (since ∣ m ∣ < 1)

2. Given:

· u = − 15 cm · f = 10 cm Using the mirror formula:

1_ v = 1/f − 1/u 1_ v = 1/10 − 1/15 1/v = 1/10 + 1/15 1_ v = 3/30 + 2/30 1_ v = 5/30 v = 6 cm Determine the Magnification m = − v/u m = − 6____

–15 m = 6/15 m = 0.4 Characteristics:

· Nature: Virtual (since v is positive) · Orientation: Upright (since m is positive) · Size: Diminished (since ∣ m ∣ < 1 )

3. v = ∞

4. v = -6.66cm Annex 4.6 - Further Information on Image Formation Spherical mirrors Graph plotting in spherical mirrors Graph plotting is a powerful tool that helps us visualise relationships between different variables. Let us dive in and explore how different distances relate in the context of mirrors Object Distance (u) vs. Image Distance (v) The diagram below shows how the distance between an object and a concave mirror (u) affects the distance of the resulting image (v) from the mirror. Each point represents a different position of the object. As the object moves closer to the mirror, notice how the image distance changes Here is a sample table of data showing the relationship between object distance and image distance. The graph of object distance (u) versus image distance (v) is shown below.

Object Distance (u/cm) Image Distance (v/cm)

50 16.67 40 20 30 24 25 30 20 40 15 60 Here is the graph of the object distance (u) on the x-axis and the image distance

(v) on the y-axis for a concave mirror.

Whilst the graph is an interesting shape, it is challenging to use it to discover the value of the focal length of the mirror. Plotting an alternative graph of 1/v against 1/u is more useful; see the results of Activity 4.15.

Annex 4.7 – Solutions to some activities

Activity 4.18

1. Bending of a pencil in Water: When you place a pencil in a glass of water, it appears bent at the surface of the water

2. It happens due to refraction.

3. Lenses in Glasses and Cameras: Lenses bend light to focus it, which helps correct vision and capture clear images.

4. Rainbow Formation: Sunlight refracts as it enters and exits raindrops, splitting into different colours and creating a rainbow.

5. Apparent Depth: Objects underwater appear closer to the surface than they actually are because light rays bend when they move from water to air.

6. Mirage: In deserts or hot roads, light bends due to temperature variations in the air, creating the illusion of water.

Annex 4.8 – Further Information on the laws of refraction

1. The First Law of Refraction:

The incident ray, the refracted ray, and the normal to the interface of two media at the point of incidence all lie in the same plane.

Explanation:

This means that if you draw a line perpendicular (normal) to the surface where the light is entering the new medium, the incoming ray (incident ray) and the outgoing ray (refracted ray) will both lie on the same flat surface.

2. The Second Law of Refraction (Snell’s Law):

The ratio of the sine of the angle of incidence (sin i) to the sine of the angle of refraction (sin r) is a constant for two given media, which is equal to the refractive index (n) of the second medium relative to the first medium.

Mathematically, this can be expressed as:

sini_ sinr = n n₁ sini = n₂ sinr where:

· n₁ is the refractive index of the first medium, · n₂ is the refractive index of the second medium, · i is the angle of incidence, · r is the angle of refraction.

Explanation:

Snell’s Law quantifies how much light will bend when entering a different medium. The refractive index is a measure of how much the speed of light is reduced inside the medium. For example, the refractive index of water is about 1.33, which means light travels 1.33 times slower in water than in a vacuum.

Review Questions 4.1

1. A light ray strikes a plane mirror at an angle of 30° to the normal to the mirror surface. What is the angle of reflection?

2. A light ray strikes a plane mirror at an angle of 45° to the normal to the mirror surface. What is the angle between the incident ray and the reflected ray (the angle of deviation)?

3. A light ray strikes a plane mirror at an angle of 60° with the normal to the mirror surface. What is the angle of reflection, and what is the angle between the incident ray and the reflected ray?

4. What is the relationship between the angle of inclination between two mirrors and the number of images formed?

5. Explain the process of formation of multiple images in a system of two inclined mirrors.

6. How can the knowledge of image formation in inclined mirrors be applied in real-world scenarios?

Review Questions 4.2

1. How would the image characteristics change if an object moves from beyond the centre of curvature (C) to the focal point (F) of a concave mirror?

2. Describe the differences between the real images formed by concave mirrors when the object is placed at different positions: beyond C, at C, and between C and F.

3. Explain how a concave mirror can be used to concentrate sunlight to a single point and what practical applications this has.

4. Compare the image formation of a concave mirror when used as a makeup mirror versus when used in a reflecting telescope.

5. Design an investigation to explore how the curvature of a concave mirror affects the focal length, and the characteristics of the images formed.

6. How would you demonstrate to your classmates the principle of image inversion in concave mirrors using everyday materials?

7. Propose a real-world scenario where the understanding of concave mirrors can solve a practical problem and explain your solution.

8. How does the image formed by a convex mirror differ from that formed by a concave mirror when the object is placed at the same distance from the mirrors?

9. Explain why convex mirrors are commonly used as side mirrors on vehicles.

10. Describe how a convex mirror can be used for security purposes in a store.

11. Compare the effectiveness of convex mirrors versus flat mirrors in providing a clear view of surroundings in public transportation systems.

12. Design an experiment to compare the field of view provided by a convex mirror to that of a flat mirror. What would you measure and what outcomes would you expect?

Review Questions 4.3

1. What is a spherical mirror?

2. Provide two examples of everyday objects that use spherical mirrors.

3. Define the terms

a) Principal axis

b) Pole

c) Principal focus in the context of spherical mirrors.

4. Describe three rays that could be used in locating the image formed in a concave mirror and state how they are reflected by the mirror.

5. With the aid of a diagram, show how an image may be produced by a convex mirror.

6. With the aid of a diagram, show how a virtual image may be produced by a concave mirror.

7. How are the laws of reflection seen in the rules applied to the formation of images in spherical mirrors?

Review Questions 4.4

1. Compare and contrast reflection and refraction of light.

2. Evaluate the impact of varying refractive indices on the behaviour of light.

3. Discuss the environmental impact of refraction-based technologies.

4. Predict the path of light through a prism and explain why white light separates into colours.

5. Explain the concept of magnification in the context of mirrors. How is it calculated using the mirror formula?

6. A concave mirror has a focal length of 20 cm. If an object is placed 40 cm from the mirror, calculate the position of the image using the mirror formula.

7. Discuss the relationship between object distance, image distance, and focal length in concave mirrors. Provide an example calculation using the mirror formula to illustrate your explanation.

Welcome back to the world of optics, or the behaviour of light. You will explore the beautiful concepts of refractive index, total internal reflection, and the interplay of real and apparent depth.

Total internal reflection is a fundamental concept in physics that occurs when a light wave hits a boundary between two media and is completely reflected back into the first medium.

The relationship between real depth, apparent depth, and refractive index can be established using the following formula: ɳ = real depth (hr)______________ apparent depth (ha) At the end of this section, you should be able to;

· Determine the refractive index of a medium.

· Explain total internal reflection.

· Establish the relationship between the real depth, apparent depth and the refractive index.

Key Ideas:

· Refractive index (ɳ ) is a measure of how much light bends when it enters a new material.

· Real depth refers to the actual distance between an object and the surface of a medium, like the distance between a coin in water and the water surface. Apparent depth is the perceived depth of an object when viewed through a medium like water. It is the depth that our eyes see and brains recognise, and it is always shallower than the real depth.

· Total Internal Reflection (TIR) occurs when light hits a boundary at a shallow angle. The light is completely reflected back into the first material.

REFRACTIVE INDEX ( ɳ )

Activity 5.1 Exploring refraction and apparent depth Read the passage below and use it to define what is meant by the refractive index of a material.

One bright afternoon, Hopeson was spending the day by the lake with his friends. As they sat on the dock, dipping their toes in the cool, clear water, Hopeson saw a shiny coin at the bottom of the water.

“Hey, look! There’s a coin down there,” Hopeson exclaimed, pointing at the water. His friend Nyarkoah tilted her head and said, “Really? I don’t see it.

Where exactly?”

Hopeson directed “Right there, in the water. It’s just a few inches down.”

Nyarkoah squinted her eyes and nodded. “I see it now. But wait, doesn’t it look like the coin is closer to the surface than it actually is?”

“You’re right!” Hopeson exclaimed. “That’s really interesting. Let’s take a closer look.”

The two friends and Aishatu carefully lowered their hands into the water, keeping their eyes on the coin. As their fingers approached the coin, they noticed that it appeared to be closer to the surface than it actually was.

Fig 5.1: Apparent depth of a coin due to refraction in water “Exactly!” Nyarkoah exclaimed. “The real depth of the coin is the actual distance from the surface to the coin, but the apparent depth is the distance it appears to be from the surface. The refractive index of the water is what causes this difference.”

Medium (M_(A)) Medium (M_(B)) Fig 5.2: Refraction of light causing the apparent bending of a pencil in water Observe the picture above carefully, does the pencil look bent?

The refractive index is a number that tells us how much light bends when it passes from one material medium into another, like from air into water or glass and vice versa. This bending of light is what makes the pencil appear bent in the picture above.

It can also be defined as the ratio of the speed of light in vacuum or free space (M_(A) in the diagram above) to the speed of light in another medium.

ɳ = s peed of light (M_(A))_______________ speed of light (M_(B)) In other words, it measures how much a medium slows down or bends light as it travels through it. It is an important concept in physics as it plays a key role in understanding various concepts (phenomena) such as refraction, diffraction, and total internal reflection. Different materials have distinct refractive indices, which can be used to identify them. Refractive index can also be defined as the ratio of real depth to the apparent depth of a medium.

ɳ = real depth (hr)______________ apparent depth (ha) Types of Refractive Index

1. Absolute refractive index

2. Relative refractive index Absolute refractive index is defined as a ratio of the speed of light in vacuum to a selected medium.

Relative refractive index is defined as the ratio of speeds of light in two different media.

If light enters any substance with a higher refractive index (such as from air into glass) it slows down its speed. The light bends towards the normal line. However, when light enters into a substance with a lower refractive index (such as from water into air) it speeds up. The light however bends away from the normal line.

Applications of Refraction

1. Designing Optical Instruments: Knowing the refractive index of different materials allows engineers to design lenses with specific focal lengths and curvatures to focus light accurately.

2. Fibre Optics: Refractive index plays a critical role in fibre optic cables, where light is guided through thin fibres by total internal reflection, which relies on the difference in refractive index between the fibre and its surroundings.

3. Creating a Rainbow Effect: Different colours of light have slightly different refractive indices. This means that when white light passes through a prism, it is separated into its constituent colours, creating a rainbow. This phenomenon is called dispersion.

4. Medical Imaging: Refractive index differences are utilised in techniques like MRI and CT scans to create images of the human body

Activity 5.2 Calculating the refractive index of various substances Complete the table by calculating the absolute refractive index of the substances listed, using the speed of light.

Table 5.1: Refractive indices and speeds of light in different substances Substance Speed of light × 10⁶(ms⁻¹) Refractive index Air 300 1.00 Water 226 Glass 200 Diamond 125

Activity 5.3 Experiment to explore the concepts of real and apparent depth using a coin and a transparent container of water.

Objective: To investigate the difference between real and apparent depth through observation of a coin submerged in water.

Materials Needed:

· Transparent container (e.g., a clear glass or plastic tank) · Coin (or any small, flat object) · Water · Cardboard · Pen and paper for recording observations Procedure:

1. Place the coin in the bottom of the transparent container, as the far side relative to you.

2. View the coin from around 45 degrees above it.

3. Place a piece of cardboard in front of the container so that the coin is now obscured.

4. Fill the transparent container with water slowly, until the coin just comes into view.

5. Record the depth of water added to the container at the point that the coin became visible.

6. Repeat the experiment, this time viewing the coin from different angles.

Record your findings and analyse the results.

7. Discuss how the angle of observation affects the perceived depth.

Activity 5.4 Solving real and apparent depth problems using refractive index Problems using the refractive index formulae:

ɳ = s peed of light (M_(A))_______________ speed of light (M_(B)) ɳ = real depth (hr)______________ apparent depth (ha)

1. A pond is 100cm deep when a coin was placed into it, it was viewed or seen at 20cm from the surface of the water. calculate the refractive index of water and hence the displacement of the coin

2. A coin was thrown into a bucket of water which was 10cm deep. An observer saw the coin to have appeared on the 5cm mark from the top of the bucket. Calculate the refractive index of the water and hence the displacement of the coin.

3. Kofi and Ama after a physics class decided to go for a swim in a swimming pool. They noticed a coin at the bottom of the pool two meters deep. The coin appeared to be 1.4 meters from the surface of the water. What was the refractive index of water in the swimming pool?

Activity 5.5 Determining the refractive index of a material using Snell’s law Fig 5.3: Refraction of light through a glass block

1. Using the diagram above, measure the angle of incidence ‘i’ and angle of refraction ‘r’ using a protractor and record it on the table below.

2. Use your calculator and find ‘sin i’ and ‘sin r’ of the measured angles.

You can also find the refractive index using the final formula, ɳ =sini___ sinr.

Complete the table below with these results.

Table 5.2: Table to record measured angles and calculated values i r sin i sin r ɳ =sini___ sinr

3. Use your calculator and press ‘sin i’ and ‘sin r ‘of the measured angles obtained and fill the empty spaces of the table below.

Table 5.3: Table to record various calculated refracted indices i r sin i sin r ɳ =sini___ sinr 20 13 30 20 40 26 50 31 60 36 70 40 80 42

4. Plot a graph of sin i (y axis) against sin r (x axis) using the results in the second table.

5. Plot a line of best fit on your graph and then find the gradient of this line.

6. What do you notice about the value of the gradient?

Activity 5.6: Finding the refractive index from experimental data

Note: in the absence of practical equipment, you can use the interactive simulation linked using the QR code below:

https://phet.colorado.edu/sims/html/bending-light/latest/bending-light_all.

html Materials Needed · Glass block · Light source (e.g., laser pointer) · Protractor · Ruler · Pencil Procedure

1. Draw a diagram of the experimental setup, including the glass block, light source, and protractor.

Fig 5.4: Schematic of the experimental setup

2. Shine the light source through the glass block at an angle.

3. Measure the angle of incidence (θᵢ) using the protractor.

4. Measure the angle of refraction (θᵣ) using the protractor.

5. Repeat steps 1-3 for different angles of incidence. Record the measurements in a table.

Table 5.4: Table to record observations made throughout activity 5.6 i r sin i sin r

6. Plot a graph of sin i against sin r and use the gradient to find the refractive index of the glass block.

Activity 5.7: Problems using the refractive index formula Refractive index formula:

n₁sinθ₁=n₂sinθ₂ where n₁is the refractive index of material 1 (which the light is initially traveling through) and n₂is the refractive index of material 2 (which the light is entering).

1. A light ray traveling in air of refractive index 1.00 strikes the surface of a block of glass at an angle of 30°. The refracted ray makes an angle of 20° with the normal. What is the refractive index of the glass?

2. A light ray traveling in water strikes the surface of a diamond at an angle of 45°. What is the angle of refraction? (n₁of water = 1.33, n₂of diamond = 2.42)

3. A light ray traveling from water into glass enters the glass at an angle of incidence of 42 degrees and emerges at an angle of 55 degrees. The refractive index of the glass is 1.42. What is the refractive index of the water?

Total internal reflection is a fundamental concept in physics that occurs when a wave, typically light, hits a boundary between two mediums and is completely reflected back into the first medium. This phenomenon happens when the angle of incidence exceeds the critical angle, causing the light to be unable to pass through the boundary and instead bounce back.

Total internal reflection is commonly observed in optics, fibre optics, and even in nature, such as in the sparkle of diamonds or the mirage effect on hot roads.

It has numerous applications, including medical imaging, telecommunications, and even in the design of optical instruments like microscopes and telescopes.

Understanding total internal reflection is crucial for harnessing the power of light and its behaviour at the boundary of different mediums.

Conditions for occurrence of total internal reflection

1. The light must travel from the denser medium to less dense medium.

2. The angle of incidence in the denser medium must be greater than the critical angle.

TIR problems can still be solved using the formula n₁sin θ₁= n₂sin θ₂, however θ₂will always be equal to 90 degrees in the case of total internal reflection (as the light does not emerge from the first material).

Optical instruments that utilise total internal reflection

1. Periscopes

2. Optical fibres

3. Binoculars

4. Telescopes

5. Microscopes

6. Periscopes

7. Spectroscopes.

Natural examples of total internal reflection

1. Mirage

2. Brilliance of diamond A mirage is a naturally occurring optical phenomenon in which light rays bend to produce a displaced image of distant objects or the sky. The word comes from English via the French mirage, from the Latin ‘mirari’, meaning “to look at, to wonder at” Definition: A mirage is an optically illusive and deceptive image formed on a tarred mac road on a hot day or image formed on the sea on the cold day as a result of total internal reflection.

Activity 5.8: Problems about TIR using the refractive index formula n₁sin θ₁= n₁sin θ₂

1. A light ray travels from a medium with a refractive index of 1.50 to a medium with a refractive index of 1.00. What is the critical angle for total internal reflection?

2. Calculate the critical angle of the medium of refractive index 1.800

Activity 5.9: Total Internal Reflection Practical

Materials needed:

· Source of light e.g. ray box or laser pointer · Plain paper · Protractor · Semi-circular glass prism · Pencil Procedure:

1. Place the semi-circular glass block in the centre of the piece of paper and draw around it. Leave it in place.

2. Shine a ray of light towards the curved surface, ensuring that it hits the surface at 90 degrees (see diagram).

3. Increase the angle of the light as it hits the far surface (angle i in the diagram below) until the point at which no light appears to emerge from the block.

4. Mark the path of the light on to the piece of paper when the above condition is met.

5. Remove the glass block and continue drawing the marked ray of light until it hits the far boundary.

6. Measure the angle between this ray of light and the normal. This is your critical angle.

Fig 5.5: Refraction of light through a semi-circular block Follow-up task:

Calculate the refractive index of the glass block using your measured value of the critical angle.

Discuss the effect that a larger refractive index would have on the critical angle.

ANNEXES Annex 5.1 – Solutions to some activities

Activity 5.1

The refractive index of a medium is a fundamental property that describes how light behaves when it passes through a particular material or substance.

Activity 5.2

Table 5.5: Solution to Activity 5.2

Substance Speed of light × 10⁶(ms⁻¹) Refractive index Air 300 1.00 Water 226 1.33 Glass 200 1.50 Diamond 125 2.40

Activity 5.4

a) Solution ɳ = hr__ ha = 100/20 = 5 Displacement = hr – ha Displacement =100 – 20 = 80cm

b) Solution ɳ = real depth (hr)______________ apparant depth (ha) ɳ = 10 cm/5 cm = 2 Displacement = hr – ha Displacement= 10cm – 5cm = 5cm

7. Solution ɳ = real depth (hr)______________ apparant depth (ha) = 2/1.4 = 1.42

Activity 5.5

iii)

Table 5.6: Solution to Activity 5.5

i r sin i sin r ɳ = sini___ sinr 20 13 0.342 0.225 1.52 30 20 0.500 0.342 1.46 40 26 0.643 0.438 1.47 50 31 0.766 0.515 1.49 60 36 0.866 0.588 1.47 70 40 0.940 0.643 1.46 80 42 0.985 0.669 1.47

vi) The value of the gradient is equal to the value of the refractive index.

Activity 5.6

Draw or plot a graph of sin i values against sin r values which passes through the origin and hence your refractive index (ɳ) using the slope of the graph.

1. ɳ = sin i___ sin r Fig 5.6

Activity 5.7

1. Solution:

Identify the knowns:

n₁= 1.00 (air) θ₁= 30° θ₂= 20° Apply the formula:

n₁sinθ₁= n₂sinθ₂ 1.00sin(30°) = n₂sin(20°) Solve for n₂:

n₂= (1.00 sin(30°))___________ sin(20°) = 1.46 Therefore, the refractive index of the glass is approximately 1.46.

2. Solution:

Identify the knowns:

n₁= 1.33 (water) n₂= 2.42 (diamond) θ₁= 45° Apply the formula:

n₁sin θ₁= n₂sin θ₂ 1.33sin (45°) = 2.42 sin θ₂ make θ₂the subject:

sin θ₂= 1.33 sin(45°)__________ 2.42 = 0.94/2.42= 0.389 sin θ₂= 0.389 θ₂= sin⁻¹(0.389) = 22.7°Therefore, the angle of refraction is approximately 22.7°3. η₁ sin θ₁ = η₂ sin θ₂ η₁ sin42 = η₂ sin55 η₁ = 1.42sin55________ sin42 = 1.74

Activity 5.8

1. At the critical angle, the angle of refraction is 90°.

Apply the formula:

η₁ sin θ₁ = η₂ sin θ₂ sin θc = 1.00 sin(90°)__________ 1.50 = 6.66 × 10⁻³θc =sin⁻¹(6 × 10⁻³) = 41.8° Therefore, the critical angle for total internal reflection is approximately 41.8°.

2. sin c = 1__ ɳ ɳ = 1.800 sin c = 1/1.800 sin c = 0.5 c = sin⁻¹(0.5) c = 30⁰Annex 5.2- Further information on refraction Deriving the apparent depth formula Consider the diagram below. An object is placed in a vessel of water and is observed at a different position by an observer.

If a coin was placed at a position C, and it was observed to have been appeared at a position B, the refractive index could be found using real depth of the object and the apparent depth.

Fig 5.7: Diagram showing refraction and depth relationships in a water” In triangle ABD and ACD tan r = AD___ AC = o__ hr and tani = o___ AB = o__ ha for small angles tan r = sin r hence sini____ sinr = tani____ tanr → O ⁄ ha___ _(O) ⁄ hr = hr___ ha ɳ = hr__ ha ɳ = real depth (hr)______________ apparent depth (ha) Explanation of the brilliance of a diamond Refractive index of diamond is 2.42, so the critical angle for diamond air interface is 24.4^(o).

Diamonds are cut in such a way that light falling on them from any surface undergoes total internal reflection at various faces and therefore remains within the diamond. Different incident rays travel along different paths and therefore come out of the diamond at different times and angles causing a sparkle effect.

Hence diamonds shine very brilliantly.

Fig 5.8: Dispersion and critical Angle in a diamond Summary of the similarities and differences between TIR, refraction and reflection

Table 5.7: Comparison of Total Internal Reflection, Regular Reflection, and Refraction Feature Total Internal Reflection Regular Reflection Refraction Definition The complete reflection of light within a medium at a boundary with another medium, occurring when the angle of incidence is greater than the critical angle.

The bouncing back of light from a surface.

The bending of light as it passes from one medium to another.

Feature Total Internal

Reflection Regular

Reflection Refraction

Occurs at

Boundary Between two media where light is passing from a more optically dense medium to a less dense one (e.g., water to air).

At any reflective surface (e.g., mirrors).

Between two media with different optical densities (e.g., air to water).

Angle of

Incidence Greater than the critical angle.

Not necessarily related to critical angle.

Usually less than the critical angle.

Critical Angle Required; if the angle of incidence exceeds this angle, total internal reflection occurs.

Not applicable. Not applicable.

Angle of

Reflection Equal to the angle of incidence.

Equal to the angle of incidence.

Not defined;

depends on Snell’s Law.

Angle of

Refraction None, as light is entirely reflected inside the medium.

Not applicable. Changes according to Snell’s Law.

Transmission of Light

No light passes through the boundary; all is reflected.

Light is reflected, and some may pass through if the surface is not perfectly reflective.

Light passes through the boundary and bends.

Medium Light travels from a denser to a less dense medium.

Can occur between any reflective surfaces.

Light travels from one medium to another with different densities.

Applications fibre optics, prisms, and certain types of lenses.

Mirrors, reflective coatings.

Lenses, glasses, water droplets (rainbows).

Review Quesiton 5.1

1. A coin is placed in a bowl of water. Identify the name for the depth to which it sinks?

2. A coin is placed in a bowl of water. Identify the name for the depth to which it is viewed?

3. How is refractive index related to real depth and apparent depth?

4. The real depth of a fish swimming in a pond is 2 metres below the water surface. The refractive index of water is approximately 1.33. Calculate where the fish will be apparently (falsely) be seen.

5. What causes total internal reflection to occur?

6. What are some observable examples of total internal reflection?