Exercises: Optical systems: thin lenses and mirrors

In the optics kit in the laboratory there are two lenses: one is thicker at the centre than at the edges, and the other is thicker at the edges than at the centre.

1. a) State which one is converging and which one is diverging.
2. b) Describe what happens to a beam of rays parallel to the optical axis when passing through each of them: do they converge at a point or diverge as if coming from a point?

An object is placed in front of a converging lens at a distance greater than twice its focal length.

1. a) State the rules for drawing the three principal rays of a converging lens (the ray parallel to the axis, the ray passing through the centre, and the ray passing through the object focus).
2. b) Draw the ray diagram in your notebook and determine the characteristics of the image: is it real or virtual, upright or inverted, larger or smaller than the object?

You can do it on paper and send a photo with your answer.

A person is standing $2\ \text{m}$ from a flat wall mirror. Observe the diagram:

1. a) At what distance behind the mirror is their image formed?
2. b) State the characteristics of the image: is it real or virtual, upright or inverted, and what size is it compared to the person?
3. c) At what distance does the person see their own image?

A watchmaker uses a magnifying glass to inspect the mechanism of a watch.

1. a) What type of lens is a magnifying glass: converging or diverging?
2. b) Where must the object be placed relative to the focus of the lens for the magnifying glass to work?
3. c) State the characteristics of the observed image (real or virtual, upright or inverted, larger or smaller).

A luminous object is placed $s = 30\ \text{cm}$ from a thin converging lens with focal length $f = 10\ \text{cm}$. Use the thin lens equation $1/f = 1/s + 1/s'$ with this sign convention: $s$ is positive for a real object, $s'$ is positive if the image is real and negative if virtual, and $f$ is positive for converging lenses and negative for diverging lenses.

1. a) Calculate the image distance $s'$.
2. b) State whether the image is real or virtual and whether it can be projected onto a screen.

Glasses for correcting myopia (short-sightedness) use diverging lenses.

1. a) Draw the ray diagram of a diverging lens with an object in front of it in your notebook.
2. b) From the drawing, justify why a diverging lens always produces virtual, upright images that are smaller than the object.

You can do it on paper and send a photo with your answer.

Match each application with the type of mirror and briefly explain why: a car's wing mirror, a dentist's mirror used to view a magnified tooth, and a car headlight mirror directing light from the bulb forward in an almost parallel beam.

1. a) Wing mirror.
2. b) Dentist's mirror.
3. c) Headlight mirror.

The power of a lens is defined as $P = 1/f$, with $f$ in metres, and is measured in dioptres ($\text{D}$).

1. a) Calculate the power of a converging lens of focal length $f = 0.5\ \text{m}$.
2. b) Calculate the focal length of a lens of $-4\ \text{D}$ and state whether it is converging or diverging.

An object $10\ \text{cm}$ high is placed $s = 60\ \text{cm}$ from a converging lens with focal length $f = 20\ \text{cm}$. Use $1/f = 1/s + 1/s'$ and the magnification $A = -s'/s$, with the usual sign convention (positive distances for real objects and real images). Observe the diagram:

1. a) Calculate the image distance $s'$.
2. b) Calculate the magnification and the height of the image.
3. c) Describe the image: real or virtual, upright or inverted, larger or smaller.

A slide projector uses a converging lens of focal length $f = 15\ \text{cm}$. The slide is placed $s = 20\ \text{cm}$ from the lens.

1. a) Calculate the distance at which the screen must be placed for the image to be in focus.
2. b) Calculate the magnification.
3. c) Explain why the slide must be placed upside down in the slide holder.

A magnifying glass is a converging lens of focal length $f = 10\ \text{cm}$. A stamp placed $s = 6\ \text{cm}$ from the lens is observed. Observe the diagram:

1. a) Calculate the image distance $s'$ and explain the meaning of its sign.
2. b) Calculate the magnification and explain the meaning of its sign.
3. c) State the characteristics of the image and why it cannot be formed on a screen.

An object $5\ \text{cm}$ high is $s = 30\ \text{cm}$ from a diverging lens of focal length $f = -20\ \text{cm}$. Observe the diagram:

1. a) Calculate the image distance $s'$.
2. b) Calculate the magnification and the height of the image.
3. c) Confirm with the results that the image is virtual, upright, and smaller, as expected for a diverging lens.

A concave spherical laboratory mirror has a radius of curvature of $R = 40\ \text{cm}$. An object $4\ \text{cm}$ high is placed $s = 30\ \text{cm}$ from the mirror. Recall that $f = R/2$ and use the same equation $1/f = 1/s + 1/s'$ with the usual convention (positive distances for real objects and images). Observe the diagram:

1. a) Calculate the focal length of the mirror.
2. b) Calculate the position of the image.
3. c) Calculate the magnification and height of the image, and state whether it can be captured on a screen.

The wing mirror of a car is a convex mirror with focal length $f = -0.5\ \text{m}$. A car $1.5\ \text{m}$ high is travelling $s = 10\ \text{m}$ from the mirror. Observe the diagram:

1. a) Calculate the position of the image.
2. b) Calculate the magnification and height of the image.
3. c) In light of the result, explain the meaning of the warning "objects in mirror are closer than they appear".

During an eye test, a hypermetropic (long-sighted) patient is prescribed lenses of $+2\ \text{D}$ and a myopic (short-sighted) patient is prescribed lenses of $-2.5\ \text{D}$. Recall that $P = 1/f$ with $f$ in metres.

1. a) Calculate the focal length of each lens and state its type.
2. b) Explain what focusing problem each patient has (where is the image of a distant object formed relative to the retina?) and why the sign of their lens corrects it.

A person $1.80\ \text{m}$ tall, with their eyes $1.70\ \text{m}$ above the ground, wants to buy the smallest vertical flat mirror that allows them to see themselves full length.

1. a) Draw the rays travelling from the feet and the top of the head to the eyes reflecting off the mirror in your notebook, and, applying the law of reflection and similar triangles, deduce that the minimum height of the mirror is half the height of the person.
2. b) Calculate this minimum height and the heights above the ground at which the bottom and top edges of the mirror should be placed.

You can do it on paper and send a photo with your answer.

It is desired to design a projector that forms a real image three times larger than the object on a screen located $4\ \text{m}$ from the object. Observe the diagram:

1. a) Given that for a real image the object-screen distance is $s + s'$, set up the system of equations with the magnification condition $|A| = s'/s = 3$ and calculate $s$ and $s'$.
2. b) Calculate the focal length of the required lens and its power in dioptres.
3. c) Reason why a diverging lens would not be suitable for this project.

A refracting telescope belonging to an astronomy club has an objective lens of focal length $f_{\text{ob}} = 90\ \text{cm}$ and an eyepiece of $f_{\text{oc}} = 3\ \text{cm}$. For a very distant object, the angular magnification of the telescope is $M = f_{\text{ob}}/f_{\text{oc}}$ and the length of the tube is approximately the sum of the two focal lengths. Observe the diagram:

1. a) Calculate the angular magnification of the telescope.
2. b) Calculate the approximate length of the tube.
3. c) Explain the role of each lens and why it is desirable for the objective to have a large diameter (think about the light it captures from a faint star).

A camera lens can be approximated as a converging lens of focal length $f = 12\ \text{cm}$. Objects located at $s = 36\ \text{cm}$, $s = 24\ \text{cm}$, and $s = 18\ \text{cm}$ from the lens are photographed.

1. a) Calculate the image distance $s'$ and the magnification in all three cases.
2. b) Describe and justify the trend: as the object is moved closer to the focus, what happens to the position and size of the image?
3. c) Analyse the limiting cases: what happens if the object is exactly at $s = f$? What if it is closer than the focal length?

A solar furnace like the one in Odeillo (France) concentrates sunlight using a concave spherical mirror of radius $R = 1.2\ \text{m}$. Observe the diagram:

1. a) Calculate its focal length.
2. b) Sunlight arrives in practically parallel rays (object at infinity): using $1/f = 1/s + 1/s'$ with $s \rightarrow \infty$, reason where the receiver should be placed to concentrate the energy.
3. c) If an object is placed $s = 0.9\ \text{m}$ from the same mirror, calculate the position of the image and the magnification, and state its characteristics.
4. d) Explain where a point light source should be located for the mirror to produce an emergent parallel beam, as in a searchlight, and justify it using the equation.

In the design of an optical instrument, two thin lenses are glued together in contact: one of $+3\ \text{D}$ and another of $-1\ \text{D}$. For thin lenses in contact, their powers add: $P = P_1 + P_2$.

1. a) Calculate the power and focal length of the combination and state whether it behaves as a converging or diverging lens.
2. b) The combination is used as a magnifying glass with an object at $s = 0.3\ \text{m}$: calculate the image position and magnification, and state the characteristics of the image.
3. c) In light of the signs, explain why the overall combination is dominated by one of the two lenses.

A convex wing mirror has $f = -0.4\ \text{m}$. A car $1.8\ \text{m}$ wide is $s = 8\ \text{m}$ away.

1. a) Calculate the position of the image, the magnification, and the apparent width of the car in the mirror.
2. b) Compare with a flat mirror of the same size, in which the image would have the actual size of the object: using your results, explain why the convex mirror allows a much larger field of view to be covered in a small mirror.
3. c) Comment on the drawback of this design for estimating distances and why regulations require the inclusion of the corresponding warning.

An optical laboratory microscope has an objective lens with $40$ magnification and an eyepiece with $10$ magnification, so that the total magnification is the product of both.

1. a) Calculate the total magnification of the microscope.
2. b) Explain the role of each lens: what type of image does the objective form of the specimen, and how does the eyepiece act on that intermediate image?
3. c) A cell measures $20\ \mu\text{m}$: calculate the apparent size with which it is observed through the microscope, expressed in $\text{mm}$.

The human eye can be modelled as a lens (the crystalline lens) whose image is always formed on the retina, located $2.4\ \text{cm}$ from the lens. The lens changes its focal length thanks to the ciliary muscles (accommodation). Observe the diagram:

1. a) Calculate the focal length of the lens when observing a very distant object ($s \rightarrow \infty$).
2. b) Calculate the focal length required to read a book $25\ \text{cm}$ from the eye.
3. c) Using both results, explain what accommodation consists of and why with age (presbyopia) it becomes difficult to focus on close objects.

Unit challenge. How can you characterise an optical system using everyday objects? Design and calibrate your own home-made optical bench.

1. Experimental measurement: Use a magnifying glass (or a transparent cylindrical glass filled with water) as a converging lens. Using a mobile phone flashlight covered with perforated card as the object, project a sharp image onto a white screen for 5 different object distances. Measure $s$ and the image distance $s'$, estimating the instrumental uncertainty ($\pm 0.1\text{ cm}$).

2. Graphical analysis: Build a table with the values of $s$, $s'$, $1/s$, and $1/s'$. Plot $1/s'$ versus $1/s$, fit the straight line, and experimentally determine the focal length $f'$ from the thin-lens equation: $\frac{1}{s'} - \frac{1}{s} = \frac{1}{f'}$.

3. Systems and reflection: Observe the formation of your reflected image on both sides of a metal spoon (concave and convex mirror) at two different distances. Draw to scale the ray diagrams for both the lens and the mirror, justifying whether the images are real or virtual, upright or inverted.

4. Final product: Prepare a brief report with the table, the best-fit plot, the ray tracings, and the physical conclusions.

Evaluation criteria:

• Precision in empirical data collection, uncertainty calculation, and graphical linearisation.

• Correct determination of the focal length $f'$ and physical interpretation of the line parameters.

• Conceptual and geometric rigour in ray tracing for refraction and reflection.