### Additional Problems

Use a ruler and a protractor to draw rays to find images in the following cases.

(a) A point object located on the axis of a concave mirror located at a point within the focal length from the vertex.

(b) A point object located on the axis of a concave mirror located at a point farther than the focal length from the vertex.

(c) A point object located on the axis of a convex mirror located at a point within the focal length from the vertex.

(d) A point object located on the axis of a convex mirror located at a point farther than the focal length from the vertex.

(e) Repeat (a)–(d) for a point object off the axis.

Where should a 3 cm tall object be placed in front of a concave mirror of radius 20 cm so that its image is real and 2 cm tall?

A 3 cm tall object is placed 5 cm in front of a convex mirror of radius of curvature 20 cm. Where is the image formed? How tall is the image? What is the orientation of the image?

You are looking for a mirror so that you can see a four-fold magnified virtual image of an object when the object is placed 5 cm from the vertex of the mirror. What kind of mirror you will need? What should be the radius of curvature of the mirror?

Derive the following equation for a convex mirror:

$\frac{1}{VO}-\frac{1}{VI}=\text{\u2212}\frac{1}{VF}$,

where *VO* is the distance to the object *O* from vertex *V*, *VI* the distance to the image *I* from *V*, and *VF* is the distance to the focal point *F* from *V*. (*Hint*: use two sets of similar triangles.)

(a) Draw rays to form the image of a vertical object on the optical axis and farther than the focal point from a converging lens. (b) Use plane geometry in your figure and prove that the magnification *m* is given by $m=\frac{{h}_{\text{i}}}{{h}_{\text{o}}}=\text{\u2212}\frac{{d}_{\text{i}}}{{d}_{\text{o}}}.$

Use another ray-tracing diagram for the same situation as given in the previous problem to derive the thin-lens equation, $\frac{1}{{d}_{\text{o}}}+\frac{1}{{d}_{\text{i}}}=\frac{1}{f}$.

You photograph a 2.0-m-tall person with a camera that has a 5.0 cm-focal length lens. The image on the film must be no more than 2.0 cm high. (a) What is the closest distance the person can stand to the lens? (b) For this distance, what should be the distance from the lens to the film?

Find the focal length of a thin plano-convex lens. The front surface of this lens is flat, and the rear surface has a radius of curvature of ${R}_{2}=\mathrm{-35}\phantom{\rule{0.2em}{0ex}}\text{cm}$. Assume that the index of refraction of the lens is 1.5.

Find the focal length of a meniscus lens with ${R}_{1}=20\phantom{\rule{0.2em}{0ex}}\text{cm}$ and ${R}_{2}=15\phantom{\rule{0.2em}{0ex}}\text{cm}$. Assume that the index of refraction of the lens is 1.5.

A nearsighted man cannot see objects clearly beyond 20 cm from his eyes. How close must he stand to a mirror in order to see what he is doing when he shaves?

A mother sees that her child’s contact lens prescription is 0.750 D. What is the child’s near point?

The contact-lens prescription for a nearsighted person is −4.00 D and the person has a far point of 22.5 cm. What is the power of the tear layer between the cornea and the lens if the correction is ideal, taking the tear layer into account?

**Unreasonable Results** A boy has a near point of 50 cm and a far point of 500 cm. Will a −4.00 D lens correct his far point to infinity?

Find the angular magnification of an image by a magnifying glass of $f=5.0\phantom{\rule{0.2em}{0ex}}\text{cm}$ if the object is placed ${d}_{\text{o}}=4.0\phantom{\rule{0.2em}{0ex}}\text{cm}$ from the lens and the lens is close to the eye.

Let objective and eyepiece of a compound microscope have focal lengths of 2.5 cm and 10 cm, respectively and be separated by 12 cm. A $\text{70-\mu m}$ object is placed 6.0 cm from the objective. How large is the virtual image formed by the objective-eyepiece system?

Draw rays to scale to locate the image at the retina if the eye lens has a focal length 2.5 cm and the near point is 24 cm. (*Hint*: Place an object at the near point.)

The objective and the eyepiece of a microscope have the focal lengths 3 cm and 10 cm respectively. Decide about the distance between the objective and the eyepiece if we need a $10\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}$ magnification from the objective/eyepiece compound system.

A far-sighted person has a near point of 100 cm. How far in front or behind the retina does the image of an object placed 25 cm from the eye form? Use the cornea to retina distance of 2.5 cm.

A near-sighted person has afar point of 80 cm. (a) What kind of corrective lens will the person need assuming the distance to the contact lens from the eye is zero? (b) What would be the power of the contact lens needed?

In a reflecting telescope the objective is a concave mirror of radius of curvature 2 m and an eyepiece is a convex lens of focal length 5 cm. Find the apparent size of a 25-m tree at a distance of 10 km that you would perceive when looking through the telescope.

Two stars that are ${10}^{9}\text{km}$ apart are viewed by a telescope and found to be separated by an angle of ${10}^{\mathrm{-5}}\phantom{\rule{0.2em}{0ex}}\text{radians}$. If the eyepiece of the telescope has a focal length of 1.5 cm and the objective has a focal length of 3 meters, how far away are the stars from the observer?

What is the angular size of the Moon if viewed from a binocular that has a focal length of 1.2 cm for the eyepiece and a focal length of 8 cm for the objective? Use the radius of the moon $1.74\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{6}\text{m}$ and the distance of the moon from the observer to be $3.8\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{8}\text{m}$.

An unknown planet at a distance of ${10}^{12}\text{m}$ from Earth is observed by a telescope that has a focal length of the eyepiece of 1 cm and a focal length of the objective of 1 m. If the far away planet is seen to subtend an angle of ${10}^{\mathrm{-5}}\phantom{\rule{0.2em}{0ex}}\text{radian}$ at the eyepiece, what is the size of the planet?