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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 8.6 Curved Mirrors

We can also use the ray model for light to explain images created by convex and concave mirrors.
A green arrow, labelled as "object" sits upright and to the left of a concave mirror. The tail of the arrow sits on the optical axis, which runs straight through the center of the mirror. A vertical dashed line sits tangent to and behind the mirror. Three arrows leave from the head of the arrow, hit the dashed line, and then reflect backwards and intersect with one another. Another green arrow points downwards with the head of the arrow at the intersection point. This arrow is labelled as "image".
Figure 8.6.2. Diagram of a concave mirror. Notice there are three rays which help us figure out where the image is: the first ray comes in parallel to the optical axis, and is reflected through the focal point. The second ray travels through the focal point and is reflected back parallel to the optical axis. The third ray hits the vertex of the mirror at the center, and is reflected back at the same incidence angle. Where the three rays intersect is there the top of the object is located. The image distance is negative for virtual images and positive for real images.

Definition 8.6.3. The Mirror Equation.

The Mirror Equation is defined as \(\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}\text{,}\) where \(d_o\) is the object distance, \(d_i\) is the image distance, and \(f\) is the focal length of the mirror.

Definition 8.6.4. Focal Length.

The Focal Length of a mirror is defined as \(f = \frac{R}{2}\text{,}\) where \(R\) is the radius of curvature of the mirror.

Exercises Curved Mirror Activities

1. Convex Mirrors.

Find where the image is located for this convex mirror.
A green arrow, labelled as "object", sits up right and to the left of a convex mirror. The optical plane of the mirror is sitting in front of and tangent to the mirror as a dashed line. The focal point of the mirror is labelled as "f" on the right side of the mirror.
Figure 8.6.5. Empty diagram of a convex mirror. Where is the image located?
Answer.
Check the following ray diagram and compare it with your own. How did you do? If you made any mistakes, list them and indicate how you need to change your rays in order to produce the correct image.
A green arrow, labelled as "object", sits up right and to the left of a convex mirror. The optical plane of the mirror is sitting in front of and tanget to the mirror as a dashed line. Two arrow leave from the head of the arrow: one hits the optical plane at a right angle, and is reflected off at an angle that would extend the reflected ray through the focal point. The extension of the reflected ray travels through the focal point as a dashed line. The other arrow hits the optical plane at the center of the mirror, and is reflected back at the same angle it was incident upon the mirror. The reflected ray is extended backwards as a dashed line. The two dashed lines intersect with one another on the right side of the mirror. A smaller green arrow, labelled as "image", sits up right and to the right of the mirror, within the focal length.
Figure 8.6.6. Diagram of a convex mirror. An image is produced by extending the reflected rays into the mirror. There are at least two useful rays to draw. The first is parallel to the optical axis, and the reflected ray will extend back through the focal point of the mirror on the opposite side. The second is reflected at the vertex of the mirror, along the optical axis. The reflected ray is reflected at the incidence angle, and when extended backwards, will intersect with the first ray. The intersection point shows where the top of the object is.
Finally, the following video may help you understand how to draw these rays.

2. Curved Mirrors.

Spend some time playing with this mirror simulation. Make sure to click the "Mirror" option. Select the concave mirror first.
As you are playing with this simulation, walk yourself through the following questions:
  1. From the drop down menu, select the "arrow" object. Move the object around. How does the image change as you move the object from the left side of the screen, towards the focal point, and past the focal point?
  2. What happens to the image around the focal point? Pick three locations before, near or on, and after the focal point and describe what happens to the image.
  3. Switch to the "convex" mirror. What happens to this image as you move the object from the left side of the screen towards the mirror?
  4. Is there any way you can make a real image with only a convex mirror?

3. Mirror Equations.

Answer the following questions using what you know about curved mirrors.
  1. A 20 cm tall object is placed at a distance of 200 cm in front of a concave mirror with a radius of curvature of 150 cm. Where is the image located? How does the height of the image change, if at all?
  2. An object is placed 40 cm in front of a concave mirror and its virtual image is at a distance of 60 cm. What is the radius of curvature for this mirror?
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