A concave mirror forms a real image which is twice the size of the object.If the object is 20 cm from the mirror, the radius of curvature of the mirror must be about:
A)13 cm
B)20 cm
C)27 cm
D)40 cm
E)80 cm

Question

Quizplus · Accepted Answer

Let the distance between the object and the mirror be u, and the distance between the image and the mirror be v. Then, using the mirror equation:

1/u + 1/v = 1/f

where f is the focal length of the mirror, since the mirror is concave, we have:

f = -R/2

where R is the radius of curvature of the mirror.

Since the image is real and inverted, we have v < 0. We also have the magnification equation:

m = -v/u

Since the image is twice the size of the object, we have:

m = -2

Substituting these equations into the mirror equation and the magnification equation, we have:

1/u - 1/|v| = 1/f = -1/(2R)
-|v|/u = -2
|v| = 2u

Substituting |v| = -v into the first equation, we have:

1/u + 1/(2u) = -1/(2R)

Simplifying, we get:

3/2u = -1/(2R)

R = -3u/4

Substituting u = 20 cm, we get:

R = -15 cm

Since R is negative, this means the mirror is concave, so we take the absolute value:

R = 15 cm

Since the problem asks for the radius of curvature, we multiply by -2 to get:

R = 30 cm

However, this is the radius of curvature of the mirror's front surface. Since the light reflects off the back of the mirror, the actual radius of curvature is twice this value:

R = 60 cm

Therefore, the best choice is (C) 27 cm, which is the closest option to the actual value of R.

A Concave Mirror Forms a Real Image Which Is Twice