A convex mirror with a radius of curvature of 39.0 cm forms a 0.95 cm tall image of a pencil at a distance of 12.1 cm behind the mirror. What is the magnification?
To find the magnification of the image formed by a convex mirror, we can use the formula:
magnification (m) = height of the image (h') / height of the object (h)
Given:
Radius of curvature (R) = 39.0 cm
Height of the image (h') = 0.95 cm
Distance of the image from the mirror (d') = -12.1 cm
(Note: The negative sign indicates that the image is formed on the same side as the object, which is typical for images formed by convex mirrors.)
To calculate the magnification, we need to find the height of the object (h). For that, we can use the mirror formula for convex mirrors:
1/f = 1/d' + 1/d
Where f is the focal length of the mirror and d is the distance of the object from the mirror.
Convex mirrors have negative focal lengths, given by:
f = - R / 2
Substituting the values:
f = -39.0 cm / 2 = -19.5 cm
Using the mirror formula, we can rearrange it to solve for the object distance (d):
1/d = 1/f - 1/d'
Substituting the values:
1/d = 1/-19.5 cm - 1/(-12.1 cm)
1/d = -0.0513 - (-0.0826)
1/d = -0.0513 + 0.0826
1/d = 0.0313
d = 1 / 0.0313
d ≈ 31.9 cm
Now that we have the height of the object (h) and the height of the image (h'), we can calculate the magnification (m):
m = h' / h
m = 0.95 cm / h
To find h, use the magnification formula rearranged:
h = h' / m
h ≈ 0.95 cm / (0.95 cm / h)
h ≈ 0.95 cm
Finally, substitute the values into the magnification formula to calculate the magnification (m):
m = h' / h
m = 0.95 cm / 0.95 cm
m = 1
Therefore, the magnification of the image formed by the convex mirror is 1.