Calculate the focal length of a concave spherical mirror that projects an image 12.0 cm in front of the mirror with a magnitude of -0.42
To calculate the focal length of a concave spherical mirror, we'll use the mirror formula:
1/f = 1/v - 1/u
where:
- f is the focal length,
- v is the image distance (distance of the image from the mirror), and
- u is the object distance (distance of the object from the mirror).
In this case, we are given:
- v = -12.0 cm (as the image is projected in front of the mirror, the distance is taken as negative),
- magnification (m) = -0.42 (the negative sign indicates the image is inverted).
To find the focal length, we need to determine the object distance (u) first. Rearranging the magnification formula, we get:
m = -v/u
Substituting the given values:
-0.42 = -12.0/u
Now we can solve for u:
12.0/u = 0.42
u = 12.0 / 0.42
u ≈ 28.57 cm
Now, we can substitute the values of u and v into the mirror formula to find the focal length:
1/f = 1/-12.0 - 1/28.57
Calculating this equation gives us:
1/f = -0.0833 - 0.035
1/f ≈ -0.1183
Finally, we can find f by taking the reciprocal of both sides:
f = 1 / -0.1183
f ≈ -8.46 cm
Therefore, the focal length of the concave spherical mirror is approximately -8.46 cm.