A shaving/makeup mirror is designed to magnify your face by a factor of 1.33 when your face is placed 20.0 cm in front of it. (a) What type of mirror is it? (b) Describe the type of image that it makes of your face. (c) Calculate the required radius of curvature for the mirror.
To answer these questions, we can use the mirror equation, which relates the object distance (distance of the face from the mirror), the image distance (distance of the image from the mirror), and the focal length of the mirror.
The mirror equation is given as: 1/f = 1/do + 1/di
Where:
f = focal length
do = object distance
di = image distance
Let's go step-by-step.
(a) To determine the type of mirror, we need to find the focal length (f) of the mirror. Given that the mirror magnifies the face by a factor of 1.33 when placed 20.0 cm in front of it, we can say that the image distance (di) is 1.33 times the object distance (do).
Since the object is placed in front of the mirror, the object distance (do) is positive (+20.0 cm).
1/f = 1/do + 1/di
Substituting the values:
1/f = 1/20 + 1/(1.33 * 20)
1/f = 1/20 + 1/26.6
Now, we need to find the value of f. To do that, we can solve for 1/f:
1/f = (20 + 26.6) / (20 * 26.6)
1/f = 46.6 / 532
1/f = 0.0872
To find f, we can take the reciprocal of 0.0872:
f = 1/0.0872 ≈ 11.47 cm
Since the focal length (f) is positive, the mirror is a converging mirror, specifically a convex mirror.
(b) The type of image produced by a convex mirror is virtual, upright, and diminished in size. This means that the image will appear smaller than the actual face and will be located behind the mirror.
(c) Now, let's calculate the required radius of curvature for the mirror. The radius of curvature (R) is related to the focal length by the formula:
R = 2f
Substituting the value for f:
R = 2 * 11.47 cm
R ≈ 22.94 cm