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 need to understand the properties and characteristics of different types of mirrors. The two primary types of mirrors are concave mirrors and convex mirrors.
(a) In this case, since the mirror magnifies the face, it is a type of mirror known as a concave mirror. A concave mirror is curved inward, like the inside of a spoon.
(b) The image formed by a concave mirror depends on the position of the object (your face) relative to the mirror's focal point. If the object is placed beyond the focal point, a real and inverted image is formed. If the object is placed between the focal point and the mirror, a virtual and magnified image is formed. Given that the face is placed 20.0 cm in front of the mirror, which is less than the focal point, the image formed will be virtual and magnified.
(c) To calculate the required radius of curvature for the mirror, we can use the mirror equation:
1/f = 1/do + 1/di
Where:
- f is the focal length of the mirror
- do is the object distance (distance of your face from the mirror)
- di is the image distance (distance of the image from the mirror)
Since the mirror magnifies the face by a factor of 1.33, we know that the image distance (di) is 1.33 times the object distance (do), which is 20.0 cm. So, di = 1.33 * 20.0 = 26.6 cm.
Now, we can rearrange the mirror equation to solve for the focal length (f):
1/f = 1/do + 1/di
1/f = 1/20.0 + 1/26.6
1/f ≈ 0.05 + 0.038
1/f ≈ 0.088
f ≈ 1/0.088
f ≈ 11.36 cm
Since the focal length is half the radius of curvature, the required radius of curvature for the mirror is approximately 2 times the focal length:
Radius of curvature ≈ 2 * 11.36
Radius of curvature ≈ 22.72 cm
Therefore, the required radius of curvature for the mirror is approximately 22.72 cm.
To determine the type of mirror and describe the type of image produced by a shaving/makeup mirror, we need to apply the mirror magnification formula:
(a) The mirror magnification formula is given by:
magnification = - (image distance / object distance) = height of image / height of object
Since the mirror magnifies the face, the magnification factor is greater than 1. Therefore, the magnification is 1.33.
For a magnification factor greater than 1, the mirror is a concave mirror.
(b) As for the type of image, since the magnification is positive, the image formed by the concave mirror is virtual and upright.
(c) To calculate the required radius of curvature, we can use the mirror magnification formula in terms of the radius of curvature:
magnification = - (distance of image from mirror / distance of object from mirror) = -(r / (r - distance of object from mirror))
Plugging in the given values:
1.33 = - ((-r) / (-r - 20.0 cm))
Solving for r:
1.33 = r / (r + 20.0 cm)
Multiply both sides by (r + 20.0 cm):
1.33r + 26.6 cm = r
0.33r = -26.6 cm
r = (-26.6 cm) / 0.33
r ≈ -80.6 cm
Therefore, the required radius of curvature for the mirror is approximately -80.6 cm.