 ## Monday, May 22, 2006

### Mirrors

• The focal length of a curved mirror is always ½ its radius of curvature (f= r/2)

• f = focal length of the mirror in meters

• The reflecting power of a mirror in diopters DM= 1/f (m)

• r = radius of curvature of the mirror in meters.

• For mirrors or reflecting surfaces: U + 2/rm= V, (rm is in meters) or U + 1/f = V

• If the mirror is convergent or plus, the focal point is to the left of the mirror.

• If the focal point is to the right of the mirror, the mirror is divergent or minus.

• Convex mirrors form virtual images on the opposite side from the object.

• Concave mirrors form real images on the same side as the object.

• A plus (concave) mirror adds positive vergence, while a minus (convex) mirror adds minus vergence.

• Convex mirrors add negative vergence like minus lenses.

• Concave mirrors add positive vergences like plus lenses.

• Plane mirrors add no vergence.

• The field of view of a plane mirror is 2 times its size.

• Holding a hand mirror farther away from the face does not enlarge the field of view.

• You need approximately a 1/2 length mirror to see your entire self.

When the object is located closer to a converging lens or a converging mirror than its focal distance, the image will be virtual and erect, not real and inverted. These are the principles applied to magnifying glasses used to read small print and a concave mirror, used as a shaving mirror.

Question: (Figure 19) Consider a concave mirror whose radius of curvature is 50cm. Therefore, the focal length of the mirror is f = r/2 = 0.5/2 = 0.25m, and the reflecting power of the mirror is 1/f = 1/0.25 = +4.00D. If an object lies 1m in front of the mirror, where is the image vergence? Click on image to enlarge.

Use the equation U + D = V where

u = -1m = -100 cm

U = object vergence = 100/u = 100/(-100) = -1.00D

D = reflecting power of the mirror = +4.00D

V = image vergence = U + D = -1.00D + (+4.00D) = +3.00D

Therefore the image is real and lies 33cm in front of the mirror.

• If an object point is 50cm in front of the mirror, which coincides with C – the center of curvature, the image vergence (-2 + 4 = +2) also coincides with C.
• If an object point coincides with F, the focal point of the mirror, the image vergence (-4 + 4 = 0) is at infinity.
• If an object point lies 20cm in front of the mirror, the image point (-5 + 4 = -1) is virtual (reflected rays are divergent) and lies 1m in back of the mirror.

Question: (Figure 20) Consider a convex mirror whose radius of curvature is 40cm. Therefore, the focal length of the mirror is f = -(r/2) = -0.4/2 = -0.20m, and the reflecting power of the mirror is 1/f = 1/-0.20 = -5.00D. If an object lies 1m in front of the mirror, what is the image vergence? Click on image to enlarge.

Use the equation U + D = V where

u = -1m = -100 cm

U = object vergence = 100/u = 100/(-100) = -1.00D

D = reflecting power of the mirror = -5.00D

V = image vergence = U + D = -1.00D + (-5.00D) = -6.00D

Therefore the image is virtual and lies 16.67cm behind the mirror.

Question: For a cornea with a radius of curvature of 8mm, what is the reflective power of the cornea?

Answer: The cornea is a convex mirror, and its reflective power is negative. The focal length of the cornea is f = -(r/2) = - (0.008/2) = -0.004m, and the reflecting power of the cornea is 1/f = 1/-0.004 = -250D.  