A convex mirror and a concave mirror are placed on the same optic axis, separate

Question

A convex mirror and a concave mirror are placed on the same optic axis, separated by a distance L = 6.500 m. The radius of curvature of each mirror has a magnitude of 0.275 m. A light source is located a distance x from the concave mirror, as shown in the figure below.

I think it may be L = .650m and not L = 6.500m

A convex mirror and a concave mirror are placed on the same optic axis, separated by a distance L 6.500 m. The radius of curvature of each mirror has a magnitude of 0.275 m. A light source is located a distance x from the concave mirror, as shown in the figure below. (a) What distance x will result in the rays from the source returning to the source after reflecting first from the convex mirror and then from the concave mirror? b) Repeat part (a), but now let the rays reflect first from the concave mirror and then from the convex one.

Explanation / Answer

given

L = 6.5 m

R = 0.275 m

a) so when light reflects first from the convex mirror

object distance = -(L - x)

image distance = v

focal length = R/2

hence

using mirror formula

1/v + 1/u = 1/f

1/v - 1/(L - x) = 2/R

1/v = 2/0.275 + 1/(6.5 - x) = 1/0.1375 + 1/(6.5 - x)

v = (6.5 - x)*0.1375/(6.6375 - x)

next reflection from the concave mirror

object distance = -(L + v)

image distance = -x

focal length = -R/2

hence

-1/x - 1/(L + v) = -2/0.275

0.1375(44.0375 - 6.6375x ) + 0.1375(6.6375 - x)x = (44.0375 - 6.6375x )x

6.05515625 - 0.91265625x + 0.91265625x - 0.1375x^2 = 44.0375x - 6.6375x^2

6.5x^2 - 44.0375x + 6.05515625 = 0

x = 0.140409 m

b) when the light reflects first from the concave mirror

object distance = -x

image ditance = v

focal length = -0.1375

hence

1/v - 1/x = -1/0.1375

1/v = 1/x - 1/0.1375

v = 0.1375x/(0.1375 - x)

for convex lens

object distance = -(L + v) = (6.3625 x - 0.89375)/(0.1375 - x)

image distance = -(L - x) = (-6.5 + x)

focal length = 0.1375

hence

1/(x - 6.5) + (0.1375 - x) / (6.3625 x - 0.89375) = 1/0.1375

0.1375*(6.3625 x - 0.89375) + 0.1375*(0.1375 - x)*(x - 6.5) = (6.3625 x - 0.89375) *(x - 6.5)

6.5x^2 -9.18125 x + 6.05515625 = 0

x = 1.3641 m

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