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A solid glass sphere of refractive index 1.5 and radius 40 cm has its entire back hemispherical surface silvered (acting as a concave mirror). A parallel beam of light enters along a diameter. Where does the beam converge after passing through the sphere?

1. 10 cm to the left of centre
2. 30 cm to the left of centre
3. 50 cm to the left of centre
4. 60 cm to the left of centre

Correct answer: 30 cm to the left of centre

Solution

The silvered back surface creates an equivalent mirror. Using power formulation (in cm units): refraction at front surface P1 = (n-1)/R = 0.5/40 = 1/80 cm⁻¹. Mirror power Pₘ = 2n/R = 3/40 cm⁻¹. Equivalent mirror power P_eq = 2P1 + Pₘ = 2/80 + 3/40 = 1/40 + 3/40 = 4/40 = 1/10 cm⁻¹. Focal length of equivalent mirror = 1/P_eq = 10 cm measured from the front surface. The centre of sphere is 40 cm from the front surface. So the focus is 40 - 10 = 30 cm to the left of centre.

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