47. A plano convex lens fits exactly into a plano concave lens. Their plane surfaces are parallel to each other. If lenses are made of different materials of refractive indices $μ_1$ and $μ_2$ and R is the radius of curvature of the curved surface of the lenses, then the focal length of the combination is ?

47. A plano convex lens fits exactly into a plano concave lens. Their plane surfaces are parallel to each other. If lenses are made of different materials of refractive indices $μ_1$ and $μ_2$ and R is the radius of curvature of the curved surface of the lenses, then the focal length of the combination is ?

\( (a) \ \frac{R}{2(\mu_{1} - \mu_{2})} \)

\( (b) \ \frac{R}{(\mu_{1} - \mu_{2})} \)

\( (c) \ \frac{2R}{(\mu_{2} - \mu_{1})} \) 

\( (d) \ \frac{R}{2(\mu_{1} + \mu_{2})} \)

Answer :  \( (b) \ \frac{R}{(\mu_{1} - \mu_{2})} \)

\( \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \)

\( = (\mu_1 - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) + (\mu_2 - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \)

\( = (\mu_1 - 1) \frac{1}{R} - (\mu_2 - 1) \frac{1}{R} \)

\( \frac{1}{f} = (\mu_1 - \mu_2) \frac{1}{R} \)