Question

A biconvex lens ($R_1 = R_2 = 30\ \text{cm}$) has focal length equal to the focal length of a concave mirror. The radius of curvature of the concave mirror is [Refractive index of material of lens $= 1.6$]

• A. $30\ \text{cm}$
• B. $40\ \text{cm}$
• C. $50\ \text{cm}$
• D. $20\ \text{cm}$

Hint:

For any symmetric biconvex lens where $R_1 = R_2 = R$, the Lens Maker's Formula simplifies directly to $f = \frac{R}{2(\mu - 1)}$. Substituting $R = 30$ and $\mu = 1.6$ gives $f = \frac{30}{2(0.6)} = \frac{30}{1.2} = 25\ \text{cm}$ in one quick step!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given a biconvex lens with equal radii of curvature and a known refractive index. Its focal length is stated to be exactly equal to the focal length of a concave mirror. We need to determine the radius of curvature ($R_m$) of that concave mirror.

Step 2: Key Formula or Approach:
1. Use the Lens Maker's Formula to determine the focal length of the biconvex lens ($f_l$): $$\frac{1}{f_l} = (\mu - 1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$ For a biconvex lens, using the standard Cartesian sign convention, $R_1 = +R$ and $R_2 = -R$. 2. Relate the focal length of a spherical concave mirror ($f_m$) to its radius of curvature ($R_m$): $$f_m = \frac{R_m}{2} \implies R_m = 2f_m$$

Step 3: Detailed Explanation:
Let's find the focal length of the lens first. Given $\mu = 1.6$, $R_1 = 30\ \text{cm}$, and $R_2 = -30\ \text{cm}$: $$\frac{1}{f_l} = (1.6 - 1) \left(\frac{1}{30} - \left(-\frac{1}{30}\right)\right)$$ $$\frac{1}{f_l} = 0.6 \times \left(\frac{1}{30} + \frac{1}{30}\right) = 0.6 \times \frac{2}{30} = \frac{1.2}{30}$$ Solving for $f_l$: $$f_l = \frac{30}{1.2} = 25\ \text{cm}$$ Since the focal length of the concave mirror is identical to that of the lens: $$f_m = f_l = 25\ \text{cm}$$ Now, compute the radius of curvature of the mirror using its geometric relation: $$R_m = 2 \times f_m = 2 \times 25 = 50\ \text{cm}$$

Step 4: Final Answer:
The radius of curvature of the concave mirror is $50\ \text{cm}$, corresponding to option (C).