Radius of Curved Surface of Plano Convex Lens from Image Properties

To find the radius of curvature of the curved surface of the plano-convex lens, we proceed step-by-step using the principles of optics:

Step 1: Determine the object distance ($u$) A real image formed by a convex lens is inverted, which means the magnification $m$ is negative. Given:

• Distance of real image behind the lens, $v = +8\text{ m}$
• Magnification, $m = -\frac{1}{3}$

Using the magnification formula for a thin lens: $m = \frac{v}{u}$ $-\frac{1}{3} = \frac{8\text{ m}}{u} \implies u = -24\text{ m}$

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Step 2: Calculate the focal length ($f$) of the lens Using the thin lens formula: $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$

Substitute the values of $v$ and $u$: $\frac{1}{f} = \frac{1}{8} - \left(-\frac{1}{24}\right) = \frac{1}{8} + \frac{1}{24} = \frac{3 + 1}{24} = \frac{4}{24} = \frac{1}{6}\text{ m}^{-1}$

Thus, the focal length of the lens is: $f = 6\text{ m}$

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Step 3: Calculate the refractive index ($\mu$) of the lens material The refractive index $\mu$ of a medium is defined as the ratio of the wavelength of light in free space ($\lambda_0$) to its wavelength in that medium ($\lambda_m$): $\mu = \frac{\lambda_0}{\lambda_m}$

Given that $\lambda_m = \frac{2}{3}\lambda_0$: $\mu = \frac{\lambda_0}{\frac{2}{3}\lambda_0} = \frac{3}{2} = 1.5$

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Step 4: Calculate the radius of curvature ($R$) using the Lens Maker's Formula For a plano-convex lens, one surface has a radius of curvature $R_1 = R$ (convex) and the planar surface has $R_2 = \infty$.

By Lens Maker's Formula: $\frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$ $\frac{1}{f} = (\mu - 1)\left(\frac{1}{R} - \frac{1}{\infty}\right) = \frac{\mu - 1}{R}$

Substitute $f = 6\text{ m}$ and $\mu = 1.5$: $\frac{1}{6} = \frac{1.5 - 1}{R}$ $\frac{1}{6} = \frac{0.5}{R}$ $R = 0.5 \times 6 = 3\text{ m}$

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Conclusion: The radius of curvature of the curved surface of the lens is $3\text{ m}$.

Correct Answer: C ($3\text{ m}$)