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Step-by-Step Solution
Step 1: Identify the Refractive Indices
Given that the wavelength of light inside the refracting medium is \tfrac{2}{3} times its wavelength in air, the refractive index of the medium ( \mu_2 ) relative to air ( \mu_1 = 1 ) is:
\mu_2 = \frac{\text{wavelength in air}}{\text{wavelength in medium}} = \frac{1}{\tfrac{2}{3}} = 1.5.
Step 2: Determine Object and Image Distances
The image is formed at v = +10 \text{ m} behind the refracting surface. It is given that the image distance is \tfrac{2}{3} of the object distance. Denote the magnitude of object distance by |u| . Then:
v = \frac{2}{3} |u|.
Since v = 10 m, we have:
10 = \frac{2}{3} |u| \quad \Rightarrow \quad |u| = 15.
By the sign convention for refraction at a convex surface (with the surface bulging towards the right and object on the left side), the object distance is taken as negative:
u = -15 \text{ m}, \quad v = +10 \text{ m}.
Step 3: Use the Refraction Formula for a Spherical Surface
For refraction from a single spherical surface separating two media with refractive indices \mu_1 and \mu_2 , the formula is:
\frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R},
where R is the radius of curvature of the refracting surface.
Step 4: Substitute the Known Values
Substitute \mu_1 = 1 , \mu_2 = 1.5 , v = +10 m, and u = -15 m into the formula:
\frac{1.5}{10} - \frac{1}{-15} = \frac{1.5 - 1}{R}.
Step 5: Simplify the Left Side
Compute each term:
\frac{1.5}{10} = 0.15, \quad \frac{1}{-15} = -\frac{1}{15} = -0.0667.
Hence,
0.15 - \left( -0.0667 \right) = 0.15 + 0.0667 \approx 0.2167.
Step 6: Simplify the Right Side
The right side is:
\frac{1.5 - 1}{R} = \frac{0.5}{R}.
Step 7: Equate and Solve for R
Set the two sides equal:
0.2167 = \frac{0.5}{R} \quad \Rightarrow \quad R = \frac{0.5}{0.2167} \approx 2.3077 \text{ m}.
The problem statement also gives R = \tfrac{x}{13} \text{ m}.
Hence,
\frac{x}{13} = 2.3077 \quad \Rightarrow \quad x = 2.3077 \times 13 = 30.
Step 8: Final Answer
Therefore, the value of x is:
x = 30.
