Question

The image of an object placed in air formed by convex refracting surface is at a distance of $10\text{m}$ behind the surface. The image is real and is at $\frac{2}{3^{rd}}$ of the distance of the object from the surface. The wavelength of light inside the surface is $\frac{3}{2}$ times the wavelength in air, The radius of the curved surface is $\frac{x}{13}\text{m}$. The value of $x$ is .

Answer 1

Since wavelength of light inside the surface is $\frac{3}{2}$ times the wavelength in air. Refractive index, $n_2=\frac{3}{2}\times n_1=\frac{3}{2}$

Image is real and it is formed at, $v=\frac{2}{3}\times 15=10\text{m}$

We know that, $$\frac{n_2}{v}-\frac{n_1}{u}=\left(\frac{n_2-n_1}{R}\right)$$$\frac{3/2}{10}-\frac{1}{(-15)}=\frac{(\frac{3}{2}-1)}{R}$

$\frac{1}{2R}=\frac{3}{20}+\frac{1}{15}=\frac{65}{300}$

$R=\frac{150}{65}=\frac{30}{13}\text{m}$
Thus, the value of $x=30$