Question

A point source of light is distance d from the surface of a 6.0-cm-diameter glass sphere. For what value of d is there an image at the same distance d on the opposite side of the sphere?

Short answer

d is equal to 6.0 cm.

Step-by-step solution

Step 1. Given information is :Diameter of spherical glass lens, D = 6.0 cmIndex of refraction of glass, n = 1.50Distance between point source and lens, s = dDistance between image of the point source and lens, s' = d

We need to find out the value of d.

Step 2. Applying Lens Maker Formula.

$\frac{1}{f}=\left(n-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

As the lens is having two equal curved surfaces,

$$\begin{gathered} R_1=Rand{R}_2=-R \\ R=\frac{D}{2} \end{gathered}$$

Putting values,

$$\begin{gathered} \frac{1}{f}=\left(n-1\right)\left(\frac{1}{R}-\frac{1}{-R}\right) \\ \frac{1}{f}=\left(n-1\right)\left(\frac{2}{R}\right) \\ \frac{1}{f}=\left(1.50-1\right)\left(\frac{2}{R}\right) \\ \frac{1}{f}=\left(0.50\right)\left(\frac{2}{R}\right)=\left(\frac{1}{R}\right) \\ f=R \end{gathered}$$

We know that,

$$\begin{gathered} \frac{1}{s}+\frac{1}{s\text{'}}=\frac{1}{R} \\ Puttingvalues, \\ \frac{1}{d}+\frac{1}{d}=\frac{1}{R}=\frac{1}{{\displaystyle \frac{D}{2}}} \\ \frac{2}{d}=\frac{2}{D} \\ d=D \\ d=6.0cm \end{gathered}$$