Question

A point source in air is placed in front of a convex spherical glass surface ($n = 1.5$ and $R = 10$ cm). Its real image is formed at the same distance from the surface as the source. The distance of the source from the surface is:

• A. 25 cm
• B. 50 cm
• C. 75 cm
• D. 100 cm

Hint:

Use n2/v - n1/u = (n2 - n1)/R with u = -d and v = +d.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Light goes from air (\(n_1 = 1\)) into glass (\(n_2 = 1.5\)) through a convex spherical surface.
We use the refraction formula for a single spherical surface and apply the sign convention.

Step 2: Key Formula:
\[ \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R} \]
The source is on the left, so the object distance is \(u = -d\).
The real image is on the right at the same distance from the surface, so \(v = +d\).
The surface is convex towards the source, so \(R = +10\) cm.

Step 3: Substitute:
\[ \frac{1.5}{d} - \frac{1}{-d} = \frac{1.5 - 1}{10} \]
\[ \frac{1.5}{d} + \frac{1}{d} = \frac{0.5}{10} \]
\[ \frac{2.5}{d} = 0.05 \]
\[ d = \frac{2.5}{0.05} = 50 \text{ cm} \]

Step 4: Check the options:
For \(d = 25\) cm, the left side is \(0.1\), twice the required \(0.05\). So (A) is wrong.
For \(d = 75\) cm, the left side is about \(0.033\). So (C) is wrong.
For \(d = 100\) cm, the left side is \(0.025\). So (D) is wrong.

Final Answer:
The source is 50 cm from the surface, which is option (B). \[ \boxed{50 \text{ cm}} \]