Question

A plano-convex lens fits exactly in to a plano-concave lens. Their plane surfaces are parallel to each other. Lenses are made up of different materials of refractive indices \( n_1 \) and \( n_2 \) and \( R \) is the radius of curvature of the curved surface of lenses. Focal length of the combination is

• A. \( \frac{R}{n_1 - n_2} \)
• B. \( \frac{R}{n_1 + n_2} \)
• C. \( \frac{R(n_1 - n_2)}{n_1n_2} \)
• D. \( \frac{R}{n_1 - n_2} \)

Hint:

For lenses in contact, the total focal length is the sum of the individual focal lengths. The focal length of a plano-convex lens depends on the refractive index and the radius of curvature.

Answer 1

The Correct Option is A

Step 1: Formula for the focal length of plano-convex lens.
The focal length \( f_1 \) of a plano-convex lens (where one surface is flat and the other is curved) is given by the lens maker’s formula:
\[ \frac{1}{f_1} = (n_1 - 1) \left( \frac{1}{R} \right), \]
where:
- \( n_1 \) is the refractive index of the plano-convex lens material, - \( R \) is the radius of curvature of the curved surface.

Step 2: Formula for the focal length of plano-concave lens.
For a plano-concave lens, the formula for focal length \( f_2 \) is:
\[ \frac{1}{f_2} = (n_2 - 1) \left( \frac{-1}{R} \right). \]
Here, the minus sign appears because the lens is concave.

Step 3: Total focal length of the combination.
Since the plano-convex and plano-concave lenses are in combination, we can use the formula for the total focal length of two thin lenses in contact:
\[ \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2}. \]
Substitute the values of \( f_1 \) and \( f_2 \) into this equation:
\[ \frac{1}{f} = (n_1 - 1) \left( \frac{1}{R} \right) + (n_2 - 1) \left( \frac{-1}{R} \right). \]

Step 4: Simplify the equation.
Simplifying the above equation:
\[ \frac{1}{f} = \frac{(n_1 - 1) - (n_2 - 1)}{R} = \frac{n_1 - n_2}{R}. \]
Thus, the focal length \( f \) of the combination is:
\[ f = \frac{R}{n_1 - n_2}. \]
Final Answer:
Thus, the focal length of the combination is:
\[ \boxed{\frac{R}{n_1 - n_2}}. \]