Problem 47 - Master Resource Book in JEE Main Physics

Question

A plano-convex lens fits exactly into a plano-concave lens. Their plane surfaces are parallel to each other. If the lenses are made of different materials of refractive indices \(\mu_{1}\) and \(\mu_{2}\) and \(R\) is the radius of curvature of the curved surface of the lenses, then focal length of the combination is (a) \(\frac{R}{2\left(\mu_{1}+\mu_{1}\right)}\) (b) \(\frac{R}{2\left(u_{1}-\mu_{1}\right)}\) (c) \(\frac{R}{\left(\mu_{1}-\mu_{2}\right)}\) (d) \(\frac{2 R}{\left(\mu_{1}+\mu_{2}\right)}\)

Step-by-Step Solution

Answer

The focal length of the combination is \( \frac{R}{\mu_1 - \mu_2} \), matching option (c).

Step 1: Understanding Lens Maker's Formula

For a lens with a refractive index \( \mu \), the lens maker's equation is given as: \( \frac{1}{f} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \). Here, \( f \) is the focal length of the lens, \( R_1 \) is the radius of curvature of the first surface, and \( R_2 \) is the radius of curvature of the second surface.

Step 2: Calculating Focal Length of Plano-Convex Lens

For a plano-convex lens, the plane surface has an infinite radius of curvature \((R_2 = \infty)\) and the curved surface has a radius \( R_1 = R \). The refractive index is \( \mu_1 \). So, the focal length \( f_1 \) is: \[ \frac{1}{f_1} = (\mu_1 - 1) \left( \frac{1}{R} - \frac{1}{\infty} \right) = \frac{\mu_1 - 1}{R} \]},

Step 3: Calculating Focal Length of Plano-Concave Lens

For a plano-concave lens, the plane surface again has an infinite radius of curvature \((R_2 = \infty)\), and the curved surface has a radius \( R_1 = -R \) (as it is concave). The refractive index is \( \mu_2 \). Thus, the focal length \( f_2 \) is: \[ \frac{1}{f_2} = (\mu_2 - 1) \left( \frac{1}{-R} - \frac{1}{\infty} \right) = \frac{1 - \mu_2}{R} \]},

Step 4: Applying Lens Combination Formula

The effective focal length \( F \) for a combination of two lenses in contact is given by: \[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \] Substituting the expressions for \( \frac{1}{f_1} \) and \( \frac{1}{f_2} \) we get: \[ \frac{1}{F} = \frac{\mu_1 - 1}{R} + \frac{1 - \mu_2}{R} = \frac{\mu_1 - 1 + 1 - \mu_2}{R} \] Simplifying, we have: \[ \frac{1}{F} = \frac{\mu_1 - \mu_2}{R} \]},

Step 5: Solving for Focal Length of the Combination

Rearranging the equation from Step 4 gives: \[ F = \frac{R}{\mu_1 - \mu_2} \] This matches option (c) from the given multiple choices.

Key Concepts

Plano-Convex Lens

A plano-convex lens is a type of lens that consists of one flat surface (plane) and one curved surface (convex). It is designed to converge or focus light rays that pass through it. The interesting thing about the plano-convex lens is that the flat side lets light rays pass without any deviation, while the convex side bends or refracts these rays toward a focal point.
The focal length of this lens, which is the distance from the lens to the point where light rays converge, can be calculated using the Lens Maker's Formula. If the refractive index of the material is denoted as \( \mu_1 \) and the radius of curvature of the curved surface is \( R \), the formula simplifies for a plano surface as:

• \( R_2 = \infty \)
• \( 1/f = (\mu_1 - 1)(1/R) \)

Thus, it produces a focus for parallel incoming light rays that have passed through its thicker central part.

Plano-Concave Lens

The plano-concave lens is somewhat the opposite of the plano-convex lens. It features one flat surface and one inwardly curved surface (concave). This type of lens is used to diverge light rays, meaning that it spreads out light that passes through it.
When considering light passing through from a dense medium, the concave surface causes the light rays to scatter as if they originated from a single point called the virtual focus. In the case of a plano-concave lens, if the refractive index is \( \mu_2 \) and the radius of curvature is also \( R \), the lens maker’s formula simplifies as follows:

• \( R_2 = \infty \)
• \( 1/f = (\mu_2 - 1)(-1/R) \)

This would make \( f \), the focal length, negative since it measures the lens's capability to diverge rays instead of converging them.

Effective Focal Length

Determining the effective focal length of a lens system, which involves more than one lens, is crucial in optical design. In our scenario with one plano-convex and one plano-concave lens in contact, the concept becomes particularly relevant.
To find the effective focal length \( F \) of the combined lens system, we use the formula:

• \( 1/F = 1/f_1 + 1/f_2 \)

For our combination, \( f_1 \) and \( f_2 \) represent the focal lengths of the plano-convex and plano-concave lenses, respectively. Substituting our previous results:

• \( 1/f_1 = (\mu_1 -1)/R \)
• \( 1/f_2 = (1 - \mu_2)/R \)

Summing these gives:

• \( 1/F = (\mu_1 - \mu_2)/R \)

Thus, the combination effectively results in an overall focal length \( F \) described by \( R/(\mu_1 - \mu_2) \), signifying how the lenses together handle converging and diverging rays.