Problem 40 - Master Resource Book in JEE Main Physics

Question

A convex lens \(A\) of focal length \(20 \mathrm{~cm}\) and a concave lens \(B\) of focal length \(56 \mathrm{~cm}\) are kept along the same axis with the distance \(d\) between them. If a parallel beam of light falling on \(A\) leaves \(B\) as a parallel beam, beam then distance, \(d\) in \(\mathrm{cm}\), will be (a) 25 (b) 36 (c) 30 (d) 50

Step-by-Step Solution

Answer

Distance \(d\) is 36 cm (option b).

Step 1: Understand the Problem

When a parallel beam of light passes through a lens system and exits as a parallel beam, the lenses act as a telescope in its afocal arrangement. This condition implies that the system's focal point aligns for collimation on both sides.

Step 2: Apply Lens Formula

The lens formula for a lens is given by \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \), where \( f \) is the focal length, \( v \) is the image distance, and \( u \) is the object distance. For a parallel beam entering the first lens, the beam focuses at its focal point, so \( u = \infty \) and \( v = f_A = 20 \text{ cm} \).

Step 3: Determine Image Distance After First Lens

Since \( v = f_A \), the image of the parallel beam after the first lens will be found at a distance equal to the focal length of the convex lens, 20 cm from lens \( A \). This forms a real image at \( v = 20 \text{ cm} \).

Step 4: Set Up Second Lens Condition

For the emerging beam to exit as a parallel beam after lens \( B \), the image formed by lens \( A \) must be at the focal point of lens \( B \). Therefore, the object distance for lens \( B \) must equal its focal length \( f_B = -56 \text{ cm} \).

Step 5: Calculate the Separation Distance

Since image formed by lens \( A \) (20 cm from \( A \)) acts as the object for lens \( B \), located 56 cm within lens \( B \), the separation distance \( d \) between the lenses is the difference in their object/image positions: \( d = 56 - 20 = 36 \text{ cm} \).

Step 6: Verify and Conclude

The required distance ensures the light exits lens \( B \) as a parallel beam, confirming the system works as an afocal telescope setup with \( d = 36 \text{ cm} \). Option (b) is correct.

Key Concepts

Lens Formula

The lens formula is a foundational equation in optics that helps us understand the relationship between the focal length, image distance, and object distance of a lens. It is usually expressed as:

• \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \)

Here, \( f \) is the focal length of the lens, \( v \) is the distance from the lens to the image, and \( u \) is the distance from the lens to the object. In the context of a parallel beam of light entering a lens, the object distance \( u \) is considered to be at infinity, which simplifies the equation since \( \frac{1}{u} \) becomes zero.
This formula allows us to determine where an image will form relative to the lens and helps in calculating the necessary distance in optical systems, such as telescopes or camera lenses.

Focal Length

The focal length of a lens is a critical property in determining how it will focus light. It is the distance from the lens to the point where it converges parallel light rays into a single focal point.

• Convex lenses have a positive focal length as they converge light.
• Concave lenses have a negative focal length as they diverge light.

In optical calculations, understanding the focal length helps in predicting the behavior of light as it passes through a lens. For a system to be afocal, the focal points of the lenses in use should align such that parallel light entering leaves as parallel light.

Afocal System

An afocal system is one where light entering and exiting the system is collimated, meaning it is parallel. This is often the setup in devices like telescopes where the objective is to view distant objects with improved clarity.
In the discussed optical setup, an afocal arrangement is achieved when the distance between the lenses is accurately set such that the system’s effective focal length becomes infinite. In practice, this means the separation between two lenses, for the system to maintain parallel light paths, is the difference in their focal lengths. The correct setup ensures that the image from one lens acts precisely as the focus for the next.

Convex Lens

Convex lenses are lenses that are thicker at the center than at the edges and are used to converge light rays. They are also referred to as converging lenses because they focus parallel light rays to a point.

• The focal length \( f_A \) of our convex lens example is 20 cm. This means light entering parallel through this lens will meet at a point 20 cm beyond the lens.
• Convex lenses are commonly used in applications requiring magnification or focusing light to a far point, such as in eyeglasses or cameras.

Convex lenses are fundamental in forming real and inverted images that can be utilized in afocal systems for producing parallel light.

Concave Lens

A concave lens is thinner at the center and thicker at the edges, and it works by diverging light rays that enter it. It is also known as a diverging lens.

• In the exercise, the concave lens has a focal length \(-56 \text{ cm}\). This negative value indicates its diverging property, spreading light instead of gathering it.
• Concave lenses are generally used to correct issues like short sightedness or to expand light waves in projectors and other optical devices.

Within an afocal system, a concave lens counterbalances the convergence by a convex lens to create a parallel exit beam.