Problem 83 - Arihant AIEEE Physics

Question

If an equiconvex lens of focal length \(20 \mathrm{~cm}\) is cut into two equal parts by a vertical plane, the focal length of each part will become: (a) \(40 \mathrm{~cm}\) (b) \(10 \mathrm{~cm}\) (c) \(20 \mathrm{~cm}\) (d) \(5 \mathrm{~cm}\)

Step-by-Step Solution

Answer

Each half of the lens will have a focal length of 40 cm, answer (a).

Step 1: Understanding the Original Lens

An equiconvex lens has equal radii of curvature for both its surfaces. Given that the focal length of the original equiconvex lens is 20 cm, we need to understand how cutting the lens affects this property.

Step 2: Applying Lens Formula to Each Half

When the lens is cut into two equal halves vertically (perpendicular to the principal axis), each half becomes a plano-convex lens. The focal length of a lens is given by the lens maker's formula: \( \frac{1}{f} = \frac{(n - 1)(1/R_1 - 1/R_2)}{1} \), where \(f\) is the focal length, \(n\) is the refractive index, and \(R_1\) and \(R_2\) are the radii of curvature. For a plano-convex lens, one surface becomes flat (\(R_2 \to \infty\)) and the formula adjusts accordingly.

Step 3: Calculating the Focal Length of Each Half

Considering the lens maker's formula for each half of the original lens, since only one face has curvature now, the formula simplifies. Each plano-convex lens now has a focal length \(f'\), which is double that of the original lens, because \(f' = 2f\) for each component when cut vertically, as the curvature effect is reduced in each component.

Step 4: Conclusion

Hence, each part will have a focal length that is double the original lens, which is 40 cm. Therefore, the correct answer from the options given is (a) 40 cm.

Key Concepts

Focal Length

The focal length is a crucial concept in optics. It defines how strongly a lens converges or diverges light. In simple terms, the focal length is the distance from the lens to the focus point, where parallel rays of light meet after passing through the lens.
A shorter focal length means a lens is more powerful, as it bends light more sharply, bringing it to a focus at a shorter distance. Conversely, a longer focal length indicates a less powerful lens, with light focusing farther away.
An equiconvex lens, like the one in our exercise, has equal curvature on both sides, which balances the light convergence.

• Positive focal length: Indicates converging lenses, which bring light rays together.
• Negative focal length: Used for diverging lenses, spreading light rays apart.

Understanding focal length helps in designing lenses for various applications, from eyeglasses to cameras.

Lens Formula

The lens formula is a fundamental equation in optics used to relate the focal length \(f\), the object distance \(o\) (distance from the lens to the object), and the image distance \(i\) (distance from the lens to the image). The formula is given by:
\[ \frac{1}{f} = \frac{1}{o} + \frac{1}{i} \]
This formula helps predict how a lens will form an image, considering the object's location in relation to the lens.
In our exercise, the lens formula is key to understanding how cutting the lens changes its characteristics. After the lens is cut, each half behaves like a plano-convex lens. Here, the lens maker's formula simplifies to consider only one curved surface, increasing the focal length.

• Origin: Developed by combining geometrical optics concepts and refraction principles.
• Usage: Determines image characteristics such as position, magnification, and orientation.

This concise tool is invaluable in optical design and correction tasks.

Equiconvex Lens

An equiconvex lens is a specific type of lens where both surfaces have the same radius of curvature. This symmetry means that the lens focuses light efficiently and equally from both directions, making it useful in devices that require precise image formation.
When you cut an equiconvex lens vertically, its symmetry is altered. The two halves become plano-convex lenses, with one side flat and the other retaining the original curvature. This change affects how each half lens bends light.
For an equiconvex lens in our problem, the focal point changes because the effective radius of curvature is modified. Each half-lens thus acts with less curvature on one side, leading to a doubled focal length as calculated using the lens maker's formula.

• Advantages: Balanced focal points for minimal image distortion.
• Commonly used in: Microscopes, cameras, and other precision optical instruments.

This design ensures that light converges sharply, crucial for clear and focused imaging.