Problem 82 - Arihant AIEEE Physics

Question

If an equiconvex lens of focal length \(30 \mathrm{~cm}\) is cut into two equal parts by a horizontal plane, then: (a) the light transmitting area of each part becomes half of the initial (b) the intensity will reduce to half (c) the aperture becomes \(\frac{1}{\sqrt{2}}\) times of tis initial value (d) all the above

Step-by-Step Solution

Answer

(d) all the above

Step 1: Understanding the Problem

An equiconvex lens is bisected horizontally, affecting its light-transmitting properties, focal length, and intensity. We need to evaluate how these parameters change and if the options (a), (b), (c), or (d) given in the problem are correct.

Step 2: Consider the Area of the Lens

Cutting the lens horizontally into two equal parts indeed halves the light-transmitting area for each piece. Initial statement (a) is accurate.

Step 3: Analyze the Intensity of Light

Since intensity depends on the amount of light passing through an area, cutting the lens in two effectively reduces the intensity by half, as only half of the lens is transmitting light. So, statement (b) is correct.

Step 4: Examine the Aperture Size Change

The aperture is effectively the diameter of the open area through which light passes. When the lens is cut into two horizontal parts, the curvature of each part remains the same, but the vertical span decreases. This can result in a change to the aperture equivalent to a \(\frac{1}{\sqrt{2}}\) factor, corroborating statement (c).

Step 5: Evaluate the Options Given

Since all three effects mentioned in statements (a), (b), and (c) are actually true for an equiconvex lens cut into two pieces, option (d), which states 'all the above', is also correct.

Key Concepts

Equiconvex Lens

An equiconvex lens is a type of lens where both surfaces are convex, meaning they bulge outwards. This symmetry gives them the name "equi," implying equal convexity on both sides. These lenses are often used in applications where reducing spherical aberrations is important, such as cameras and telescopes.
When light passes through an equiconvex lens, it converges due to the shape of the lens. This means that light rays that initially travel parallel to the optical axis will meet at a point called the focus. The distance from the lens to this focus point is called the focal length, a key characteristic that we'll discuss further.
Equiconvex lenses are preferred in optical systems because they can minimize optical distortions when used with light coming from or going into symmetrical setups. They are made of materials that affect how light bends. This property is known as refractivity.

Focal Length

The focal length of a lens is a central concept in optics. It defines how strongly the lens converges or diverges light. In an equiconvex lens, the focal length is determined by the curvature of its surfaces and the material's refractive index.
Mathematically, the focal length \( f \) of an equiconvex lens can be calculated using the lensmaker's equation:\[\frac{1}{f} = (n - 1) \left(\frac{1}{R_1} + \frac{1}{R_2}\right)\]where \( n \) is the refractive index of the lens material, and \( R_1 \) and \( R_2 \) are the radii of curvature of the lens surfaces.
When you cut an equiconvex lens horizontally, you're not changing its focal length directly because both halves retain the same curvature. However, this affects how you use the lens in practice, especially influencing light intensity and aperture, which we'll delve into in the sections below.

Light Intensity

Light intensity relates to the amount of light passing through a lens and is usually measured in terms of power per area, such as watts per square centimeter. It's important because it determines how bright or dim an image will appear after light passes through the lens.
When an equiconvex lens is cut into two parts, the area through which light can pass is reduced. Since intensity is proportional to this area, cutting the lens reduces the light intensity to half. Therefore, each half of the lens lets through only half the amount of light compared to the whole lens.
In practical terms, reducing the intensity can affect image brightness. For optical applications that depend on precise lighting conditions, adjustments may need to be made to compensate for this drop in intensity.

Lens Aperture

The aperture of a lens refers to the diameter of the lens's opening that allows light to pass through. It is crucial in controlling the amount of light entering the optical system and consequently affects depth of field and brightness.
When cutting the equiconvex lens horizontally, the change in physical dimensions affects the aperture size. While the overall area of the aperture is reduced by half, the effective aperture becomes \( \frac{1}{\sqrt{2}} \) times its original size. This factor comes from the geometric realization of splitting the plane of incidence.
Understanding aperture changes is essential, as it influences not just light quantity, but also attributes like optical resolution and focus depth in imaging systems. This modification to the aperture size may lead to adjustments in settings or configurations in practical optics, such as photography or cinematography.