Problem 66 - Arihant AIEEE Physics

Question

On the axis of a spherical mirror of focal length \(f\), a short linear object of length \(L\) lies on the axis at a distance \(u\) from the mirror. Its image has an axial length \(L^{\prime}\) equal to: (a) \(L\left[\frac{f}{(u-f)}\right]^{1 / 2}\) (b) \(L\left[\frac{(u+f)}{f}\right]^{1 / 2}\) (c) \(L\left[\frac{(u-f)}{f}\right]^{2}\) (d) \(L\left[\frac{f}{(u-f)}\right]^{2}\)

Step-by-Step Solution

Answer

The answer is option (d) \(L\left(\frac{f}{u-f}\right)^2\).

Step 1: Understand the Mirror Formula

To solve the problem, we start by using the mirror formula which relates object distance \(u\), image distance \(v\), and focal length \(f\). The formula is: \[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \] This formula will help us find the image distance \(v\).

Step 2: Rearrange the Mirror Formula

Rearrange the mirror formula to find \(v\): \[ \frac{1}{v} = \frac{1}{f} - \frac{1}{u} \] Solving for \(v\), we get: \[ v = \frac{uf}{u - f} \] This equation gives us the distance from the mirror to the image.

Step 3: Calculate Magnification

The linear magnification \(m\) of the mirror is given by the ratio of the image distance \(v\) to the object distance \(u\): \[ m = \frac{v}{u} = \frac{uf}{u(u - f)} = \frac{f}{u - f} \] Thus, magnification is expressed as \( \frac{f}{u-f} \).

Step 4: Determine the Image Axial Length

Using magnification, the axial length of the image \( L' \) is given by: \[ L' = m^2 \cdot L = \left( \frac{f}{u-f} \right)^2 \cdot L \] Therefore, \(L' = L\left(\frac{f}{u-f}\right)^2\).

Step 5: Identify the Correct Answer from Options

Comparing the expression for \(L'\) to the given options, we see that option (d) \(L\left(\frac{f}{u-f}\right)^2\) matches our calculation.

Key Concepts

Mirror Formula

In the study of spherical mirrors, a fundamental concept is the mirror formula. This equation establishes a relationship between three crucial variables: object distance (\( u \)), image distance (\( v \)), and focal length (\( f \)). The formula is written as:

\[\frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]This equation is essential in mirror-related problems as it allows us to calculate the unknown distances if at least two values (\( u, v, f \)) are known.

Let's break it down:

• Focal Length (\( f \)): Represents the distance from the mirror's focal point to its surface.
• Object Distance (\( u \)): The distance from the object to the mirror.
• Image Distance (\( v \)): The distance from the mirror to the image.

The mirror formula aids in determining the type and position of the image produced by the mirror, which is pivotal in optics. By understanding how to manipulate and rearrange this formula, you can solve numerous optical problems efficiently, whether it concerns concave or convex mirrors.

Linear Magnification

Linear magnification is another key concept in the context of spherical mirrors. It describes how much larger or smaller the mirror makes the image appear relative to the object's actual size. The formula for linear magnification (\( m \)) is:

\[m = \frac{v}{u} \]Substituting the image distance (\( v \)) derived from the mirror formula, it can be expressed as:

\[m = \frac{f}{u-f} \]Here’s what each component means:

• Image Distance (\( v \)): How far the image is from the mirror.
• Object Distance (\( u \)): Distance between the object and the mirror.

Magnification is a unit-less number, indicating the factor of size transformation:

• A magnification greater than 1 means the image is larger than the object.
• Less than 1 means the image is smaller.
• A negative magnification indicates an inverted image.

Understanding magnification helps predict how the image will appear and can influence the design and application of optical devices.

Focal Length

Focal length is a significant measurement in the analysis of mirrors and lenses. It essentially defines how strongly the mirror converges or diverges light rays:

• Concave Mirror: Positive focal length because it converges light.
• Convex Mirror: Negative focal length as it diverges light.

The focal length (\( f \)) is the distance between the mirror and the point where parallel rays of light converge. Here's why it's important:

• Determines how the image is formed relative to the mirror.
• Helps in calculating both image and object distances using the mirror formula.
• Directly influences the magnification and type (real or virtual) of the image.

Focal length can be affected by the mirror's shape and size. For practical purposes, knowing the focal length is essential as it impacts the overall performance of optical instruments.