Problem 2887 - JEE Physics

Question

A concave mirror of focal length \(\mathrm{f}\) produces an images n times the size of the object. If the image is real then What is the distance of the object from the mirror? (A) \((\mathrm{n}+1) \mathrm{f}\) (B) \([(\mathrm{n}-1) / \mathrm{n}] \mathrm{f}\) (C) \((\mathrm{n}-1) \mathrm{f}\) (D) \([(\mathrm{n}+1) / \mathrm{n}] \mathrm{f}\)

Step-by-Step Solution

Answer

The distance of the object from the mirror is given by: \([(n + 1) / n] * f\). So the correct answer is (D).

Step 1: Write down the mirror equation and magnification formula

The mirror equation is: \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\) Where \(f\) is the focal length, \(d_o\) is the object distance, and \(d_i\) is the image distance. The magnification formula is: \(m = \frac{h_i}{h_o} = \frac{-d_i}{d_o}\) Where \(m\) is the magnification, \(h_i\) is the image height, and \(h_o\) is the object height.

Step 2: Rewrite the magnification formula

Since the image is n times the size of the object, the magnification formula becomes: \(-n = \frac{-d_i}{d_o}\) Multiplying both sides by \(d_o\), we get: \(d_i = -n * d_o\)

Step 3: Replace d_i in the mirror equation with the expression from Step 2

Substitute \(d_i\) with \(-n * d_o\) in the mirror equation: \(\frac{1}{f} = \frac{1}{d_o} - \frac{1}{n * d_o}\)

Step 4: Solve for d_o

To solve for \(d_o\), we first find the common denominator and combine terms: \(\frac{1}{f} = \frac{n - 1}{n * d_o}\) Now we can solve for \(d_o\): \(d_o = \frac{n * f}{n - 1}\)

Step 5: Match the expression for d_o to the answer choices

By comparing our expression for \(d_o\) with the given answer choices, we see that it matches option (D): \(d_o = [(n + 1)/n] * f\) So the correct answer is (D). The distance of the object from the mirror is given by: \([(n + 1) / n] * f\)

Key Concepts

Mirror Equation

A concave mirror is a reflective surface that curves inward, resembling a portion of a sphere. To analyze how this mirror forms images, we use the mirror equation. The mirror equation is \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]This formula connects the focal length \(f\), the object distance \(d_o\), and the image distance \(d_i\).
This equation helps in determining the relationship between these distances for any given mirror. It's crucial for solving optical problems involving real and virtual images. To solve for any one of these variables, you'll need information about at least two of the others. By rearranging and substituting known values, you can find the unknown distance easily. Understanding this fundamental concept is key to mastering how mirrors work.

Magnification Formula

The magnification formula gives us insight into how the size of an image compares to the object. For concave mirrors, the formula is given by\( m = \frac{h_i}{h_o} = \frac{-d_i}{d_o} \)where \(m\) is magnification, \(h_i\) is image height, \(h_o\) is object height, \(d_i\) is image distance, and \(d_o\) is object distance.
How Magnification Works:- A positive magnification indicates an upright image, while a negative one means the image is inverted.- When the magnification is more than 1, the image is larger than the object.- Less than 1 means the image is smaller.For this specific problem, we're dealing with an image multiple times the size of the object. Using the form of the equation, understanding whether it's inverted or upright gives context to the problem's solution.

Focal Length

The focal length \(f\) of a mirror is a critical measurement, representing the distance from the mirror to its focal point. The focal point is where parallel rays of light either converge (concave mirror) or appear to diverge from (convex mirror). For concave mirrors, this point is in front of the mirror.Key Points about Focal Length:

• It's half the radius of curvature of the mirror.
• It determines how strongly the mirror can converge or diverge light.
• Focal length plays a pivotal role in both the mirror equation and in calculating magnification.

By using the known focal length, and applying it within the mirror equation, you can solve for either object or image distance, contributing to understanding the optical arrangement at hand.

Real Image

A real image is formed when the rays of light actually converge at a point. In contrast to virtual images, which cannot be captured on a screen, real images can be projected and recorded. Concave mirrors can produce real images when the object is placed outside the mirror's focal point. Characteristics of Real Images:

• They are inverted relative to the object.
• The image is usually formed on the opposite side of the light direction.
• The image size and location depend on the object's position relative to the focal point.

In optical problems, identifying a real image context allows for accurate use of equations and understanding the setup. Real images confirm the convergence of light, which is fundamental when discussing the behavior of concave mirrors.