Question

An object is placed beyond the centre of curvature C of the given concave mirror. If the distance of the object is \(d_1\) from C and the distance of the image formed is \(d_2\) from C, the radius of curvature of this mirror is :

• A. \(\frac{d_1d_2}{d_1 - d_2}\)
• B. \(\frac{d_1d_2}{d_1 + d_2}\)
• C. \(\frac{2d_1d_2}{d_1 - d_2}\)
• D. \(\frac{2d_1d_2}{d_1 + d_2}\)

Hint:

This problem can also be solved using Newton's formula for spherical mirrors, which states \(x_1 x_2 = f^2\), where \(x_1\) and \(x_2\) are the distances of the object and image from the focal point. Here, distances are from C, so deriving from the basic mirror formula is the safest approach. Always be careful with sign conventions.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given the distances of an object and its image from the center of curvature (C) of a concave mirror. We need to find the radius of curvature (R) in terms of these distances.
Step 2: Key Formula or Approach:
We will use the mirror formula, which relates object distance (\(u\)), image distance (\(v\)), and focal length (\(f\)). All distances are measured from the pole (P) of the mirror. The sign convention is crucial. \[ \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \] We also know that for a spherical mirror, \(R = 2f\).
Step 3: Detailed Explanation:
Let's define the distances from the pole (P) using the given information. Let R be the magnitude of the radius of curvature.
- The center of curvature C is at a distance R from the pole. By sign convention, its coordinate is \(-R\). - The focal point F is at a distance \(f = R/2\) from the pole. Its coordinate is \(-R/2\). Object Position (u):
The object is placed at a distance \(d_1\) from C, beyond C. So, the distance of the object from the pole is \(u = - (R + d_1)\).
Image Position (v):
For a concave mirror, when the object is beyond C, the real image is formed between C and F. The distance of the image from C is \(d_2\). So, the distance of the image from the pole is \(v = - (R - d_2)\).
Now, substitute \(u\) and \(v\) into the mirror formula \(\frac{1}{v} + \frac{1}{u} = \frac{1}{f}\). And we use \(f = -R/2\). \[ \frac{1}{-(R - d_2)} + \frac{1}{-(R + d_1)} = \frac{1}{-R/2} \] \[ \frac{1}{R - d_2} + \frac{1}{R + d_1} = \frac{2}{R} \] Now, we solve for R. Find a common denominator for the left side: \[ \frac{(R + d_1) + (R - d_2)}{(R - d_2)(R + d_1)} = \frac{2}{R} \] \[ \frac{2R + d_1 - d_2}{R^2 + Rd_1 - Rd_2 - d_1d_2} = \frac{2}{R} \] Cross-multiply: \[ R(2R + d_1 - d_2) = 2(R^2 + Rd_1 - Rd_2 - d_1d_2) \] \[ 2R^2 + Rd_1 - Rd_2 = 2R^2 + 2Rd_1 - 2Rd_2 - 2d_1d_2 \] Cancel \(2R^2\) from both sides: \[ Rd_1 - Rd_2 = 2Rd_1 - 2Rd_2 - 2d_1d_2 \] Rearrange the terms to isolate R: \[ 2d_1d_2 = 2Rd_1 - Rd_1 - 2Rd_2 + Rd_2 \] \[ 2d_1d_2 = Rd_1 - Rd_2 \] \[ 2d_1d_2 = R(d_1 - d_2) \] \[ R = \frac{2d_1d_2}{d_1 - d_2} \] Step 4: Final Answer:
The radius of curvature of the mirror is \(R = \frac{2d_1d_2}{d_1 - d_2}\).