Question

A concave mirror has a radius of curvature 20 cm. Calculate the distance of an object from the mirror so as to form an image of magnification -2. Also, find the location of the image.

Hint:

For concave mirrors, the focal length is half of the radius of curvature. In this case, removing the silver coating would prevent the mirror from reflecting light, rendering it ineffective for image formation.

Answer 1

(a) Finding the Distance of the Object:
We know that the magnification \( m \) is related to the object distance \( u \) and the image distance \( v \) by the equation: \[ m = \frac{-v}{u} \] Also, the magnification is given as \( m = -2 \), so we have: \[ -2 = \frac{-v}{u} \] which gives us: \[ v = 2u \] The mirror equation relates the focal length \( f \), object distance \( u \), and image distance \( v \) as follows: \[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \] The focal length \( f \) of a concave mirror is related to the radius of curvature \( R \) by the equation: \[ f = \frac{R}{2} \] Given that the radius of curvature \( R = 20 \, \text{cm} \), we get: \[ f = \frac{20}{2} = 10 \, \text{cm} \] Substituting the value of \( f \) into the mirror equation: \[ \frac{1}{10} = \frac{1}{u} + \frac{1}{2u} \] Simplifying: \[ \frac{1}{10} = \frac{3}{2u} \] Solving for \( u \): \[ u = 15 \, \text{cm} \] Now, using the relation \( v = 2u \), we get: \[ v = 2 \times 15 = 30 \, \text{cm} \] Thus, the object distance is \( u = 15 \, \text{cm} \) and the image distance is \( v = 30 \, \text{cm} \). (b) Effect of Removing Silver Coating:
If the silver coating around the center of a concave mirror is removed, the mirror will no longer reflect light at the center. The silver coating is what enables the mirror to reflect light, and without it, the concave mirror will lose its reflective properties. Thus, the mirror will no longer form an image, as the reflective surface is essential for focusing light and forming an image. Therefore, removing the silver coating would mean that the mirror will not form any image. The reflection would not occur at the center and no image will be formed at all.