Question

The outer surface of a hollow sphere of aluminium of radius 50 cm is to be used as a mirror. What will be the focal length of this mirror and which type of spherical mirror will it provide?

Answer 1

Hint : A mirror is a piece of glass that reflects light. When focussed via the lens of the eye or a camera, light that bounces off a mirror reflects a picture of whatever is in front of it. Mirrors reflect light in an equal but opposite direction, reversing the direction of the picture.

Complete Step By Step Answer:
The focal length is the inverse of the optical power of a system and indicates how strongly it converges or diverges light. A system with a positive focus length converges light, whereas one with a negative focal length diverges light. A system with a shorter focal length bends the rays more sharply, bringing them closer to the focus or diverging them faster. A positive focal length indicates how far in front of the lens a point source must be located to form a collimated beam for the special case of a thin lens in air, or alternatively, a negative focal length indicates how far in front of the lens a point source must be located to form a collimated beam.
The magnitude of the focal length for a spherically curved mirror in air is equal to the mirror's radius of curvature divided by two. A concave mirror has a positive focal length, while a convex mirror has a negative focal length. A concave mirror has a negative radius of curvature in the sign convention used in optical design, thus
$f =\dfrac{R}{2}$
Here it is R = 50 cm
So, $f =\dfrac{R}{2} = \dfrac{50}{2} = 25$
It will provide a convex mirror.

Note :
In optical design, the radius of curvature (ROC) has a specific meaning and sign convention. The centre of curvature of a spherical lens or mirror surface is either parallel to or decentered from the system's local optical axis. The local optical axis passes via the vertex of the lens surface. The radius of curvature of the surface is the distance between the vertex and the centre of curvature.
The following is the sign convention for the optical radius of curvature:
The radius of curvature is positive if the vertex is to the left of the centre of curvature.
The radius of curvature is negative if the vertex is to the right of the centre of curvature.