Relation between R and f

Terms Used in Spherical Mirrors

Pole: The center of the spherical reflecting surface is called the pole (P) of the mirror.

Center of curvature: The center of the sphere which is a part of the mirror is known as the center of curvature.

The radius of curvature: The radius of the sphere which is the part of the sphere is known as the radius of curvature.

Principal axis: It is the line joining the center of curvature and pole of the mirror.

Principal focus: When two light rays parallel to the principal axis and coming from infinity strike the reflecting surface, they meet or appear to meet at a point. The point where they meet or appear to meet is called principal focus.

Focal length: The distance between principal focus and pole of the mirror is known as focal length.

Aperture: The diameter of the boundary of the mirror is called its aperture.

Image of a Point Object Formed by the Mirror: Concave Mirror

Consider a point object O placed on the principle axis of a concave mirror. The mirror forms an image I in the same side of the object is beyond focus. So the image formed by the mirror is real as shown in the figure. But when the object O lies within the principle focus and pole of the mirror image will be formed on the next side of the object at I which virtual as shown in the figure.

If a point object O is placed on the principle axis of a convex mirror its image will be formed always in the side at point I. so the image by the convex mirror is always virtual as shown in the figure.

Relation between R and f

Concave Mirror

Consider a concave mirror of the small aperture. When a ray of light OA parallel to the principle axis is incident at point A on the mirror, it will be reflected along AB passing through the focus F as shown in the figure. Join AC is normal at A.

From the laws of reflection of light,

\begin{align*} \angle OAC = \angle FAC \dots (i) \\ \text {and} \: \angle OAC = \angle ACF \dots (ii) \\ \text {Hence} \: \Delta \: AFC \: \text {is an isosceles triangle and in such triangle,} \\ AF = FC \dots (iii) \\ \text {If the aperture of the mirror is small, then points A and P are very close to each other, and we will have } \: AF \approx PF. \\ \text {Thus equation} \: (ii) \text {becomes} \\ PF = FC = PC – PF \\ \text {or} \: 2PF = PC \\ \text {or} \: 2f &= R \\ \text {or} \: f &= \frac {R}{2} \\ \end{align*}

where f is focal length and R is radius of curvature of the mirror. Thus the focal length of a concave mirror is one half of its radius of curvature.

Convex Mirror

Consider a convex of focal length f and small aperture. A ray of light OA parallel to the principle axis is incident at point A on the mirror and it passes along AB after reflection as shown in the figure. The virtual image will be formed at F in the next side of the object. Join CA and produce outward. Here C is the centre of curvature and P is the pole of the mirror. We have,

\begin{align*} \angle OAN = \angle NAB \: (i=r) \dots (i) \\ \angle OAN = \angle ACF \: (\text {corresponding angles} ) \dots (ii) \\ \angle CAF = \angle NAB \: (\text {vertically opposite angles}) \dots (iii) \\ \text {From equation } \: (iii) \: \text {and equation} \: (iv), \text {we have} \\ \angle CAF = \angle ACF \\ \text {Hence} \: \Delta ACF \: \text {is an isosceles triangle. So} \\ AF = FC \dots (v) \\ \text {If the aperture of the mirror is small, then points A and P will lie very close to each other.} \\ \text {So,} AF \approx PF \: \text {and equation} \: (v) \: \text {becomes} \\ PF = FC = PC – PF \\ \text {or,} \: 2PF = PC \\ \text {or,} \: 2f = R \\ \text {or,} \: f = \frac {R}{2} \\ \end{align*}

where f is focal and R is radius of curvature of the mirror. Thus the focal length of convex mirror is one half of its radius of curvature.