Notes on Relation between R and f | Grade 11 > Physics > Reflection at Plane and Curved Surfaces | KULLABS.COM

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#### 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.

•  The distance between principal focus and pole of the mirror is known as focal length.
• The diameter of the boundary of the mirror is called its aperture.
• The center of the spherical reflecting surface is called the pole (P) of the mirror.
• The center of the sphere which is a part of the mirror is known as the center of curvature.
• The radius of the sphere which is the part of the sphere is known as the radius of curvature.
•  It is the line joining the center of curvature and pole of the mirror.
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