Parabolic Mirror and Magnification

Parabolic Mirror

Parabolic Mirror a mirror which has reflecting surface in the shape parabola is called is called parabolic mirror. As the mirror can converge accurately a wide parallel beam of light at the focus, it is used in search light. The intensity of the reflected beam is thus undiminished at a long distance. These mirrors have no spherical aberration.

Wide Beam in Concave Mirror: The Caustic Curve

When a narrow beam of light is incident on a concave mirror parallel and close to principle axis, all reflected rays converge at a single point F called the principle focus. Thus, a concave mirror is a real focus.

When a wide beam of light parallel to the principle axis is incident on a concave mirror of wide aperture, the reflected rays do not pass through a single point. They cross at different points on the axis as shown in the figure. The rays near to axis, called the paraxial rays, converge at a point farther from the pole of the mirror while, the rays farther from the axis, called the marginal rays, converge at a point near the pole. The reflected ray appear to touch a surface known as caustic curve.

In the same way, when a wide beam of light on a convex mirror is incident parallel to the principle axis, the reflected rays do not appear to diverge from a single point. So, when a point source of light is placed at the focus of a concave mirror the reflected rays will not be parallel and have less intensity as they lose energy due to crossing. But, if the point source is placed at the focus of a parabolic mirror the reflected rays will be parallel and have much intensity in a long distance. So, the parabolic mirrors are used in search light and head lights of cars, buses etc.

Magnification

It is defined as the ratio of image size formed by the spherical mirror to the object size. It is denoted by m.

$$ m = \frac {\text {size of image}}{\text {size of image}} $$

1. Magnification in Case of Concave Mirror

Producing Real Image

Let us consider a concave mirror and an object OB is placed on the principle axis as shown in the figure. When the rays of light are incident on the mirror at point A and P, they are reflected and meet at point B’ in front of the mirror. So real image O’B’ is formed by the mirror.

\begin{align*} \text {Here,} \: \Delta s \text {BPO and B’PO’ are similar. So for similar triangle we can write} \\ \frac {O’B’}{OB} &= \frac {O’P}{OP} = \frac vu \\ \text { From definition,} \\ m &= \frac {O’B’}{OB} \\ \text {or,} &= \frac {v}{u} \\ \end{align*}

2. Magnification in Case of Convex Mirror

Producing virtual Images

Let us consider a convex mirror and an object OB is placed on the principle axis. When the rays of light are incident on the mirror at points A and P as shown in the figure, they are reflected and appeared to meet at point B’ on the next side of the object. So the virtual image O’B’ is formed by the mirror.

\begin{align*} \text {Here,} \: \Delta s \text {OBP and PO’B’ are similar. So for similar triangle we can write} \\ \frac {O’B’}{OB} &= \frac {B’P}{BP} = \frac {-v}{u} \\ \text { Since image is virtual. From definition,} \\ m &= \frac {O’B’}{OB} \\ \text {or,} &= \frac {-v}{u} \\ \text {Expression for magnification in other form} \\ \frac 1f &= \frac 1u + \frac 1v \\ \text { Multiplying both sides by u, in above equation we get} \\ \frac uf &= 1 + \frac uv = 1 + \frac {1}{v/u} \\ \text {or,} \: \frac uf &= 1 + \frac 1m \\ \frac 1m &= \frac uf – 1 \\ \therefore m &= \frac {f}{u-f} \\ \end{align*}

\begin{align*} \text {We have} \\ \frac 1f &= \frac 1u + \frac 1v \\ \text { Multiplying both sides by v, in above equation we get} \\ \frac vf &= \frac vu + 1= m +1 \\ \text {or,} \: m &= \frac vf – 1 \\ &= \frac {v – f}{f} \\ \therefore m &= \frac {v – f}{f} \\ \end{align*}