Question

Obtain the mirror formula and write the expression for the linear magnification.

Answer 1

Mirror Formula

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The above figure shows the ray diagram for image formation by a concave mirror.

In figure triangle A'B'F and ENF are similar.

`∴ (A'B')/(NE) = (A'F)/(NF)`

As the aperture of the concave mirror is small, the points N and P lie very close to each other.

∴ NF ≈ PF and NE = AB

`(A'B')/(AB) = (A'F)/(PF)`

Since all the distances are measured from the pole of the concave mirror, we have

A'F  = PA' - PF

`:. (A'B')/(AB) = (PA' - PF)/(PF)`  ...(i)

Also, triangles ABP and A'B'P are similar

`:. (A'B')/(AB) = (PA')/(PA)`  ....(ii)

Applying the new Cartesian sign conventions, we have

PA = -u (Q distance of the object is measured against incident ray)

PA' = -v (Q distance of image is measured against incident ray)

PF = -f(Q focal length of the concave mirror is measured against incident ray)

Substituting these values in equation (iii),

We have

`(-v-(-f))/(-f) = (-v)/(-u)`

`(v-f)/f = v/u`

`v/f - 1 = v/u` or `1/f - 1/v = 1/u`

`1/u + 1/v = 1/f`

Linear magnification

The ratio of the size of the image formed by a spherical mirror to the size of the object is called the linear magnification produced by the spherical mirror.

It is denoted by m.

`m = I/O = (-v)/u`

Where,

I → Size of the image

O → Size of the object