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Question
Obtain the mirror formula and write the expression for the linear magnification.
Solution
Mirror Formula
`
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
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