Q: The distance between the object and the real image formed by a convex lens is d. If the linear magnification is m, find the focal length of the lens in terms of d and m.

Sol: The convex lens formula for a real image is

$\large \frac{1}{v} + \frac{1}{u} = \frac{1}{f} $ …(i)

Multiplying by u we get

$\large \frac{u}{v} + 1 = \frac{u}{f} $

$\large \frac{1}{m} = \frac{u}{f} – 1 = \frac{u-f}{f} $

or, m(u – f) = f

$\large u = \frac{(1 + m)f}{m}$ ….(ii)

Multiplying (i) by v we get

$\large 1 + \frac{v}{u} = \frac{v}{f} $

$\large 1 + m = \frac{v}{f} $

or, v = f (1 + m) … (iii)

Now u + v = d. Using (ii) and (iii) we have

$\large d = \frac{(1 + m)f}{m} + f (1 + m) $

which gives , $\large f = \frac{m d}{(1+m)^2} $