The distance of two points on the axis of a magnet from its centre...

The magnetic field on the axis of a bar magnet of length $2l$, having a pole strength $m$, at a distance $d$ is

$B = \dfrac{\mu0}{4\pi}\dfrac{m}{\left[d - l\right]2} - \dfrac{\mu0}{4\pi}\dfrac{m}{\left[d + l\right]2}$

$B = \dfrac{{μ}_{0}m}{4\pi}\dfrac{\left[d + l\right]2 - \left[d - l\right]2}{\left[d^{2} - l^{2}\right]2} = \dfrac{{μ}_{0}mld}{\pi\left[d^{2} - l^{2}\right]2}$

$⇒ \dfrac{B1}{B2} = \dfrac{d1}{d2}\left[\dfrac{{d}_{2}^{2} - l^{2}}{{d}_{1}^{2} - l^{2}}\right]2$

$⇒ \dfrac{12.5}{1} = \dfrac{10}{20}\left[\dfrac{400 - l^{2}}{100 - l^{2}}\right]2$

$⇒ l = 5 cm$

Hence, length of magnet $= 2l = 10 cm$