A bar magnet has total length $2l = 20$ units and the field point $P$ is at a distance $d = 10$ units from the centre of the magnet, as shown in the figure. If the relative uncertainty of length measurement is $1\%$, then the uncertainty of the magnetic field at point $P$ is
Answer: (A) $4\%$
$P$ is on the perpendicular bisector (equatorial line) of the magnet. With pole strength $q_m$ and $m = q_m(2l)$:
$$B = \frac{\mu_0}{4\pi}\frac{q_m(2l)}{(d^2 + l^2)^{3/2}}$$
Taking logarithms and the maximum error, with $\dfrac{\Delta l}{l} = \dfrac{\Delta d}{d} = 1\%$:
$$\frac{\Delta B}{B} = \frac{\Delta l}{l} + \frac{3}{2}\cdot\frac{2d\,\Delta d + 2l\,\Delta l}{d^2 + l^2} = 1\% + 3\cdot\frac{d^2(1\%) + l^2(1\%)}{d^2 + l^2} = 1\% + 3\% = 4\%$$
Solution by Sreeraj P, M.Sc Physics