Q 12-04-229JEE MainJEE Main 2018 (15 Apr, Shift 1)Medium
A Helmholtz coil has a pair of loops, each with $N$ turns and radius $R$. They are placed coaxially at distance $R$ and the same current $I$ flows through the loops in the same direction. The magnitude of the magnetic field at $P$, midway between the centres $A$ and $C$, is given by:
Answer: (D) $\dfrac{8N\mu_0I}{5^{3/2}R}$
On the axis of a coil of $N$ turns, at distance $x$ from its centre,
$$B = \frac{\mu_0NIR^2}{2(R^2 + x^2)^{3/2}}$$
$P$ is at $x = \dfrac R2$ from each coil:
$$B_1 = \frac{\mu_0NIR^2}{2\left(\frac54R^2\right)^{3/2}} = \frac{\mu_0NIR^2}{2}\cdot\frac{8}{5^{3/2}R^3} = \frac{4\mu_0NI}{5^{3/2}R}$$
The currents are in the same direction, so the fields add:
$$B = 2B_1 = \frac{8N\mu_0I}{5^{3/2}R}$$
Solution by Sreeraj P, M.Sc Physics