Question

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 magnetic field at $P$, midway between the centres $A$ and $C$, is given by [Refer to figure given below] :

• A. $\frac{8 \, N \, \mu_0 I}{5^{1/2} R}$
• B. $\frac{8 \, N \, \mu_0 I}{5^{3/2} R}$
• C. $\frac{4\, N \, \mu_0 I}{5^{1/2} R}$
• D. $\frac{4 \, N \, \mu_0 I}{5^{3/2} R}$

Answer 1

Answer: (B)

Magnitude of the magnetic field due to one coil is given by
$B=\frac{\mu_{0} i R^{2}}{2\left(R^{2}+(R / 2)^{2}\right)^{3 / 2}}$
$=\frac{\mu_{0} i R^{2}}{2\left(\frac{4 R^{2}+R^{2}}{2^{2}}\right)^{3 / 2}}=\frac{8 \mu_{0} i R^{2}}{2\left(5 R^{2}\right)^{3 / 2}}$
$\Rightarrow B=\frac{4 \mu_{0} i R^{2}}{5^{3 / 2} R^{3}}=\frac{4 \mu_{0} i}{5^{3 / 2} R}$
Now net magnetic field $B_{\text {net }}=2 B=\frac{8 \mu_{0} i}{5^{3 / 2} R}$
Now $i=N I$, therefore,
$B_{\text {net }}=\frac{8 \mu_{0} N I}{5^{3 / 2} R}$