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. $\frac{8N{\mu }_{0}I}{5^{1/2}R}$
• B. $\frac{8N{\mu }_{0}I}{5^{3/2}R}$
• C. $\frac{4N{\mu }_{0}I}{5^{1/2}R}$
• D. $\frac{4N{\mu }_{0}I}{5^{3/2}R}$

Answer 1

Answer: (B)

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