Question

A Helmholtz coil has 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):

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

Answer 1

The correct option is B $\frac{8N{\mu }_{0}I}{5^{3/2}R}$

The magnetic field due to both the coils are in the same direction and equal in magnitude

The magnitude of the magnetic field due to one coil is give as

$B=\frac{{\mu }_{o}i{R}^2}{2(R^2+(R/2{)}^2{)}^{3/2}}$

$B=\frac{4{\mu }_{o}i{R}^2}{5R^2}$

$B_{net}=2B$

$B_{net}=\frac{8{\mu }_{o}i{R}^2}{5R^2}$