Question
Two similar coils of radius $$R$$ are lying concentrically with their planes at right angles to each other. The currents flowing in them are $$I$$ and $$2I,$$ respectively. The resultant magnetic field induction at the centre will be
A.
$$\frac{{\sqrt 5 {\mu _0}I}}{{2R}}$$
B.
$$\frac{{3{\mu _0}I}}{{2R}}$$
C.
$$\frac{{{\mu _0}I}}{{2R}}$$
D.
$$\frac{{{\mu _0}I}}{R}$$
Answer :
$$\frac{{\sqrt 5 {\mu _0}I}}{{2R}}$$
Solution :
The magnetic field $$\left( B \right)$$ at the centre of circular current carrying coil of radius $$R$$ and current $$I$$ is $$B = \frac{{{\mu _0}I}}{{2R}}$$
Similarly, if current is $$2I,$$ then
Magnetic field $$ = \frac{{{\mu _0}2I}}{{2R}} = 2B$$
So, resultant magnetic field $$ = \sqrt {{B^2} + {{\left( {2B} \right)}^2}} = \sqrt {5{B^2}} $$
$$ = \sqrt 5 B = \frac{{{\mu _0}I\sqrt 5 }}{{2R}}$$