Question

Two circular coils of wire each having a radius of 4 cm and 10 turns have a common axis and are 6 cm apart. If a current of 1 A passes through each coil in the opposite direction find the magnetic induction.
(a) At the centre of each coil;
(b) At a point on the axis, midway between them.

Answer 1

We know Magnetic field at centre of a coil having $\text{Current I, radius r and number of turns N}$ is given as $B=\frac{{\mu }_{0}NI}{2r}$

and at a distance x from the centre and on the axis is $B=\frac{{\mu }_{0}NIx}{2(x^2+r^2)}$ and its direction is given by right-hand screw rule i.e.

if the observer sees that the current is anticlockwise then the field will be towards the observer and vice-versa, as shown in the $figure-1$

$Part$ $a$

Field at center $O_1$ will be $\text{difference of field due to loop-1 and loop-2}$

$\text{At}{O}_1$ $B_{net}$ = $\frac{{\mu }_{0}NI}{2r}$ - $\frac{{\mu }_{0}NIx}{2(x^2+r^2)}$ $\text{towards observer}$

putting ${\mu }_0=4\pi \times {10}^{-7}$ ,$\text{N=6, r=.04 meter ,x= .06 meter and I = 1}$ all units in S.I. system

we get $B_{net}=5.07\times {10}^{-5}$ Tesla towards the observer

Similarly, $\text{At}{O}_1$ the filed due to circle 2 will dominate so net filed will be

$B_{net}=5.07\times {10}^{-5}$ Tesla $\text{in a direction away}$ from the observer.

$Part$ $b$

According to the above theory the field at the midpoint on the axis of joining the centres will be equal and opposite as current in loops is opposite so $\text{net field is zero there at O}$