Question

Two current loops similar are placed with their planes one along x-axis and the other along y-axis. Then
the ratio of resultant magnetic field at a common point on the axis to the individual magnetic field at the
same point is​

Answer 1

Ratio of resultant magnetic field to the individual magnetic field is $\bold{\frac{\mu_{0} I}{2 a} \times \sqrt{2}}$

The magnetic field intensity due to a loop placed along x axis is given by the formula:

$B_{y}=\frac{\mu_{0} I 2 \pi a^{2}}{4 \pi\left(a^{2}+y^{2}\right)^{\frac{3}{2}}}$

When y = 0, then the magnetic field intensity is:

$B_{y 0}=\frac{\mu_{0} I}{2 a}$

The magnetic field intensity due to a loop placed along y axis is given by the formula:

$B_{x}=\frac{\mu_{0} I 2 \pi a^{2}}{4 \pi\left(a^{2}+x^{2}\right)^{\frac{3}{2}}}$

When x = 0, then the magnetic field intensity is:

$B_{x 0}=\frac{\mu_{0} I}{2 a}$

Now, the magnetic field intensity at common point is:

$B=\sqrt{B_{x 0}^{2}+B_{y 0}^{2}}$

$B=\sqrt{\left(\frac{\mu_{0} I}{2 a}\right)^{2}+\left(\frac{\mu_{0} I}{2 a}\right)^{2}}$

$\therefore B=\frac{\mu_{0} I}{2 a} \times \sqrt{2}$