A circular loop of radius 𝑟 is carrying current I A. The ratio of magnetic field at the center of circular loop and at a distance 𝑟 from the center of the loop on its axis is

Answer (1)

We know that,

Magnetic field at the centre of a circular loop,

$\displaystyle B_1=\frac{\mu _0I}{2r}$

Magnetic field at axial distance x,

$\displaystyle B_2=\frac{\mu _0Ir^2}{2(r^2+x^2)^\frac{3}{2} }$

Here x=r,

$\displaystyle \begin{aligned}\therefore B_2 &=\frac{\mu _0Ir^2}{2(r^2+r^2)^\frac{3}{2} } \\&=\frac{\mu _0Ir^2}{2(2r^2)^\frac{3}{2} }\\&=\frac{\mu _0Ir^2}{2\times 2\sqrt{2} r^3}=\frac{\mu _0I}{4\sqrt{2} r} \end{aligned}$

$\begin{aligned}\therefore \frac{B_1}{B_2} &=\frac{\displaystyle \frac{\mu _0I}{2r} }{\displaystyle \frac{\mu _0I}{4\sqrt{2} r} } =2\sqrt{2} \end{aligned}$