The magnetic field due to a current-carrying loop of radius 3 cm at a point on its axis at the distance of 4 cm from the centre is 54$\mathrm{\mu }$T. What will be its value at the centre of the loop?

Given data:

 radius$3 \mathrm{cm}$ and distance $4 \mathrm{cm}, centre is 54 \mu T$

Concept used: magnetic dipole movement

Detailed solution:

When a circular loop of radius is carrying current, the magnetic field generated at a point on the axis located X distance from the center is

$\mathrm{B}=\frac{{\mu }_0}{4\pi }\mathrm{I}\times 2\pi \frac{{\mathrm{a}}^2}{{\left({\mathrm{a}}^2+{\mathrm{x}}^2\right)}^{3/2}}$

$\Rightarrow 54\times {10}^{-6}=\frac{{10}^{-7}\times \mathrm{I}\times 2\pi \times {\left(3\times {10}^{-2}\right)}^2}{{\left[{\left(3\times {10}^{-2}\right)}^2+{\left(4\times {10}^{-2}\right)}^2\right]}^{3/2}}$

$\Rightarrow \mathrm{I}=11.94 \mathrm{A}$

Magnetic field at center 

$=\frac{{\mu }_{\mathrm{o}}\mathrm{I}}{4\pi }\times \frac{2\pi }{a}$

$\frac{{10}^{-7}\times 11.94\times 2\pi }{3\times {10}^{-2}}$

$=250\mu \mathrm{T}.$