Question

A steady current flows in a long wire. It is bent into a circular loop of one turn and the magnetic field at the centre of the coil is b b. If the same wire is bent into a circular loop of n n turns, the magnetic field at the centre of the coil is

Answer 1

Given that,

Magnetic field = B

Current = I

We know that,

The circumference of the first loop is the length of the wire is

$circumference=2\pi r$

We need to calculate the magnetic field at the center

Using formula of magnetic field

$B=\dfrac{\mu_{0}I}{2\pi R}$

Put the value into the formula

$B=\dfrac{\mu_{0}I}{2\pi r}$

The same wire is bent into n circular coils.

We need to calculate the new radius

Using formula of circumference

$circumference=2\pi r$

$n\times2\pi r'=2\pi r$

$r'=\dfrac{r}{n}$

We need to calculate the magnetic field at the center of loopes

Using formula of magnetic field

$B'=\dfrac{\mu_{0}I}{2\pi R}$

Put the value into the formula

$B'=\dfrac{n\mu_{0}I}{2\pi r'}$

Put the value of r'

$B'=\dfrac{n^2\mu_{0}I}{2\pi r}$

$B'=n^2B$

Hence, The magnetic field at the centre of the coil is n²B