# Magnetic Field at an axial point of a circular loop

Consider a circular loop of radius R, carrying current in yz plane with centre at origin O. Let P be a point on the axis of the loop at a distance ‘x’ from the centre ‘O’ of the loop.

Consider a conducting element dl of loop. According to Biot-Savart’s law, the magnitude of magnetic field due to the current element is

$\large dB = \frac{\mu_0}{4\pi} \frac{I dl sin\theta}{r^2}$

Here angle θ between the element $\vec{dl}$ and $\vec{r}$ is π/2

$\large dB = \frac{\mu_0}{4\pi} \frac{I dl}{r^2}$

The direction of $\vec{dB}$ is perpendicular to the plane formed by $\vec{r}$ and $\vec{dl}$ .

In case of a point P on the axis of circular coil, for every current element ‘idl’ there is a symmetrically situated opposite element. The component of the field dB perpendicular to the axis cancel each other while component of the field dB along the axis add up and contributes to the net magnetic field.

$\large B = \int dB sin\phi = \int \frac{\mu_0}{4\pi} \frac{I dl }{r^2} . \frac{R}{r}$

$\large B = \frac{\mu_0}{4\pi} \frac{I R}{r^3} \int dl$

$\large B = \frac{\mu_0}{4\pi} \frac{I R}{r^3} (2 \pi R)$

$\large B = \frac{\mu_0}{4\pi} \frac{2 \pi I R^2}{(R^2 + x^2)^{3/2}}$

If coil has N number of turns , then

$\large B = \frac{\mu_0}{4\pi} \frac{2 \pi N I R^2}{(R^2 + x^2)^{3/2}}$

$\large B = \frac{\mu_0}{4\pi} \frac{2 N I A}{(R^2 + x^2)^{3/2}}$ ; ( Where A = π R2)

$\large B = \frac{\mu_0}{4\pi} \frac{2 M}{(R^2 + x^2)^{3/2}}$ ; (Where M = Magnetic dipole moment)

If x >> R ;

$\large B = \frac{\mu_0}{4\pi} \frac{2 M}{x^3}$

### Circular current loop as magnetic dipole:

From the above expression $\large B = \frac{\mu_0}{4\pi} \frac{2 M}{x^3}$

(i)Magnetic moment of the circular current carrying coil is M = NiA;

(ii)M is independent of shape of the coil
Hence , Current loop behaves like a magnetic dipole with poles on either side of its face and it is known as “magnetic shell”.

(iii)SI unit of magnetic moment (M) is A-m² .

(iv)Magnetic moment of a current loop is a vector perpendicular to the plane of the loop and the direction is given by right hand thumb rule.