Magnetic Field on the Axis of a Circular Current Loop: Class 12 Physics Notes, Definition & Working Principle

Magnetic Field due to a circular loop refers to the magnetic field at any point on the axis of the loop. The axis passes through the loop and is perpendicular to the plane containing the loop. This can be determined using the Biot-Savart Law and the right-hand thumb rule. 

Below is the derivation of the magnetic field due to a circular loop.

Magnetic Field on the Axis of a Circular Current Loop ( $\vec{\mathit{B}}$  due to circular loop)

Due to each $\overrightarrow{d\mathcal{l}}\mathrm{}\mathrm{}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{}\mathrm{o}\mathrm{f}\mathrm{}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{p}\mathrm{}\vec{B}$ at ‘c’ is inwards (in this case).

Direction of $\vec{B}$ : The direction of the magnetic field at the centre of a circular wire can be obtained using the right-hand thumb rule. If the fingers are curled along the current, the stretched thumb will point towards the magnetic field (figure).

Another way to find the direction is to look into the loop along its axis. If the current is in an anticlockwise direction, the magnetic field is towards the viewer. If the current is in the clockwise direction, the field is away from the viewer.

Semicircular and quarter of a circle:

Magnetic Field on the Axis of a Circular Loop

: $B=\frac{{\mu }_{0}N{\rm I}{R}^2}{2(R^2+x^2{)}^{\frac{3}{2}}}$

N = No. of turns (integer)

The right-hand thumb rule is used to determine the direction. Curl your fingers in the direction of the current, then the direction of the thumb points in the direction of the points on the axis.

The magnetic field at a point not on the axis is mathematically difficult to calculate. We show qualitatively in the figure the magnetic field lines due to a circular current, which will give some idea of the field.

 

The magnetic field pattern is comparable to the magnetic field produced by a bar magnet.

The side ‘I’ (the side from which the $\vec{B}\mathrm{}\mathrm{}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{s}\mathrm{}\mathrm{o}\mathrm{u}\mathrm{t}\left)\mathrm{}\mathrm{o}\mathrm{f}\mathrm{}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{p}\mathrm{}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{s}\mathrm{}\mathrm{a}\mathrm{s}\mathrm{}\mathrm{‘}\mathrm{N}\mathrm{O}\mathrm{R}\mathrm{T}\mathrm{H}\mathrm{}\mathrm{P}\mathrm{O}\mathrm{L}\mathrm{E}\mathrm{’}\mathrm{}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{}\mathrm{I}\mathrm{I}\mathrm{}\right(\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{}\mathrm{i}\mathrm{n}\mathrm{}\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{h}\mathrm{}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{}\vec{B}$ enters) acts as the ‘SOUTH POLE’. It can be verified by studying the force on one loop due to a magnet or another loop.

Mathematically

 B_axis = $\frac{{\mu }_{0}N{\rm I}{R}^2}{2(R^2+x^2{)}^{\frac{3}{2}}}\cong \frac{{\mu }_{0}N{\rm I}{R}^2}{2x^3}$  for x >> R 

$=2\left(\frac{{\mu }_0}{4\pi }\right)\left(\frac{IN\pi R^2}{x^3}\right)$                                       

it is similar to B_axis due to magnet = 2 $\left(\frac{{\mu }_0}{4\pi }\right)\frac{m}{x^3}$

Magnetic dipole moment of the loop  M = INπR²

M = INA for any other shaped loop.

Unit of M is Amp. m²_.

Unit of m (pole strength) = Amp. m                            {in magnet M = ml}

To be determined by the right-hand thumb rule, which is also used to determine direction of $\vec{B}$ on the axis. It is also from the ‘S’ side to the ‘N’ side of the loop.