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Using the Biot - Savart Law to Find the Magnetic Field on the Axis of a Circular Loop

The Biot – Savart Law makes it possible to find the magnetic field at each point in space due to a distribution of electric currents. The Law is completely general and can in principle be used whatever the distribution of currents. In many situations however, the problem has symmetry which makes much simplification possible. For the example of the magnetic field on the axis of a current loop, all the components of the magnetic field directed off the axis of the loop cancel out and the net magnetic field on the axis is directed along the axis – this result is due to the circular symmetry of the loop.

The Biot – Savart Law states

whereis a tiny piece of current loop and hat r is the unit vector in the direction of

For the point on the axis of the current loop aboveandHence

Whenat the centre of the coil,