# 0.4 Magnetic field due to current in a circular wire

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Magnetic field due to current in circular wire is largely axial. It means that we need to concentrate our investigation of magnetic field on axial positions. One of the important axial positions is center of the circular wire itself. We shall limit our discussion in this module to this case of magnetic field at the center of circular coil. The procedure for deriving expression for the magnetic field due to current in circular wire is same as that of current carrying straight wire. Here also, we make use of superposition principle whereby we combine the small magnetic fields due to each of the small current elements composing the circular coil.

Circular wire is considered to be composed of small linear current elements. We determine magnetic field due to each of the linear current elements applying Biot-Savart law. Finally, we determine net magnetic field using superposition principle (i.e. by determining vector sum of magnetic fields due to all current elements).

In general, the bending of current carrying wire in circular shape has the effect of strengthening or localizing magnetic field in narrower region about the axis.

## Direction of magnetic field (right hand thumb rule)

Let us consider two diametrically opposite small current elements on the circular wire. The magnetic field lines are compressed inside the circle as it accommodates all the circular closed lines drawn outside. This compression of magnetic field lines is maximum at the center. In the figure here, we consider the circular coil in horizontal plane. The magnetic field lines being perpendicular to current elements are in the plane of drawing.

Such is the case with any other pair of current elements as well. This means that magnetic field line passing though axis is reinforced by all such diametrically opposite pairs of current element. The magnetic field due to current in circular wire, therefore, is nearly axial.

The observations as above are the basis of Right hand thumb rule for current in circular wire. If we orient right hand such that curl of fingers follows the direction of current in the circular wire, then extended thumb points in the direction of magnetic field at its center.

Right hand thumb rules for straight wire and circular wire are opposite in the notations. The curl of hand represents magnetic field in the case of straight wire, whereas it represents current in the case of circular wire. Similarly, the extended thumb represents current in the case of straight wire, whereas it represents magnetic field in the case of circular wire.

There is yet another simple way to find the direction of axial magnetic field at the center. Just look at the circular loop facing it. If the current is clockwise, then magnetic field is away from you and if the current is anticlockwise, then magnetic field is towards you.

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