Magnetic field intensity (H) on the Axis of a Circular Loop

  on the Axis of a Circular Loop

AU : Dec.-02, 03, 08, 09, 10, 17, May-06, 09, 16, 18

• Consider a circular loop carrying a direct current I, placed in xy plane, with z axis as its axis as shown in the Fig. 7.7.1. The magnetic field intensity   at point P is to be obtained. The point P is at a distance z from the plane of the circular loop, along it's axis.

• The radius of the circular loop is r. Consider the differential length  of the circular loop as shown in the Fig. 7.7.1.

• In the cylindrical co-ordinate system,

• The unit vector  is in the direction along the line joining differential current element to the point P.

From the Fig. 7.7.2, it can be observed that,

• Note that while calculating cross product |R₁₂| is neglected for convenience, which must be considered in further calculations.

• According to Biot-Savart law, the differential field strength  at point P is given by,

• Note that || which was neglected while obtaining the cross product is now considered in  .

• The total  is to be obtained by integrating  over the circular loop i.e. for ϕ = 0 to 2 π .

Note : It can be observed that though  consists of two components , due to radial symmetry all  components are going to cancel each other. So  exists only along the axis in  direction.

where          r = Radius of the circular loop

z = Distance of point P along the axis

Note : If point P is shifted at the centre of the circular loop i.e. z = 0, we get the result obtained in earlier section.

where  is the unit vector normal to xy plane in which the circular loop is lying. 

Ex. 7.7.1 Two narrow circular coils A and B have common axis and are placed 10 cm apart. The coil A has 10 turns of radius 5 cm with a current of 1 A passing through it. The coil B has a single turn of radius 7.5 cm. If the magnetic field at the centre of coil A is to be zero, what current should be passed through coil B.

Sol. : The at the centre of the circular coil with N turns is given by,

The  at the centre of coil A i.e. point P due to coil

B is,

Where r = r₂ = 7.5 cm, I = I₂

and z = Distance between point P and coil = 10 cm

The total   at P = 100 + 1.44 I₂ which must be zero,

I₂ = -100 / 1.44 = - 69.44 A

The negative sign indicates direction of I₂, is opposite to that of I₁.

Ex. 7.7.2 A circular loop located on  x² + y² = 9, z = 0 carries a direct current of 10 A along  Determine   at (0,0,4) and (0,0,- 4).

 AU: Dec.-09, Marks 16

Sol. : The loop in x-y plane is shown in the Fig. 7.7.4 (a)

In the cylindrical co-ordinates,

r = √x² + y² = √9 = 3

Thus the radius of the circular loop is 3.

Consider the differential length   at point P on the circular loop. This   is in the plane for which r is constant and z = 0. Thus I  is tangential at point P in   direction.

Key Point : Due to the radial symmetry, all the radial (-   ) components at A are going to cancel each other. Hence  exist only in z direction.

Thus at both the points A (0, 0, 4) and B (0, 0, - 4), the  remains same.

Examples for Practice

Ex. 7.7.3 A circular loop located on x2 + y2 = 9, z = 0 carries a current of 10 A. Determine H at (0, 0, 5) and (0, 0, - 5). Take the direction        of current in anti-clockwise direction.

Ex. 7.7.4 A thin ring of radius 5 cm is placed on the plane z = 1 cm, so that its center is at (0, 0, 1 cm). If the ring carries a direct current of 50 mA along direction, find  at P (0, 0, - 1 cm) using Biot-Savart’s law.

Review Question

1. What is Biot-Savart law 1 Find the magnet flux density at a point P on the axis of a circular loop of radius 'a' that carries a direct current I.

AU : Dec.-02, 03. 08, 10, 17, Hay-06, 09, 16, 18. Harks 16