Question

Calculate magnetic field induction at the centre of coil

Answer 1

Answer:

A current carrying wire generates a magnetic field. According to Biot-Savart’s law, the magnetic field at a point due to an element of a conductor carrying current is,

   Directly proportional to the strength of the current, i

   Directly proportional to the length of the element, dl

   Directly proportional to the Sine of the angle θ between the element and the line joining the element to the point and

   Inversely proportional to the square of the distance r between the element and the point.

Thus, the magnetic field at O is dB, such that,

                                                                                 

Then,

                                                                             

where,

                                                                                    is the proportionality constant and

                                                                               is called the permeability of free space.

Then,

                                                                               

In vector form,

                                                                               

Consider a circular coil of radius r, carrying a current I. Consider a point P, which is at a distance x from the centre of the coil. We can consider that the loop is made up of a large number of short elements, generating small magnetic fields. So the total field at P will be the sum of the contributions from all these elements. At the centre of the coil, the field will be uniform. As the location of the point increases from the centre of the coil, the field decreases.

By Biot- Savart’s law, the field dB due to a small element dl of the circle, centered at A is given by,

                                                                           

This can be resolved into two components, one along the axis OP, and other PS, which is perpendicular to OP. PS is exactly cancelled by the perpendicular component PS’ of the field due to a current and centered at A’. So, the total magnetic field at a point which is at a distance x away from the axis of a circular coil of radius r is given by,

                                                                                   

If there are n turns in the coil, then

                                                               

where µ0  is the absolute permeability of free space.

Since this field Bx from the coil is acting perpendicular to the horizontal intensity of earth’s magnetic field, B0, and the compass needle align at an angle θ with the vector sum of these two fields, we have from the figure

                                                   

The horizontal component of the earth’s magnetic field varies greatly over the surface of the earth. For the purpose of this simulation, we will assume its magnitude to be B0 = 3.5x10-5 T. The variation of magnetic field along the axis of a circular coil is shown here.    

Explanation: