# Magnetic field

A small/thin circular conducting disk that can carry total current, as represented by 'I'. The current path is circular at every distance from the center of the disk, and the each circle center is the disk center. Find the magnetic field at the disk center, assuming the current density is:

1. Constant

2. Inversely proportional to the distance from the center.

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A small/thin circular conducting disk that can carry total current 'I'. The current path is circular at every distance from the center of the disk, and the each circle center is the disk center. Find the magnetic field at the disk center, assuming the current densit y is:

1. Constant

2. inversely proportional to the distance from the center.

Solution:

(please see the attached file)

Case 1: Current density is constant

The fig. shows a conducting disc of radius R and thickness t carrying the circularly flowing current. Let us consider an annular ring of inner radius r and outer radius r+dr with its center coinciding with the center of the disc. As the current density y is constant, the current density in the annular ring is given by:

y = k ...(1) where k is a constant

Area of cross section of the annular ring = t dr ...(2)

From (1) and (2): Circular current flowing through the annular ring = dI(r) = Current density x Area of cross section of the annular ring = kt dr ...(3)

Magnetic field at the centre of a current carrying loop is given by: B = Î¼_0i/2a ...(4)

where i = Current flowing in the loop, a = Radius of the loop

Using (4) we determine the magnetic field dB at the centre of the annular ring due to current flowing through it as follows: Radius of the ring = r, Current flowing through it dI = kt dr

Substituting in (4) we get: dB = (Î¼_0/2r)(kt dr) = (Î¼_0kt/2) (1/r) dr ...(5)

To determine the magnetic field B at the center of the disc due to the current flowing through the complete disc, we integrate (5) in the limits r = 0 and r = R.

Case 2: Current density is inversely proportional to the distance from the center

As the current density y is inversely proportional to the distance from the centre, the current density in the annular ring is given by:

y = k/r ...(1) where k is a constant

Area of cross section of the annular ring = t dr ...(2)

From (1) and (2): Circular current flowing through the annular ring = dI(r) = Current density x Area of cross section of the annular ring = kt (1/r) dr ...(3)

Magnetic field at the centre of a current carrying loop is given by: B = Î¼0i/2a ...(4)

where i = Current flowing in the loop, a = Radius of the loop.....

(Please see the attached file)

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