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B, H and M due to cylindrical conductor

An infinitely long solid cylindrical conductor of radius R carries a free current density J(s) = Cs^3z distributed over its cross section. The z axis is the long axis of the cylinder. The conductor has a permeability 'mu' which does not equal 'mu-0'. Outside the conductor is a vacuum.

A. Find H, B, M inside the conductor and discern between diamagnetic and paramagnetic materials.
B. Find H,B,M outside the conductor.
C. Sketch the situation for diamagnetic and paramagnetic materials.

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(A)

By Ampere's circuital law,
mu * I = close line Integral [ vector(B) . vector(dl) ]

At distance 's' (< R) from the axis of the conductor (z direction), draw a circle (Ampere circuit) of radius 's', concentric to the axis and in XY plane:

Current within the area of interest,
I = integral (0 to s) [ vector(J) . vector(ds) ]

=> I = integral (0 to s) [ C*s^3 * 2*pi*s*ds] = 2*pi*C*s^5/5 = (2/5)*pi*C*s^5

Hence, by Ampere's circuital law,

mu * (2/5)*pi*C*s^5 = B * 2*pi*s [ Because, B will be constant at all ...

Solution Summary

Due to an infinitely long straight cylindrical conductor, carrying current, magnetic field is estimated inside and outside the conductor.

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