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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.


Solution Preview


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.