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A long conducting cylinder having a radius ‘b’ is placed along the z axis. The current density is J = Jₐ r³ ẑ for the region r < b where r is the distance in the radial direction. The magnetic field intensity (H) for the region inside the conductor (i.e. for r < b) is

1. Jₐ/4 r⁴
2. Jₐ/3 r³
3. Jₐ/5 r⁴
4. Jₐ r³

Correct answer: Jₐ/5 r⁴

Solution

Enclosed current I(r) = integral_0^r (Ja r'^3)(2*pi*r') dr' = 2*pi*Ja*r^5/5. Then H = I/(2*pi*r) = Ja*r^4/5. Stored answer (Ja/4 r^4) is wrong; the correct one is Ja r^4 /5.

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