A coaxial cable has the cross section shown in the figure. The shaded region is insulated. The regions in which r < a and b < r < c are conducting. A uniform dc current density of total current I flows along the inner part of the cable (r < a) and returns along the outer part of the cable ( b< r < c) in the directions shown. The radial dependence of the magnitude of the magnetic field, H, is shown by which of the following?
Know your College Admission Chances Based on your Rank/Percentile, Category and Home State.
Get your JEE Main Personalised Report with Top Predicted Colleges in JoSA
For a coaxial cable with inner conductor (r < a) carrying current I outwards and outer conductor (b < r < c) returning current I, the magnetic field H is determined by Ampere's circuital law: ∮H·dl = I_enc.
Inside inner conductor (r < a): I_enc = I × (πr²)/(πa²) = I(r²/a²), so H = (I r)/(2π a²) ∝ r.
Between conductors (a < r < b): I_enc = I, so H = I/(2πr) ∝ 1/r.
Inside outer conductor (b < r < c): I_enc = I - I × (π(r²-b²))/(π(c²-b²)) = I[1 - (r²-b²)/(c²-b²)], so H decreases with r.
Outside (r > c): I_enc = 0, so H = 0.
Final answer: The third option (H ∝ r for r < a, H ∝ 1/r for a < r < b, decreasing for b < r < c, and zero for r > c).