A straight conductor of uniform square cross-section lies along the axis

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Vector calculus and field geometry

Electricity, Magnetism and Electromagnetism, Ch Divergence, curl and the local form of the field laws

Problem

A straight conductor of uniform square cross-section lies along the xx axis. At one instant its current density is everywhere parallel to the axis,J=J0(1+xd)x^,Jy=Jz=0,\mathbf{J} = J_0\left(1 + \frac{x}{d}\right)\hat{\mathbf{x}},\qquad J_y = J_z = 0,with J0=2.0×106J_0 = 2.0\times10^{6} A m−2^{-2} and d=0.010d = 0.010 m, and xx measured in metres from one end. ((a)) Starting from charge conservation for a closed surface, ∮SJ⋅dA=−dQencdt\oint_S \mathbf{J}\cdot d\mathbf{A} = -\dfrac{dQ_{\text{enc}}}{dt} with Qenc=∫Vρ dVQ_{\text{enc}} = \int_{\mathcal{V}}\rho\,dV, use the divergence theorem to derive the law at a point. ((b)) Compute ∇⋅J\nabla\cdot\mathbf{J} for this current density. ((c)) Find ∂ρ/∂t\partial\rho/\partial t at this instant and say whether charge is piling up or draining away. ((d)) In a steady state a tapered wire also has JxJ_x growing along it, yet ∂ρ/∂t=0\partial\rho/\partial t = 0. Explain in one or two sentences what is different there.

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