An infinite straight wire lies along the -axis and carries a steady current with…

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Relativity & Gravitation · Relativistic dynamics and fields in flat spacetime

The Geometry of Gravity, Ch Tensors in flat spacetime: index notation and covariant field equations

Problem

An infinite straight wire lies along the zz-axis and carries a steady current with uniform current density jz=2.0×106 A m−2j_z = 2.0\times10^6\ \mathrm{A\,m^{-2}} inside the wire. All fields are time-independent (∂/∂t=0\partial/\partial t = 0 everywhere) and there is no free charge density. Take FμνF^{\mu\nu} to be assembled from E⃗\vec E and B⃗\vec B as usual, obeying ∂μFνμ=μ0Jν\partial_\mu F^{\nu\mu} = \mu_0 J^\nu. ((a)) Suppose only that Jμ∂μfJ^\mu\partial_\mu f is known to be the same number in every inertial frame for every scalar field ff (that is, for every covector ∂μf\partial_\mu f that can be realised as a gradient at a point). Use the quotient rule to show that JμJ^\mu transforms as a type (1,0)(1,0) vector. ((b)) Part (a) makes JμJ^\mu a four-vector, so it transforms under a boost like any other. In the laboratory the wire carries no charge density. An observer moves at speed vv along +z+z, parallel to the wire. Transform JμJ^\mu to show that this observer measures a charge density ρq′=−γvjz/c2\rho_{\rm q}' = -\gamma vj_z/c^2 inside the wire. ((c)) Setting ν\nu equal to the zz-index in ∂μFνμ=μ0Jν\partial_\mu F^{\nu\mu} = \mu_0 J^\nu recovers the Ampere-Maxwell law, which reduces here (static fields) to (∇×B⃗)z=μ0jz(\nabla\times\vec B)_z = \mu_0 j_z. Find (∇×B⃗)z(\nabla\times\vec B)_z inside the wire, using μ0=4π×10−7 T m A−1\mu_0 = 4\pi\times10^{-7}\ \mathrm{T\,m\,A^{-1}}.

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