Inside a region threaded by a current loop

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · 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

Inside a region threaded by a current loop, a magnetic field is decaying uniformly in time: ∂Bz/∂t=−3.0×10−4 T s−1\partial B_z/\partial t = -3.0\times10^{-4}\ \mathrm{T\,s^{-1}}, uniform over the region, with Bx=By=0B_x = B_y = 0 and ∂Bx/∂t=∂By/∂t=0\partial B_x/\partial t = \partial B_y/\partial t = 0. Take FμνF^{\mu\nu}, JμJ^\mu as usual, obeying ∂μFνμ=μ0Jν\partial_\mu F^{\nu\mu} = \mu_0 J^\nu and ∂[αFβγ]=0\partial_{[\alpha}F_{\beta\gamma]}=0. ((a)) Suppose only that ημνAμ\eta^{\mu\nu}A_\mu is known to transform as a type (1,0)(1,0) vector for every covector AμA_\mu. Use the quotient rule to show that ημν\eta^{\mu\nu} is a type (2,0)(2,0) tensor. ((b)) Take ∂[αFβγ]=0\partial_{[\alpha}F_{\beta\gamma]} = 0 with (α,β,γ)=(0,1,2)(\alpha,\beta,\gamma) = (0,1,2) (that is, t,x,yt,x,y) to show ∂tBz=−(∂xEy−∂yEx)=−(∇×E⃗)z\partial_tB_z = -(\partial_xE_y - \partial_yE_x) = -(\nabla\times\vec E)_z. Hence find (∇×E⃗)z(\nabla\times\vec E)_z for the decaying field above.

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