In a source-free region

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Relativistic electrodynamics

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

Problem

In a source-free region, with E⃗=0\vec E = 0 everywhere, a physicist proposes the magnetic field B⃗(x,y,z)=B0L(x x^+y y^−2z z^),\vec B(x,y,z) = \frac{B_0}{L}\big(x\,\hat x + y\,\hat y - 2z\,\hat z\big), with B0=1.0×10−4 TB_0 = 1.0\times10^{-4}\ \mathrm{T} and L=1.0 mL = 1.0\ \mathrm{m}. Take FμνF^{\mu\nu} to be built from E⃗\vec E and B⃗\vec B as usual. ((a)) Suppose only that FμνAμBνF^{\mu\nu}A_\mu B_\nu is known to be frame-independent for every pair of covectors Aμ,BνA_\mu, B_\nu that can be chosen completely independently of one another. Use the quotient rule to show that FμνF^{\mu\nu} must be a type (2,0)(2,0) tensor. ((b)) An observer moves at speed vv along +x+x. Using the type (0,2)(0,2) transformation law of FμνF_{\mu\nu}, as in the chapter's field-boost calculation, show that at the event with laboratory coordinates (x,y,z)(x,y,z) she measures Ey′=2γvB0z/LE_y' = 2\gamma vB_0z/L and Ez′=γvB0y/LE_z' = \gamma vB_0y/L, although E⃗=0\vec E=0 in the laboratory. ((c)) The homogeneous equation ∂[αFβγ]=0\partial_{[\alpha}F_{\beta\gamma]} = 0, with all three indices spatial, reduces to ∇⋅B⃗=0\nabla\cdot\vec B = 0. Find ∇⋅B⃗\nabla\cdot\vec B for the proposed field above and say whether it satisfies this law.

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