A student claims that in vacuum the field E = (40.0 V ,m^ -1 ) ! [(0.210 rad ,m^ -1 )…

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Maxwell's equations and conservation laws · Electromagnetic waves and their propagation

Electricity, Magnetism and Electromagnetism, Ch Plane electromagnetic waves: polarisation, energy flux and radiation pressure

Problem

A student claims that in vacuum the fieldE=(40.0 V m−1)cos⁡ ⁣[(0.210 rad m−1) y−(6.30×107 rad s−1) t]z^\mathbf{E} = (40.0\ \mathrm{V\,m^{-1}})\cos\!\left[(0.210\ \mathrm{rad\,m^{-1}})\,y - (6.30\times10^{7}\ \mathrm{rad\,s^{-1}})\,t\right]\hat{\mathbf{z}}is part of a plane electromagnetic wave, with a magnetic field of the form B=Bx(y,t) x^\mathbf{B} = B_x(y,t)\,\hat{\mathbf{x}}. Maxwell's equations in vacuum are ∇⋅E=0\nabla\cdot\mathbf{E} = 0, ∇⋅B=0\nabla\cdot\mathbf{B} = 0, ∇×E=−∂B/∂t\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t and ∇×B=μ0ε0 ∂E/∂t\nabla\times\mathbf{B} = \mu_0\varepsilon_0\,\partial\mathbf{E}/\partial t; take 1/μ0ε0=3.00×108 m s−11/\sqrt{\mu_0\varepsilon_0} = 3.00\times10^{8}\,\mathrm{m\,s^{-1}}. ((a)) State the direction of travel, and check the two divergence equations. ((b)) Show that the wave equation obtained from the two curl equations requires ω/k=1/μ0ε0\omega/k = 1/\sqrt{\mu_0\varepsilon_0}, and test the claim by computing ω/k\omega/k for the numbers given. ((c)) Use Faraday's law to find Bx(y,t)B_x(y,t), taking the constant of integration to be zero, give the amplitude B0B_0, and state the direction of B\mathbf{B} when E\mathbf{E} points along +z^+\hat{\mathbf{z}}. Verify that the Ampère–Maxwell law is then also satisfied.

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