A plane wave travels through vacuum along with its electric field along : E = E_0 (kx - t)

PHYSICS · Subjective · 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 plane wave travels through vacuum along +x+x with its electric field along zz:E=E0cos⁡(kx−ωt) z^,E0=75.0 V m−1,\mathbf{E} = E_0\cos(kx - \omega t)\,\hat{\mathbf{z}},\qquad E_0 = 75.0\ \mathrm{V\,m^{-1}},and its magnetic field is of the form B=By(x,t) y^\mathbf{B} = B_y(x,t)\,\hat{\mathbf{y}}. The wavelength is λw=0.400 m\lambda_{\mathrm{w}} = 0.400\,\mathrm{m}. 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 c=1/μ0ε0=3.00×108 m s−1c = 1/\sqrt{\mu_0\varepsilon_0} = 3.00\times10^{8}\,\mathrm{m\,s^{-1}}. (a) Show that both divergence equations are satisfied by these fields, and say in one sentence what they rule out. (b) Using Faraday's law in Cartesian components, find By(x,t)B_y(x,t) (take the constant of integration to be zero) and show that the Ampère–Maxwell law then forces ω=ck\omega = ck. (c) Give the numerical values of kk, of ω\omega, and of the magnetic-field amplitude B0B_0, and state the direction of B\mathbf{B} at an instant when E\mathbf{E} points along +z^+\hat{\mathbf{z}}.

Browse the problem library