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2. (a) Starting with Maxwell's equations in differential form, show that the electric field must obey a three-dimensiona
Posted: Tue May 17, 2022 9:02 pm
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2. (a) Starting with Maxwell's equations in differential form, show that the electric field must obey a three-dimensional wave equation. Assume a region of space where there are no charges or currents. (b) Show that a possible solution to the three-dimensional wave equation is the real part of) E(x,y,z, t) = Eo exp (i (k.z - wt)). The terms have their usual meanings. (c) Write down, without further proof, the related wave equation for the magnetic field and suggest a similar possible solution to this wave equation. (d) Find the divergence of the E field and, recalling that we consider free space, show that electromagnetic waves must be transverse waves. (e) Consider the curl of the E field to show that the electric and magnetic field components of an electromagnetic wave are mutually perpendicular. (f) Hence, find an expression relating the amplitudes of the electric and mag- netic fields in an electromagnetic wave and justify the statement that they are in phase with each other. (9) Provide a sketch of a travelling electromagnetic wave that highlights the above features.