Plane Electromagnetic Waves I

The existence of combined electric and magnetic fields that propagate through space as electromagnetic (EM) waves is the most significant prediction of Maxwell's equations. As Maxwell's equations hold in free space, the predicted electromagnetic waves do not require a medium for their propagation. An EM wave comprises an electric field, defined as the force per charge on a stationary charge, and a magnetic field, which is the force per charge on a moving charge.

The EM field is assumed to be a function of only the x-coordinate and time. Thus, the y-component of the electric field is then written as E_y(x, t), and the z-component of the magnetic field is written as B_z(x, t). Due to the assumption of free space, there are no free charges or currents, and hence, the charge enclosed and the current in Maxwell's equation are set to be zero.

The applicability of Gauss's law for electric fields is examined by assuming a rectangular Gaussian surface with a square cross-section of side l and whose third side has length Δx. The y-component of the electric field is the same on the box's top and bottom sides. Therefore, the fluxes on both sides cancel each other. Similarly, the net flux from the z-component of the electric field through the two lateral sides is also canceled out. Any net flux through the surface, therefore, comes entirely from the x-component of the electric field. As the electric field has no y- or z-dependence, E_x(x, t) is constant over the face of a box with area A and has a possibly different value of E_x(x + Δx, t) that is constant over the opposite face of the box. Applying Gauss's law gives the net flux of the x-component.

Since q_enc, the charge enclosed, is zero, the net flux of the component is also zero, and the above equation implies E_x(x, t) = E_x(x + Δx, t) for any Δx. Such a component E_x(x, t) would not be part of an electromagnetic wave propagating along the x-axis, so E_x(x, t) = 0 for this wave. Therefore, the only non-zero components of the electric field are E_y(x, t) and E_z(x, t), perpendicular to the direction of propagation of the wave.

A similar argument holds by substituting E for B and using Gauss's law for magnetism instead of Gauss's law for electric fields. This shows that the B field is also perpendicular to the direction of propagation of the wave. The electromagnetic wave is, therefore, a transverse wave, with its oscillating electric and magnetic fields perpendicular to its direction of propagation.