It is difficult to visualize time-dependent electromagnetic fields in three dimensional space. However, for fields that propagate in the z-direction, it is interesting to focus on the locus of points where the transverse (x and y) components of the fields vanish simultaneously. For E(r, t), this is the one-dimensional curve where the surface E_x (x, y, z, t) = 0 intersects the surface E_y (x, y, z, t) = 0. The same condition with B(r, t) gives the curve of magnetic zeroes.

(a) Show that the scalar function ψ(r, t) = k {gx + kx² sin δ + iy'} exp i(kz − ωt) solves the wave equation where g and δ are parameters and y' = y cos δ + z sin δ.

(b) Let E = x̂ψ + i ŷψ + ẑE_z. Use Gauss’ law and Faraday’s law to deduce E_z(r, t) and B(r, t) so that E and B solve the Maxwell equations in free space.

(c) If the parameter g >> 1, there is a large region of space where the x² piece of the fields can be dropped with very little loss of accuracy. Do this and show that the curve of electric zeroes is the line y = −z tan δ in the x = 0 plane.

(d) In the same g >> 1 limit, show that the locus of zeroes for the transverse magnetic field is an elliptic helix that rotates in time around its axis of symmetry. The latter (which we will call the z axis) is the line of electric zeroes found in part (c). Notice that y' and z' are orthogonal and use polar coordinates (r, θ) defined by gx = r cos θ and y' = r sin θ. Make a sketch that shows where both sets of zeroes occur.