The Traveling Electromagnetic Wave, Qualitatively — Question 5

Question 5

A dipole antenna radiates only its transverse acceleration component

Use a supplied far-zone dipole model $\mathbf E=C[\ddot{\mathbf p}-(\ddot{\mathbf p}\cdot\hat n)\hat n]$ at equal distance, with constant C and $\ddot{\mathbf p}=a\hat z$. The observation direction makes angle theta with +z.

1. Show that the field is transverse and find its magnitude.

2. Compare the field amplitudes on the axis, at 30 degrees, and in the equatorial plane.

Question 5 - Solution

A dipole antenna radiates only its transverse acceleration component

(a) Method and calculation

$$\hat n\cdot\mathbf E=C[(\ddot{\mathbf p}\cdot\hat n)-(\ddot{\mathbf p}\cdot\hat n)]=0.$$$$|\mathbf E|^2=C^2a^2(1-\cos^2\theta)\Rightarrow\boxed{|E|=|Ca|\sin\theta}.$$

(b) Interpretation and further calculation

$$\boxed{|E(0)|=0,\quad|E(30^\circ)|=|Ca|/2,\quad|E(90^\circ)|=|Ca|}.$$$$\boxed{|E(30^\circ)|/|E(90^\circ)|=1/2}.$$

Physical check

The angular null follows from projecting out the longitudinal component; it is not a slower propagation speed along the axis.