The Traveling Electromagnetic Wave, Quantitatively — Question 3

Question 3

A Gaussian electric pulse is an exact one-direction vacuum field profile

Let $\xi=x-ct$ and $E_y=F(\xi)=E_0e^{-\xi^2/a^2}$ with a positive. Seek a +x vacuum wave without static field offsets. Use the one-dimensional curl equations $\partial_xE_y=-\partial_tB_z$ and $-\partial_xB_z=c^{-2}\partial_tE_y$.

1. Find Bz and verify both curl equations for arbitrary differentiable F.

2. Find the full width at half maximum of the electric-field amplitude in space and in a fixed-point time trace.

Question 3 - Solution

A Gaussian electric pulse is an exact one-direction vacuum field profile

(a) Method and calculation

$$\boxed{B_z=F(\xi)/c};\quad\partial_xE_y=F^{\prime},\quad-\partial_tB_z=-(-c)F^{\prime}/c=F^{\prime}.$$$$-\partial_xB_z=-F^{\prime}/c=c^{-2}(-cF^{\prime})=c^{-2}\partial_tE_y.$$

(b) Interpretation and further calculation

$$F=E_0/2\Rightarrow\xi^2/a^2=\ln2\Rightarrow\boxed{\Delta x=2a\sqrt{\ln2}}.$$$$\boxed{\Delta t=\Delta x/c=2a\sqrt{\ln2}/c}.$$

Physical check

These widths describe field amplitude; the intensity half-maximum width is smaller by a factor of square root of two.