The Traveling Electromagnetic Wave, Quantitatively — Question 1

Question 1

A fully specified field reveals its carrier scales and first maximum

In SI units a vacuum wave is $\mathbf E=4.00\cos(20\pi x-6.00\pi\times10^9t+\pi/3)\hat y$ V/m. Use $c=3.00\times10^8$ m/s.

1. Find frequency, wavelength, propagation direction, and magnetic-field expression.

2. At x=0 find the first positive time at which Ey reaches its positive maximum.

Question 1 - Solution

A fully specified field reveals its carrier scales and first maximum

(a) Method and calculation

$$k=20\pi\Rightarrow\lambda=2\pi/k=0.100\,\mathrm m,\quad f=\omega/(2\pi)=3.00\,\mathrm{GHz}.$$$$\boxed{\mathbf B=(1.3333\times10^{-8}\,\mathrm T)\cos(20\pi x-6.00\pi\times10^9t+\pi/3)\hat z};\quad\hat n=+\hat x.$$

(b) Interpretation and further calculation

$$x=0:\quad-\omega t+\pi/3=0\Rightarrow t=\pi/(3\omega).$$$$\boxed{t_{\max}=5.5556\times10^{-11}\,\mathrm s=55.56\,\mathrm{ps}}.$$

Physical check

The phase coefficient is angular frequency, not cyclic frequency; omitting 2pi would give the wrong period.