The Traveling Electromagnetic Wave, Quantitatively — Question 2

Question 2

A phase measurement needs a wavelength bound to resolve cycle ambiguity

At one instant, the phase increase between two probes separated by d=0.300 m along +x is pi/2 modulo 2pi. The wave travels +x and its wavelength is known to lie between 0.80 m and 1.50 m. A fixed probe measures period T=4.00 ns.

1. List the possible positive wave numbers and use the wavelength bound to select one.

2. Find the wavelength and phase speed, and test consistency with vacuum propagation.

Question 2 - Solution

A phase measurement needs a wavelength bound to resolve cycle ambiguity

(a) Method and calculation

$$kd=\pi/2+2\pi m\Rightarrow k_m=\frac{\pi/2+2\pi m}{d},\quad m=0,1,2,\ldots.$$$$\lambda_m=\frac{2\pi}{k_m}=\frac{4d}{1+4m}\Rightarrow\boxed{m=0}.$$

(b) Interpretation and further calculation

$$\boxed{\lambda=1.20\,\mathrm m,\quad k=5\pi/3\,\mathrm{rad/m}}.$$$$\boxed{v_\phi=\lambda/T=3.00\times10^8\,\mathrm{m/s}}\quad\text{consistent with vacuum}.$$

Physical check

Without the wavelength bound, the phase difference alone cannot distinguish an added whole number of cycles.