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Do these two electromagnetic waves interfere constructively or destructively?

Physics Asked on March 15, 2021

let’s consider two electromagnetic waves with same frequency that meet at the same point P in the figure.

enter image description here

where:

  1. $vec{S_1}$ and $vec{S_2}$ are the Poynting vectors of the two waves

  2. the ring symbol for the magnetic field indicates it exits from the plane of this page

Now I’m thinking in terms of phasors. Suppose that $|vec{E_1}| = |vec{E_2}|$ and $|vec{H_1}| = |vec{H_2}|$ (and so also $|vec{S_1}| = |vec{S_2}|$. In the previous picture I’m considering the situation in which the magnetic fields have the same phase ($angle{H_1} = angle{H_2}$), while the electric field have opposite phase ($angle{E_1} = – angle{E_2}$).

So, my questions are:

  1. Do the two waves interfere constructively or destructively? I don’t know because the sum of the two electric fields is 0 at any time, while the sum of the two magnetic field is double with respect to a single magnetic field.

  2. The magnetic field seems to be doubled and the electric field seems to be neglected. What will I measure if I put a device which measures H and E at the point P?

One Answer

It's just a standing wave.

At point P there would be no electric field but an oscillating H-field. It is a node of the standing wave in E. The nodes of the H-field are displaced by a quarter of a wavelength.

For example, if P is at the point $x=0$ and your waves are travelling along the x-axis, then the combined electric field is $$ E = 2E_0 sin(kx)sin(omega t),$$ whereas the combined H-field will be $$ H = 2H_0cos(kx)cos(omega t).$$

Correct answer by ProfRob on March 15, 2021

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