EM wave orthoganality

It is interesting and very useful to think about two signals passing one another in a coax cable (e.g., RG-8) without interference. I show cartesian coordinates below, but in a coax, E is radial, and H is azimuthal. Also, for RG-8, E/H = 50 ohms

E_x = E₀ exp[j(k z - w t)]
H_y = H₀ exp[j(k z - w t)]

and
E_x = E₀ exp[j(-k z - w t)]
H_y = -H₀ exp[j(-k z - w t)]

So one pulse goes from left to right, and the other from right to left, without a "collision". There are instruments (directional couplers) that can measure the amplitude AND direction of each pulse simultaneously.

Edit: changed k x to k z four places

 

Have you tried taking a dot product?

Edit- I just noticed the form you wrote for E and B- spherical wavefronts. That can complicate the maths, so it's easier to use plane waves.

 

For a plane wave, we can use the vector identity (ignoring any currents J)
div(E x H) = H curl E - E curl H
= -E dD/dt - H dB/dt = -1/2 (d/dt)(e₀E² + u₀ H²). (= energy loss per meter). So
S= E x H is a vector (called the Poynting vector), whose direction is the direction, and magnitude is the average power flow, in watts per square meter. Both E and H are orthogonal to direction of power flow.

 

Actually, the OP is using plane waves (and just not specifying the direction of propagation in Cartesian coords, explicitly). Spherical waves have a much more complicated form.

 

If you are having trouble working out the derivatives, it will help you to look at my responses to question 2. in this thread since you are essentially both trying to work out the same curls....Are you also having trouble with the time derivatives?

 

You're welcome!

The scalar [itex]k[/itex] is the wavenumber, the vector [itex]\vec{k}[/itex] which appears in your plane waves (it's a good idea to get into the habit of explicitly writing vectors with some sort of marking to distinguish them from scalars!) , is defined as the product of the wavenumber with the unit vector in the direction of propagation. In other words, [itex]\vec{k}[/itex] points in the direction of propagation and its magnitude is the wavenumber.