# Prove that a capacitor driven by an AC voltage radiates EM

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1. Dec 23, 2016

### zhouhao

1. The problem statement, all variables and given/known data
When the capacitor driven by DC voltage $V_0$,it has the electric field distribution $f(x,y,z)$
When $V=V_0e^{iwt}$,how to show the EM travel in the space forever like the light?
2. Relevant equations
$-{\nabla}^2E-u{\epsilon}\frac{{\partial}^2{E}}{{\partial}t^2}=0 (1)$
derived from Maxwell equation.

3. The attempt at a solution
I firstly think $E(x,y,z,t)=e^{iwt}f(x,y,z)$.But this is not right,since $f(x,y,z)$ makes electric field not travel too long.

2. Dec 23, 2016

### tech99

A capacitor driven by AC does not radiate as there are no charges (electrons) being accelerated, except in the connecting wires. EM radiation cannot be obtained by just creating electric and magnetic fields. The only case I can think of where a capacitor radiates is where there is a thick dielectric in which charges can be accelerated.

3. Dec 24, 2016

### Cutter Ketch

What? Why would you think there isn't charge being moved on and off the plates of the capacitor? There is a great deal of charge being accelerated and moved back and forth between the plates of a capacitor. Further the charge constitutes an oscillating dipole. The arrangement is very similar to the classic and ubiquitous dipole antenna invented by Heinrich Hertz and used to prove the existence of propagating EM waves.

Also you say EM radiation can't be created simply by creating EM fields. How do you think they are created? I will agree that not all fields radiate, but oscillating dipole fields do. Sloshing charge back and forth between sides of a dipole antenna to create an oscillating dipole field and is how we make radio.

4. Dec 24, 2016

### zhouhao

What I want to know is that which apparatus could radiate EM and whether the EM could travel like the light travelling forever in the vacuum space.

5. Dec 24, 2016

### tech99

Well, I enjoyed reading your reply! I do agree that there is a lot of charge sloshing back and forth on the plates of a capacitor, but any radiation that is happening arises from the conductor and circuit joining them together. This is because to obtain radiation we need to accelerate charges over a significant distance. This was the mechanism of radiation propose by J J Thompson and formalised by J Larmor. If Hertz had shortened his doublet to a small length, then for a given capacitor charge, the radiation would have diminished. With a closely spaced capacitor the radiation would be negligible.
You mention radiation from a dipole, and in this case you will notice that charges are being accelerated over a significant distance. Further, if we increase the strength of the E-field at the end of the dipole by using a thinner conductor, then the radiation is unaltered, so this E-field is evidently not related to radiation.