Electronic – Capacitor in presence of an external electric field

capacitorchargeelectric fieldelectromagneticelectromagnetism

Let's consider a capacitor made of a couple of parallel metal strips (suppose they are made of perfect electric conductor) as shown in the figure, which represents a little capacitor.

Suppose an external (time variant) electric field is applied in the space in which the capacitor is placed, as shown in figure.
enter image description here

Such an electric field may be for instance that of an incident orthogonal electromagnetic wave, as shown in the following picture.

enter image description here

What happens? I don't know which are cause and effects in this situation.

I'd say that:

  1. The external electric field induces separation of charges in the two strips. So, plus charges on one strip and minus charges on the other.

  2. The previous situation is like that happens on a capacitor when supplied by a voltage source. But, in such a case, I'd say that the voltage source provides charges to the capacitor metal plates, and then the resulting charge distribution generates the electric field. In this case, however, the electric field is already existing since it's the external electric field (the incident wave).

In other words, is the electric field between the plates the original external incident E field or is it the E field generated by the charges separated by the original incident E field?

Best Answer

In other words, is the electric field between the plates the original external incident E field or is it the E field generated by the charges separated by the original incident E field?

Using Superposition theorem

Electric field between the plates is due to

1.Electric field of electromagnetic wave 2.induce charges of 1st plate 3.induce charges of 2nd plate .

Net electric field between the plates $$=\vec(E)+\vec(E_1)+\vec(E_2)+\vec(E_3)+\vec(E_4)$$

But if plates are very very close to each other then we can assume that they will behave as parllel sheet of infinite length and we get net electric field between plates is $$\vec(E)$$ because between plates $$\vec(E_1)+\vec(E_2)+\vec(E_3)+\vec(E_4)=0$$

and final distribution will look like -

enter image description here

And from uniqueness theorem this distribution is unique(but function of time ) for a given wave and charges on conductor (=0).

But what if plates are not very close to each other ?

Then we cannot assume as parllel sheet of infinite length and hence a distorted field will be obtained in between of plates and for that we have to solve Laplace equations with suitable boundary conditions which is too complicated

Note-$$\vec(E),\vec(E_1),\vec(E_2),\vec(E_3),\vec(E_4) $$ are electric field due to electromagnetic wave, due to induce charges on 1st plate (2 surface) and due to induce charges on 2nd plates (2surface) respectively.