# What's the electric field inside a capacitor with AC current?

In DC-circuits the Electric field can be easily calculated under the conditions the field is homogeneous: $$U = \int E \,\mathrm{ds} = E\,d$$. Now I wonder what if you apply an alternating Voltage $$U(t) = U_0 \,\sin(\omega t)$$? To the best of my knowledge that'd mean $$U_C = U_0 \,\sin(\omega t) = E\,d$$ what'd be outrageous, because the field ain't homogeneous. However, I see no other way around.

Ultimately my aim is to determine the force $$F = E\,q$$ on a charged particle inside.

• Special relativity says that information cannot travel instantly. We come to the concept of retarded potentials. Commented May 28, 2021 at 12:31
• But the frequency of the $AC$ current won't be that high. Isn't there a way to predict the electric field inside the capacitor then?
– Leon
Commented May 28, 2021 at 12:42
• The very fact that you say the field won't be homogeneous means that the voltage is changing sooo fast that the information cannot travel quickly between the plates and establish a homogeneous field. Commented May 28, 2021 at 12:47

To the best of my knowledge that'd mean $$U_C = U_0 \,\sin(\omega t) = E\,d$$ what'd be outrageous, because the field ain't homogeneous.

Electric field inside capacitor is still homogeneous even if the applied voltage is oscillating harmonically (except at boundaries of capacitor plates, but that is so even in DC).

Total electric field is composed of electrostatic component $$\mathbf E_C$$ (the Coulomb integral of charge density in all space) and induced component $$\mathbf E_i$$ (connected to magnetic field changing in time):

$$\mathbf E = \mathbf E_C + \mathbf E_i.$$

In general, voltage by definition is integral of the electrostatic part $$\mathbf E_C$$. So we can already determine electrostatic part of the field from $$E_C d = U_0\sin \omega t.$$

Induced part $$\mathbf E_i$$ in real capacitor won't be exactly zero, but it is often negligible in a well constructed capacitor (due to geometry of conductors). It would take coiled wire (loops) to create substantial induced field near the wire (like in inductor). Capacitors don't have such loops. Wires that the capacitor is connected to produce some induced field, but usually we can ignore it as it is very small. So the above equation holds very well even for total field $$E$$.

You can set up a force equilibrium with the drag force and gravitational force in between a capacitor with an electric force too and with the DC case it's the Millikan's oil-drop experiment.

I found your question by asking the same thing and coming to the same conclusion with the electric field oscillating with lower amplitude than your input voltage due to the negative line integral of voltage dotted with the plate separation vector giving you the same oscillation in electric field as voltage supplied but with your distance division.It would be cool to see an oscillating electron charged oil droplet