I am having a lot of trouble trying to figure out how to find all the electric fields in and around a DC circuit. Although I have studied electric circuits for years, not much discussion has been given to these circuits' electric fields. And when I started to think about them a bit, I ran into a paradox that I am hoping someone on this forum can help resolve for me.

Now, I am aware of the relation between the electric field and voltage:

-grad V = E
​

and of the point form of Ohm's Law:

E = Jρ​

but something has to give rise to E in the first place. This means that unpaired charges need to be present within the circuit. However, since

div J = 0
​

within a conductor at steady state, these charges must be present on the surfaces of the conductors. So, from my understanding, a simple DC circuit with a series connection of three resistors of equal resistance connected to a battery would look like this with its surface charges shown:

​

But you attach a capacitor across R3 in the circuit above and you let the circuit go to steady state, you would get the following charge distribution over the circuit:

​

!?! But shouldn't the charges on capacitor plates in circuits be equal and opposite? I seem to have a paradox here. Did I mislearn something somewhere, or what? Could someone offer me some more explanation here?

 

The line integral of the E vector of the electric field is zero around any closed path, external to the battery you have shown.

It is not, however zero within the battery.

 

Hello there. In your first diagram, the charge distribution is incorrect. It needs to be red between R2 and R3. Only blue at direct connection to battery. You can measure the field by measuring the potential(voltage) at any point in the circuit. This gives the potential for the cap. I'm curious, what kind of circuits do you study?

 

how do you get a reference to measure the fields? most circuits are referenced to one side of the power supply or the other. case in point, the airconditioning repair crew here was having a problem and I found they were using the case of the system as a ground for their meter, niether side of the control transformer was grounded to the case, so all their measurements were meaningless.

 

Yes, studiot, I am aware of the fact that all closed line integrals around any E field is zero (except in the case of changing magnetic fields, which is not happening here).