# Finding displacement current in an AC capacitor circuit

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1. Mar 13, 2017

### Peter Andrews

1. The problem statement, all variables and given/known data
A capacitor is made of two parallel plates of area A, separation d. It is being charged by an AC source. Show that the displacement current inside the capacitor is the same as the conduction current.

2. Relevant equations
Idisp = ε(dΦE/dt)
Q = CV
C = Aε/d
Xc = 1/(2πƒC)
Q(t) CV(1-e-t/τ)

3. The attempt at a solution
First of all, as it's an AC circuit so we won't be completely charging the capacitor, so I don't think we will use the exponential charging equation for a capacitor.
The displacement current can be obtained by differentiating Q=CV.
That gives (dQ/dt) = C(dV/dt). Assuming V = Vosin(2πft), we get a suitable expression for Id. This leads to a current answer. But of we want to use the electric flux equation then how can we do that?

2. Mar 13, 2017

### cnh1995

What is the equation for electric flux in terms of charge Q?

3. Mar 13, 2017

### Peter Andrews

Can use the Guass theorem, qenclosed/ε = ∫E.ds (closed integral, I don't know how to insert that sign). But what Guassian surface to choose?

4. Mar 13, 2017

### cnh1995

You can assume the electric field to be uniform. So, electric flux will be simply electric field*area. What is the relevant formula for electric field here?

5. Mar 13, 2017

### Peter Andrews

But how can we assume electric field to be constant when the charge is changing? The electric field in a capacitor is σ/ε. σ arial charge density. If we use that, then:
Φ = A.E
= Q/ε
Now differentiating,
(dΦ/dt) = (DQ/dt)/ε
So ε(dΦ/dt) = (DQ/dt)
Hence Idisplacement is dq/dt following which I come back to the process given in my solution above. Is there no other way via which I can get the result, without having to apply dq/dt on capacitor?

6. Mar 13, 2017

### cnh1995

Right.
The field is changing but it is uniform at any instant.

7. Mar 13, 2017

### Peter Andrews

Exactly. Constant and uniform have different meanings, at least in fields.

8. Mar 13, 2017

### cnh1995

That is why I immidiately changed it to 'uniform'.