# Alternate current problem

1. Nov 12, 2009

### fluidistic

1. The problem statement, all variables and given/known data
I'm trying to solve the problem 8.12 in Purcell's book on Electricity and Magnetism.

The circuit is like that :

|--------------|------------|
|...................|..................R
|...................C................|
emf................|................C
|....................R...............|
|--------------|-------------
(The points represent nothing, I had to write them because otherwise the circuit wouldn't appear as I'd like).
1)Find the current passing through the emf.
2)Demonstrate that if $$V_{AB}=V_B-V_A$$ then $$|V_{AB}|^2=V_0^2$$ for all $$\omega$$.
3)Find the phase difference between the current that passes through the emf and a capacitor.
2. Relevant equations
None given.

3. The attempt at a solution
I'm currently trying to do part 1).
I forgot to mention that $$\omega$$ is the angular frequency and $$V_0$$ is the amplitude of $$V(t)$$.
What I did so far : I notice that the current through both loops is the same and is worth $$I=\frac{V(t)}{Z}$$ where $$Z=R-\frac{i}{\omega C} \Rightarrow I(t)=\frac{2V_0 \cos (\omega t + \phi)\cdot \omega C}{\omega C R-i}$$.
How is that possible that the current is has an imaginary part? I guess I made an error, could you confirm?

2. Nov 12, 2009

### Pythagorean

currents and voltages both can have an "imaginary part" (I hate the name "imaginary", so misleading).

It's usually written in phasor notation, so you'd convert the cartesian form:

$$x + iy$$
to the polar form
$$r^{i\theta}$$

and then write it in phasor notation:
$$r \angle \theta$$

3. Nov 12, 2009

### fluidistic

Ok thank you, I understand.

4. Nov 12, 2009

### Pythagorean

that I can't say for sure, but I can tell you that it's what I would have done.

5. Nov 12, 2009

### fluidistic

Thank you once again.