# Alternate current problem

Gold Member

## Homework Statement

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.

None given.

## 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?

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Pythagorean
Gold Member
How is that possible that the current is has an imaginary part? I guess I made an error, could you confirm?
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:
$$r \angle \theta$$

Gold Member
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:
$$r \angle \theta$$
Ok thank you, I understand.

Pythagorean
Gold Member
Ok thank you, I understand.