What is the current through the emf in a circuit with capacitors and resistors?

[fluidistic]

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.

Homework Equations

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?

 

[Pythagorean]

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:
$$r \angle \theta$$

 

[fluidistic]

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.
Is my answer correct though?

 

[Pythagorean]

Ok thank you, I understand.
Is my answer correct though?

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

 

[fluidistic]

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

Thank you once again.