Magnitude of a current

Question

According to the book Halliday et al., we can define the magnitude of a current as the flow of charge passing through a cross-section of a conductor in the time unit. The book states also we can choose any kind of section (it may be not perpendicular to the conductor) to define the magnitudine of the current. I don't completely agree with this last statement because if we don't choose a perpendicular section, I guess the charge amount in the unit time can change, but I don't know if I am wrong.

Answer 1

Assuming there's no charge buildup (i.e., no one stuck a capacitor in the circuit anywhere) it doesn't matter what cross-section you choose. Mathematically, this is captured by the divergence theorem (here $J$ is current density):

$$\int_V (\nabla \cdot J) dV = \int_S (J \cdot n) dS.$$

Here's the idea; take two surfaces $S_1$ and $S_2$ with the same boundary; choose these to be the two different cross-sections you have in mind when calculating current. The difference between the two is then the right-hand side of the equation above. Using the divergence theorem, it's equal to the integral of the divergence of the current in the volume enclosed by the surfaces (left-hand side). Assuming no charge buildup that integral is zero. For a conductor there is no charge buildup, so that assumption is justified.

Answer 2

Suppose we divide a length of wire into three sections, A, B and C by two imaginary planes, P and Q. The plane, P, separating A from B is at 90° to the longitudinal axis of the wire, and the plane, Q, separating B from C is not at 90° to the axis. If more charge (or less charge) per second passes from A to B through P than passes through from B to C through Q then charge is going to pile up in (or be removed from) B. But we haven't done anything to bring this about – the planes P and Q are imaginary! We have demonstrated a reductio ad absurdum.