# Magnitude of a current

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.

• If the current density is uniform in the conductor, it really doesn't matter. Think about how a surface integral works - you take the dot product of the current density and the area vector. Mar 4, 2020 at 20:27

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.

• But here I am assuming we are not choosing a cross-section.
– user248666
Mar 4, 2020 at 21:23
• @AngeloGiannuzzi, you can choose an surface that cuts through the wire completely. It doesn't have to be perpendicular to the axis of the wire, and it doesn't have to be a flat surface. It does have to completely cut through the wire. Whether you call that a "cross-section" or not might depend on some subtlety of the terminology that you know about and others aren't familiar with. Mar 4, 2020 at 22:00
• Indeed when the book Halliday et al. defines the current i = dq/dt, it states that dq is the amount of charge that passes in time dt through a hypothetical surface that cuts across the conductor.
– user248666
Mar 5, 2020 at 10:41
• But lots of books always use a plane perpendicular to the conductor's axis, so for me it was difficult to immagine a different situation.
– user248666
Mar 5, 2020 at 10:43

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.