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When learning about electric current in circuits, I have learned that at each point in the circuit there is a current associated with it which is defined as the charge per unit time passing through that point. Because we assume usually that circuits are made of wires that have zero cross-sectional area.

Now, by learning further, I have found that we want to explain current using motion of charged particles in a wire and then we define something called current density vector which is defined as the one that satisfies the following property.

$$ \int \limits_{S} \boldsymbol{j} \cdot \boldsymbol{n} dS = I(S),$$ where $I(S)$ is current through surface $S$, which has normal vector at each point $\boldsymbol{n}$. $\boldsymbol{j}$ is the current density vector.

So basically we define current through a surface. Then, using this definition one can show how current density vector is related to the charge density or number density of charged particles, and also how it is related to the speed of particles.

My question is how to interpret the definition of current at points in the circuit in terms of definition using microscopic variables such as densities and velocities. Also, I feel that using different surface areas in the wire one could obtain different current values at each of the point, but experimentally one knows that there are physical quantities, such as Joule heating, which are true for current definition for circuits. How to make sure that both definitions agree and give the same physical results?

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If the current is constrained to an infinitely thin wire, then it has the following form (assuming, the wire is oriented along the $z$ axis):

$$j(\vec r) = \delta(x)\delta(y)\begin{pmatrix}0\\0\\I\end{pmatrix}$$

Then the integral over such a current will, up to a sign, give $I$ along any surface which is intersected by the $z$ axis (the wire) no matter the orientation.

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In the usual electrical circuit description one assumes that the wires are so thin that their cross section can be neglected compared to the length of the wire, e.g. concerning magnetic effects. Thus the cross section at an arbitrary location can be approximated by a "point". In reality, of course, they have a finite thickness and cross section so that your current density integral applies to obtain the total current $I$ through the wire. Due to the current continuity the current will be constant along the wire independently of the detailed shape of the surface $S$ cutting across it at a chosen location.

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