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When voltage is applied across a conductor, current will start to flow and its value is determined by Ohm's law. Since current is constant across a conductor, electric potential must have a gradient across a conductor because of its resistance.

Why is it that electric potential has a gradient or why is it that current is constant across a conductor?

We know from electrostatics that electric potential is a scalar quantity and that its value at any point in electric field is thus scalar sum of electric potentials of all charges at that point.

In context of electric circuits, how is electric potential developed at some point or why is it different at different points within a circuit? What physically happens so that electric potential at some point is different than on the other one? Is there some kind of charge build up at different points within a circuit?

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    $\begingroup$ When voltage is applied across a conductor, current will start to flow and its value is determined by Ohm's law. Be careful, not everything follow Ohm's law. $\endgroup$ Jun 22 '21 at 18:21
  • $\begingroup$ @BioPhysicist It does, thank you. Surface charges are responsible for potential difference created on circuit elements like resistor. However, why is it that current is the same through resistors in series? Why is it that if resistor has a bigger R that bigger voltage is created on it such that current is the same through all resistors in series? $\endgroup$ Jun 25 '21 at 14:26
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Memory is shaky, I think this covers it.

Current is constant because no charge accumulates anywhere in the circuit. Over time, as much charge leaves a region as enters it. If there's no entering and leaving, there's no current.

At it's most fundamental, Ohm's Law asserts that $\vec{J}=\sigma\vec{E}$, i.e. The current density vector is proportional to the applied electric field, where that constant of proportionality is the conductance. If we assume that electric field is constant, as implied by constant current and Ohm's Law, then it must be the negative of the potential difference, divided by the distance between field points. In other words, the potential difference causes the constant electric field and not the resistance.

Now $\vec{J}$ is on average in the direction of the current with magnitude $I/A$ where $I$ is current, and $A$ is cross-sectional area of the circuit. The electric field also points in the direction of the current and has magnitude $V/l$ where $V$ is the potential difference between two points in the circuit and $l$ is the length between them.

So: $\vec{J}=\sigma \vec{E}\implies \frac{I}{A}=\sigma\frac{V}{l}\implies I=\frac{\sigma A}{l} V$.

The reciprocal of the conductance($\sigma$) is the resistivity($\rho$) and we have $R=\frac{\rho l}{A}$ and $V=IR$.

Given $\vec{J}=\sigma \vec{E}$, take the divergence of both sides.

By the continuity equation for charge, we have that $\nabla \cdot \vec{J}=-\partial\rho/\partial t$ where $\rho$ is charge density. By Gauss' Law, $\nabla \cdot \vec{E}=\rho/\epsilon_0$.

So: $-\partial \rho/\partial t = \frac{\sigma}{\epsilon_0}\rho$

So $\rho=\rho_0e^{-\sigma t/\epsilon_0}$ by solving the differential equation.

So within the wire net charge exponentially decays when there's an applied field.

The foregoing analysis fails on the surface and in non-Ohmic materials, but for some reason that isn't covered before grad school.

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  • $\begingroup$ Yes. It seems that surface charges are responsible for creating potential difference on different circuit elements like resistors. $\endgroup$ Jun 25 '21 at 14:32
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Why is it that electric potential has a gradient or why is it that current is constant across a conductor?

Let's look at what happens at the atomic level.

On one side of the source there is an excess of electrons and on the other side they are missing. If you connect the poles with an electrically conductive wire, the electrons start moving along the wire. The potential difference (electrical voltage) between the poles determines the number of moving electrons along the wire.

  • Anyway, the number of starting electrons must be equal to the number of arriving electrons on the other side. That is what you said about the constant current.

The moving electrons constantly meet atomic trunks and other electrons and are repeatedly slowed down in the process. This characteristic of the material together with its smallest cross-section is its electrical resistance.

  • Shorten the poles of the source, you are not able to measure the potential difference. Disconnect the poles and the potential difference will be infinite. Imagine now the flow of electrons with the flow of water from a higher to a lower point. You are able to install a lot of water turbines and each turbine generates the power of its part of the potential difference (the high between the turbines). The same is the electric potential gradient.
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