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What is the difference between electrostatic and electric potential in a circuit?

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    $\begingroup$ If you have a circuit with a current then you don't have an electrostatic situation. It is either one or the other. But the potential difference has the same definition in both cases: work per unit charge. $\endgroup$
    – nasu
    Jun 13 at 2:26

4 Answers 4

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They're usually considered identical when speaking, but there are some details to keep in mind.

Electrostatic potential is defined because, with the static regime hypothesis:

$$\vec{\nabla}\times\vec{E}=\vec{0} \quad\Rightarrow\quad\vec{E}=-\vec{\nabla}V$$

This electrostatic potential satisfies Kirchhoff's voltage law as is well-known.

But outside of the static regime, $\vec{\nabla}\times\vec{E}$ is no longer zero, so at first glance $V$ no longer exists. However:

$$\vec{\nabla}.\vec{B}=0 \quad\Rightarrow\quad\vec{B}=\vec{\nabla}\times\vec{A}$$

Because of that:

$$\vec{\nabla}\times\left(\vec{E}+\frac{\partial\vec{A}}{\partial t}\right)=\vec{0} \quad\Rightarrow\quad \vec{E}=-\vec{\nabla}V-\frac{\partial\vec{A}}{\partial t}$$

This new electric potential $V$ has different properties, in particular Kirchhoff's voltage law seems to be lost. There's a trick, however: if there's magnetic induction going on, simply add a (fictitious) potential difference $e$ to the circuit, and Kirchhoff's voltage law is restored if $e$ is taken into account.

This voltage $e$ is the electromotive force from Faraday's law.

As long as there's no induction in the circuit, this extra term doesn't appear, and the electric potential satisfies normal Kirchhoff's voltage law just like the electrostatic one.

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    $\begingroup$ Employment of KVL demands accounting for induction. Much confusion results if this isn't done. $\endgroup$
    – John Doty
    Jun 14 at 9:48
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Usually They both are same. There is no difference, while studying electrostatics sometimes mentors prefers to use the word Electrostatic potential instead of saying Electric potential else both are the same.

Potential difference is defined as the negative of workdone by conservative coulombic force per unit positive charge or workdone by an external force per unit positive charge from one point to another.

That's the definition for both the words.

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The electric potential and electrostatic potential are different words for the same electric phenomena in a circuit. It is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in an electric field.

See: @ https://en.wikipedia.org/wiki/Electric_potential

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    $\begingroup$ Just a nitpick: your answer correctly covers "ordinary" electricity, but as the article you linked explains, this definition is only approximately valid outside of electrostatic. I think the very point of the question was to discuss what happens outside of electrostatic (or very low frequence regime). $\endgroup$
    – Miyase
    Jun 13 at 2:21
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Electrostatic has electropotential.

In other physical terms, you might think of electrostatic as a collection of marbles being held above the ground and electropotential as being that collection's height above the ground, giving it a potential energy and a potential max velocity. Its not EXACTLY like that, but is almost like that.

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