Circulation theorem

The circulation theorem expresses in integral form a structural property of the electrostatic field: its inability to do net work along a closed path. It is the counterpart, for the line integral, of what Gauss’s theorem is for flux.

Law — Circulation of the electrostatic field

The line integral of $\vec{E}$ along any closed path is zero: $\ev{\oint \vec{E}\cdot \dd\vec{\ell} = 0}$

This result expresses the fact that the electrostatic field is conservative: the work needed to move a charge from $A$ to $B$ does not depend on the path followed, only on the initial and final points. It is precisely this path-independence that makes it possible to define a potential $V$: if the work depended on the chosen route, it would not be possible to assign a unique value of potential energy to each point.

Three equivalent statements

$\text{conservative field} \iff \text{zero circulation} \iff \text{a potential } V \text{ exists}$ The three properties are faces of the same fact: proving one is equivalent to proving them all.

The link with equipotential surfaces is immediate. Travelling a closed loop is equivalent to returning to the starting point without ever “descending” more than one has “climbed” on the potential: the net work is zero and the integral of $\vec{E}$ vanishes. Every uphill stretch of the potential is exactly compensated by downhill stretches, and on return the balance is zero.

Zero of the potential for extended distributions

The potential of the infinite plane and of the wire diverges as $d\to\infty$: for extended charge distributions the zero cannot be placed at infinity. In these cases a reference point is arbitrarily chosen and assigned $V = 0$, without this altering the potential differences, which are the only physically measurable quantities.

Collegamenti

Argomenti: Campo elettrico e potenziale Concetti: Potenziale elettrico Metodi: Teorema della circuitazione

Esercizi collegati: Two connected conducting spheres · Ranking potentials · From the graph to the sign of the charges