Pre-Quantum Electrodynamics
The Curl of \({\bf E}\)ems.es.ef.cE
As a consequence of \ref{Gr(2.22)}, we immediately see that the work done when travelling in any closed path vanishes, that is:
\[ \oint {\bf E} \cdot d{\bf l} = 0 \tag{ointE0}\label{ointE0} \]
which by Stokes' theorem implies that the electrostatic field is curlless,
\[ {\boldsymbol \nabla} \times {\bf E} = 0. \tag{curlE0}\label{curlE0} \]
By the superposition principle, this therefore applies to any combination of electrostatic fields.
One important point to make is that this fact is only related to the fact that electrostatic forces are central forces, i.e. they act purely radially along the line connecting the point charges. It has nothing to do with the fact that the force falls off with \(1/r^2\): any central force, irrespective of how it falls off, has the property that the work done is independent of the path.
Created: 2022-02-07 Mon 08:02