# Equipotential Lines

In 2D why exactly do lines of equal potential exist on a point charge at rest? This doesn't make sense to me if the point charge is hypothetically a lone charge undisturbed by other charges.

• Can you elaborate a bit on what you're asking? Typically we say you can't define the potential at the actual location of a point charge, so we wouldn't draw any equipotential line through the point charge. – The Photon Nov 10 '19 at 0:03

In 3 dimensions, equipotential surfaces are a family of surfaces such that Electrostatic Potential $$\phi$$ is $$\textbf{constant}$$ along them. A direct consequence of the fact that $$\vec{E}=-\nabla\phi$$ is, using this result from Vector Calculus, that the electric field $$\vec{E}$$ points in some direction $$\textbf{normal}$$ (or perpendicular) to surfaces $$\phi=constant$$, i.e., equipotential surfaces. This way, in the case of a point charge, the electric field produced is radial according to Coulomb's Law, so the equipotential surfaces are spherical shells centered at the position of the charge. A physical consequence of the existence of equipotential surfaces is that the Work exerted upon a charge immersed in some electric field is zero along some path contained in those surfaces.