Electric Scalar Potential

Hence, Equation (152) yields

where

is the scalar potential. (See Section 1.3.) It follows from Equation (154) that

In other words, an electric field generated by (stationary) charges is irrotational.

According to Equation (153), we can write Equation (152) in the form

(157) |

Hence,

(158) |

where use has been made of Equations (23)-(25). We deduce that

which we recognize as the first Maxwell equation. (See Section 1.2.) The integral form of this equation, which follows from the divergence theorem,

(160) |

is known as

Equations (154) and (159) can be combined to give

which we recognize as Poisson's equation, with . (See Section 2.3.) Hence, Equation (147) yields

which is equivalent to Equation (155). Incidentally, according to the analysis of Sections 2.2 and 2.3, the previous expression represents the unique solution to Equation (161), subject to the boundary condition

as | (163) |