Poisson's Equation

We have seen that the electric field generated by a set of stationary charges can be written as the gradient of a scalar potential, so that

$${\bf E} = - \nabla\phi.$$ (2.98)

[See Equation (2.17).] The previous equation can be combined with the field equation (2.54) to give a partial differential equation for the scalar potential:

$$\nabla^2 \phi = -\frac{\rho}{\epsilon_0}.$$ (2.99)

(See Section A.21.) This equation is known as Poisson's equation. (See Section 1.8.5.)