The Poynting vector describes electromagnetic energy flux. Maxwell's equations imply a deeper result: a local conservation law that tracks how electromagnetic field energy changes, flows, and is transferred to matter.

In vacuum the electromagnetic field energy density is

$$u=\frac{\varepsilon_0}{2}E^2+\frac{1}{2\mu_0}B^2.$$

The Poynting theorem is

$$\boxed{\frac{\partial u}{\partial t}+\nabla\cdot\mathbf S=-\mathbf J\cdot\mathbf E},$$

where

$$\mathbf S=\frac{1}{\mu_0}\mathbf E\times\mathbf B$$

is the Poynting vector and $\mathbf J$ is current density.

Meaning of the three terms

The term

$$\frac{\partial u}{\partial t}$$

is the rate of change of electromagnetic energy stored per unit volume.

The divergence

$$\nabla\cdot\mathbf S$$

measures the net electromagnetic energy flux leaving a small region.

The term

$$\mathbf J\cdot\mathbf E$$

is the power per unit volume delivered by the electric field to charged matter. For example, in a resistive conductor it represents conversion of electromagnetic energy into internal energy.

The equation can therefore be read as

$$\text{increase of field energy} +\text{energy flowing out} =-\text{energy delivered to matter}.$$

Integral form

Integrating over a fixed volume $V$ gives

$$\frac{d}{dt}\int_Vu,dV +\int_V\nabla\cdot\mathbf S,dV =-\int_V\mathbf J\cdot\mathbf E,dV.$$

Applying the divergence theorem to the flux term,

$$\int_V\nabla\cdot\mathbf S,dV =\oint_{\partial V}\mathbf S\cdot d\mathbf A,$$

so

$$\boxed{\frac{d}{dt}\int_Vu,dV +\oint_{\partial V}\mathbf S\cdot d\mathbf A =-\int_V\mathbf J\cdot\mathbf E,dV}.$$

This states that field energy inside the volume can decrease in two ways: energy can cross the boundary, or the field can do work on matter inside.

Example: steady energy delivery to a resistor

Consider a volume enclosing a resistor in a steady DC circuit. Once the electromagnetic fields have reached steady values,

$$\frac{d}{dt}\int_Vu,dV=0.$$

The theorem becomes

$$\oint_{\partial V}\mathbf S\cdot d\mathbf A =-\int_V\mathbf J\cdot\mathbf E,dV.$$

The right-hand side is negative because the field transfers energy to the resistor. Consequently there is a net inward Poynting flux through the surface surrounding it.

The electrical energy dissipated in the resistor is therefore supplied through the electromagnetic field around the circuit, not by an instantaneous transfer of energy along an abstract circuit connection.

The Poynting theorem is the electromagnetic energy-conservation law in local form. It extends the introductory interpretation of $\mathbf S$ as energy flux into a precise balance among stored field energy, transported field energy, and work done on matter.