Q21Engineering Physics

In the study of classical electrodynamics, it is not enough to simply know the magnitude of electric and magnetic fields; we must fundamentally understand how energy is stored in these fields and how it propagates through space. Poynting's theorem, formulated by John Henry Poynting in 1884, serves as the ultimate mathematical statement of the conservation of energy applied to electromagnetic fields.

1. Energy Stored in Electromagnetic Fields

It is a well-established fact that assembling an electric charge distribution requires mechanical work to overcome electrostatic repulsion. This work is stored as potential energy entirely within the resulting electric field. The energy density (energy per unit volume, $u_E$) of an electric field $E$ in a medium with permittivity $ϵ$ is: $u_E=\frac{1}{2}ϵE^2$

Similarly, establishing a steady current against self-inductance requires work, which is stored as potential energy in the resulting magnetic field. The energy density ($u_B$) of a magnetic field $H$ in a medium with permeability $\mu$ is: $u_B=\frac{1}{2}\mu H^2$

The total electromagnetic energy density $u$ at any point in space is simply the sum of these two components: $u=u_E+u_B=\frac{1}{2}ϵE^2+\frac{1}{2}\mu H^2$

The total electromagnetic energy $U$ stored within a finite macroscopic volume $V$ is the volume integral of this energy density: $U={\int }_V\left(\frac{1}{2}ϵE^2+\frac{1}{2}\mu H^2\right)dV$

2. Derivation and Statement of Poynting's Theorem

If the electromagnetic fields within this volume $V$ are changing over time (for instance, if an electromagnetic wave is passing through), the total stored energy $U$ will change. According to the universal law of conservation of energy, if the stored energy decreases, that energy must go somewhere. It can either be dissipated as heat (work done on charges) or it can physically flow out through the boundary surface $S$ enclosing the volume $V$.

Consider the rate at which the fields do electrical work on the free charges moving within the volume. The power dissipated per unit volume (Joule heating or Ohmic loss) is given by the dot product of the electric field $\vec{E}$ and the current density $\vec{J}$: $\text{Power Dissipated Density}=\vec{J}\cdot \vec{E}$

Using Maxwell's equations (specifically Ampere's Law with Maxwell's addition, $\nabla \times \vec{H}=\vec{J}+\frac{\partial \vec{D}}{\partial t}$, and Faraday's Law, $\nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}$), and applying complex vector calculus identities, we can strictly derive Poynting's theorem. The theorem is expressed mathematically as:

Physical Interpretation of the Theorem: The equation reads perfectly as a conservation law: - The Left-Hand Side: Represents the rate of decrease of the total stored electromagnetic energy within the volume $V$ (the negative sign indicates a decrease). - The First Term on the Right-Hand Side: Represents the total electromagnetic power flowing outward through the closed surface $S$ bounding the volume. - The Second Term on the Right-Hand Side: Represents the rate at which energy is dissipated as heat (Joule heating) due to the conduction current $\vec{J}$ flowing through the material's resistance within the volume.

3. The Poynting Vector ($\vec{S}$)

The surface integral term ${\oint }_S(\vec{E}\times \vec{H})\cdot d\vec{A}$ dictates the energy flux. The integrand itself, the cross product of the electric and magnetic field vectors, is defined as the Poynting Vector $\vec{S}$.

The Poynting vector is an extraordinarily important concept in physics. Its magnitude ($|S|$) gives the instantaneous rate of energy flow per unit area (Power/Area, measured in $W/m^2$). Its direction (determined by the right-hand rule of the cross product) indicates the precise direction in which the electromagnetic wave is propagating and transporting that energy through space.