Energy in the Electromagnetic Field: Poynting's Theorem
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You should be comfortable with the macroscopic Maxwell equations in vacuum, the Lorentz force law, and basic vector calculus identities involving the divergence and curl of vector fields.
Why this matters
Energy conservation in electromagnetism is not an assumption but a consequence of the field equations.
In mechanics, we often postulate a potential energy function to ensure energy conservation. In electromagnetism, the situation is more profound: the conservation of energy emerges directly from the interplay between Maxwell's equations and the Lorentz force law. This result, known as Poynting's theorem, provides a local conservation law, meaning energy is conserved at every point in space and time, not just globally. It identifies the energy density stored in the fields and the flux of energy flowing through space, fundamentally changing our view of how energy is transported in physical systems.
The Idea in Plain Language
Imagine the electromagnetic field as a fluid that carries energy. Just as water flows through a pipe, electromagnetic energy flows through space. Poynting's theorem tells us that if the amount of energy in a specific region of space changes, it is either because energy is flowing in or out of that region, or because the field is doing work on the charges within that region. There is no "hidden" energy source or sink; the field itself is the carrier. This local conservation law is the electromagnetic analogue of the continuity equation for mass in fluid dynamics, ensuring that energy is neither created nor destroyed, only transferred between the field and the matter it interacts with.
Local Conservation and the Continuity Equation
The core result of this lecture is the local energy conservation law. It states that the rate of change of electromagnetic energy density \( u \) within a volume, plus the net flux of energy out of that volume, equals the negative of the work done by the field on the charges per unit volume. Mathematically, this is expressed as:
Here, \( u \) is the energy density and \( \vec{S} \) is the Poynting vector. The term \( \vec{J}\cdot\vec{E} \) represents the power per unit volume delivered to the charges by the electric field. The negative sign indicates that if the field does positive work on the charges (accelerating them), the field energy must decrease. This equation is a continuity equation, structurally identical to the charge conservation law \( \partial \rho / \partial t + \nabla\cdot\vec{J} = 0 \), but for energy rather than charge.
Field Energy Density
The energy density \( u \) is not arbitrary; it is derived from the field equations. It consists of two additive contributions: one from the electric field and one from the magnetic field. The expression is:
The first term, \( \frac{\varepsilon_0 E^2}{2} \), represents the energy stored in the electric field. The second term, \( \frac{B^2}{2\mu_0} \), represents the energy stored in the magnetic field. Both terms are positive definite, ensuring that the field energy is always non-negative. This form is analogous to the energy stored in a capacitor (\( \frac{1}{2}CV^2 \)) and an inductor (\( \frac{1}{2}LI^2 \)), generalised to every point in space. The derivation shows that these terms arise naturally when combining Maxwell's equations with the Lorentz force, confirming that the fields themselves possess energy.
The Poynting Vector: Energy Flux
The Poynting vector \( \vec{S} \) describes the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic field. It is defined as:
The direction of \( \vec{S} \) is given by the right-hand rule for the cross product \( \vec{E}\times\vec{B} \). In a plane electromagnetic wave, \( \vec{E} \) and \( \vec{B} \) are perpendicular to each other and to the direction of propagation, so \( \vec{S} \) points in the direction the wave is traveling. The magnitude of \( \vec{S} \) is proportional to the product of the electric and magnetic field magnitudes. This vector is crucial for understanding how energy is transported by light and radio waves. It is not merely a mathematical artifact; it has physical dimensions of power per unit area (\( \mathrm{W/m^2} \)), representing the actual flow of energy through space.
The derivation
Poynting's Theorem and Field Energy
Starting from the microscopic Maxwell equations and the Lorentz force law, we derive the local conservation law for electromagnetic energy: the rate of change of the field energy density \( u = \tfrac{1}{2}\varepsilon_0 E^2 + \tfrac{1}{2\mu_0} B^2 \) plus the divergence of the Poynting vector \( \vec{S} = \tfrac{1}{\mu_0}\,\vec{E}\times\vec{B} \) equals minus the rate at which the fields do work on charges per unit volume, \( \partial u/\partial t + \nabla\cdot\vec{S} = -\,\vec{J}\cdot\vec{E} \). Both \(u\) and \(\vec{S}\) emerge from the derivation rather than being postulated.
A common misconception is that the Poynting vector \( \vec{S} \) always points in the direction of energy flow in a simple, intuitive sense. While true for plane waves, in complex near-field situations (such as around a static charge or a steady current), \( \vec{S} \) can circulate or point in counter-intuitive directions. For instance, around a static electric charge, \( \vec{B} = 0 \), so \( \vec{S} = 0 \), yet energy is stored in the field. The local conservation law still holds, but the interpretation of "flow" requires careful consideration of the boundary conditions and the global integral form of the theorem. C
Worked example
Problem. A uniform plane electromagnetic wave propagates in the \( +x \)-direction in vacuum. The electric field is given by \( \vec{E} = E_0 \cos(kx - \omega t) \hat{y} \). Determine the time-averaged Poynting vector \( \langle \vec{S} \rangle \) and the time-averaged energy density \( \langle u \rangle \). Assume \( E_0 = 10\ \mathrm{V/m} \).
- For a plane wave in vacuum, \( \vec{B} = \frac{1}{c} \hat{x} \times \vec{E} \). Thus, \( \vec{B} = \frac{E_0}{c} \cos(kx - \omega t) \hat{z} \).
- Calculate the instantaneous Poynting vector: \( \vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B} = \frac{1}{\mu_0} \left( E_0 \cos(kx - \omega t) \hat{y} \right) \times \left( \frac{E_0}{c} \cos(kx - \omega t) \hat{z} \right) \).
- Simplify the cross product: \( \hat{y} \times \hat{z} = \hat{x} \). So, \( \vec{S} = \frac{E_0^2}{\mu_0 c} \cos^2(kx - \omega t) \hat{x} \).
- Time-average \( \cos^2(kx - \omega t) \) over one period. The average value is \( \frac{1}{2} \). Thus, \( \langle \vec{S} \rangle = \frac{E_0^2}{2 \mu_0 c} \hat{x} \).
- Substitute \( c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \) to get \( \langle \vec{S} \rangle = \frac{1}{2} \varepsilon_0 c E_0^2 \hat{x} \).
- Calculate the numerical value: \( \langle S \rangle = \frac{1}{2} (8.85 \times 10^{-12}\ \mathrm{F/m}) (3.00 \times 10^8\ \mathrm{m/s}) (10\ \mathrm{V/m})^2 \).
- Compute: \( \langle S \rangle = 0.5 \times 8.85 \times 10^{-12} \times 3.00 \times 10^8 \times 100 = 1.3275 \times 10^{-1}\ \mathrm{W/m^2} \approx 0.133\ \mathrm{W/m^2} \).
Units check. \( \varepsilon_0 \) is \( \mathrm{F/m} = \mathrm{C/(V \cdot m)} \). \( c \) is \( \mathrm{m/s} \). \( E_0^2 \) is \( \mathrm{V^2/m^2} \). Product: \( \frac{\mathrm{C}}{\mathrm{V \cdot m}} \cdot \frac{\mathrm{m}}{\mathrm{s}} \cdot \frac{\mathrm{V^2}}{\mathrm{m^2}} = \frac{\mathrm{C \cdot V}}{\mathrm{s \cdot m^2}} = \frac{\mathrm{J}}{\mathrm{s \cdot m^2}} = \mathrm{W/m^2} \). The units are consistent.
Time-averaged energy density: \(\langle u \rangle = \tfrac12 \varepsilon_0 E_0^2 = 0.5 \times 8.854 \times 10^{-12} \times 100 = 4.43 \times 10^{-10}\ \mathrm{J\,m^{-3}}\) (electric and magnetic parts contribute equally), and \(c\langle u \rangle = 3.00 \times 10^{8} \times 4.43 \times 10^{-10} = 0.133\ \mathrm{W\,m^{-2}} = \langle S \rangle\), as it must for a plane wave.
Three quick tests
- What is the physical meaning of the term \( \vec{J}\cdot\vec{E} \) in Poynting's theorem?
Answer
It represents the rate at which the electromagnetic field does work on the charges per unit volume (power density transferred to matter). - If \( \vec{E} \) and \( \vec{B} \) are parallel, what is the Poynting vector?
Answer
The Poynting vector is zero because the cross product \( \vec{E}\times\vec{B} \) vanishes when the vectors are parallel. - How does the time-averaged energy density \( \langle u \rangle \) relate to the time-averaged Poynting vector magnitude \( \langle S \rangle \) for a plane wave?
Answer
For a plane wave, \( \langle S \rangle = c \langle u \rangle \), meaning the energy flux is the energy density multiplied by the speed of light.
Problem set for PU-204 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes checked line by line by the editor; every number in the worked example was recomputed.