How is energy transported through empty space from the sun to the earth? The answer lies in the Poynting vector, a mathematical construct that quantifies the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic field. In this post, we will rigorously derive Poynting's Theorem starting from the work-energy principle and Maxwell's equations.
[Image of electromagnetic wave energy flow]
Derivation of the Poynting Vector and Poynting's Theorem
To derive the Poynting vector, we begin by considering the work done by electromagnetic fields on a distribution of charge. We will link the mechanical work done on charges to the decrease in energy stored within the electromagnetic fields.
1. The Rate of Work Done by Fields
Consider a volume \( V \) containing a charge density \( \rho \) and current density \( \mathbf{J} \). The force on a charge \( q \) is given by the Lorentz force law:
\[ \mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \]The magnetic force \( q(\mathbf{v} \times \mathbf{B}) \) acts perpendicular to the velocity and therefore does no work. The work is done solely by the electric field. The rate of work done (power) per unit volume is:
\[ \frac{dW}{dt} = \mathbf{F} \cdot \mathbf{v} = q\mathbf{E} \cdot \mathbf{v} \]Generalizing to a continuous charge distribution where \( \mathbf{J} = \rho \mathbf{v} \), the total power dissipated in volume \( V \) is:
Our goal is to express the term \( \mathbf{E} \cdot \mathbf{J} \) entirely in terms of the fields \( \mathbf{E} \) and \( \mathbf{B} \), thereby eliminating the source terms.
2. Substituting Maxwell’s Equations
We use the Ampère-Maxwell law to substitute for \( \mathbf{J} \):
\[ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \]Rearranging for \( \mathbf{J} \):
\[ \mathbf{J} = \frac{1}{\mu_0} (\nabla \times \mathbf{B}) - \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \]Now, take the dot product with \( \mathbf{E} \):
\[ \mathbf{E} \cdot \mathbf{J} = \frac{1}{\mu_0} \mathbf{E} \cdot (\nabla \times \mathbf{B}) - \epsilon_0 \mathbf{E} \cdot \frac{\partial \mathbf{E}}{\partial t} \]3. Applying Vector Identities
We need to transform the term \( \mathbf{E} \cdot (\nabla \times \mathbf{B}) \). We use the standard vector identity:
\[ \nabla \cdot (\mathbf{E} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{E}) - \mathbf{E} \cdot (\nabla \times \mathbf{B}) \]Rearranging this identity gives:
\[ \mathbf{E} \cdot (\nabla \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{E}) - \nabla \cdot (\mathbf{E} \times \mathbf{B}) \]Now, substitute Faraday’s Law \( \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \) into the equation above:
\[ \mathbf{E} \cdot (\nabla \times \mathbf{B}) = \mathbf{B} \cdot \left( -\frac{\partial \mathbf{B}}{\partial t} \right) - \nabla \cdot (\mathbf{E} \times \mathbf{B}) \] \[ \mathbf{E} \cdot (\nabla \times \mathbf{B}) = - \mathbf{B} \cdot \frac{\partial \mathbf{B}}{\partial t} - \nabla \cdot (\mathbf{E} \times \mathbf{B}) \]4. Assembling the Energy Terms
Return to our expression for \( \mathbf{E} \cdot \mathbf{J} \) from Section 2 and substitute the result from Section 3:
\[ \mathbf{E} \cdot \mathbf{J} = \frac{1}{\mu_0} \left[ - \mathbf{B} \cdot \frac{\partial \mathbf{B}}{\partial t} - \nabla \cdot (\mathbf{E} \times \mathbf{B}) \right] - \epsilon_0 \mathbf{E} \cdot \frac{\partial \mathbf{E}}{\partial t} \]We can simplify the time derivatives using the chain rule \( \frac{1}{2} \frac{\partial (\mathbf{X}^2)}{\partial t} = \mathbf{X} \cdot \frac{\partial \mathbf{X}}{\partial t} \):
- \( \mathbf{B} \cdot \frac{\partial \mathbf{B}}{\partial t} = \frac{1}{2} \frac{\partial B^2}{\partial t} \)
- \( \mathbf{E} \cdot \frac{\partial \mathbf{E}}{\partial t} = \frac{1}{2} \frac{\partial E^2}{\partial t} \)
Substituting these back gives:
\[ \mathbf{E} \cdot \mathbf{J} = -\frac{1}{2\mu_0} \frac{\partial B^2}{\partial t} - \frac{1}{\mu_0} \nabla \cdot (\mathbf{E} \times \mathbf{B}) - \frac{\epsilon_0}{2} \frac{\partial E^2}{\partial t} \]Rearranging the terms to group the time derivatives together:
\[ \mathbf{E} \cdot \mathbf{J} = - \frac{\partial}{\partial t} \left( \frac{\epsilon_0 E^2}{2} + \frac{B^2}{2\mu_0} \right) - \nabla \cdot \left( \frac{\mathbf{E} \times \mathbf{B}}{\mu_0} \right) \]5. The Poynting Vector Definition
The equation derived above is the differential form of Poynting's Theorem (Energy Conservation). We can identify two distinct terms:
Energy Density (u)
The term inside the time derivative represents the total energy density stored in the electromagnetic fields:
\[ u = \frac{1}{2}\epsilon_0 E^2 + \frac{1}{2\mu_0} B^2 \]The Poynting Vector (S)
The vector inside the divergence term represents the energy flux density. This is the Poynting Vector:
This vector represents the energy per unit time, per unit area, transported by the electromagnetic fields.
6. Integral Form and Interpretation
Integrating our final equation over a volume \( V \) bounded by surface \( A \):
\[ \int_V (\mathbf{E} \cdot \mathbf{J}) d\tau = - \frac{d}{dt} \int_V u \, d\tau - \oint_A \mathbf{S} \cdot d\mathbf{a} \]Physical Interpretation:
- Left Side: The work done by the fields on the charges (mechanical energy gain).
- Right Side (First Term): The rate of decrease of stored electromagnetic energy in the volume.
- Right Side (Second Term): The rate of energy flowing out of the volume across the boundary surface \( A \).
This confirms that \( \mathbf{S} \) is indeed the energy flux vector.
7. Visualizing the Poynting Vector
To fully grasp the concept, visualize the orthogonal relationship between the fields and the energy flow.
Key Diagram Description:
- The Axes: Imagine a standard 3D Cartesian system.
- Electric Field (\(\mathbf{E}\)): Vectors pointing along the Y-axis.
- Magnetic Field (\(\mathbf{B}\)): Vectors pointing along the Z-axis.
- Resultant Energy Flow (\(\mathbf{S}\)): According to the cross product \(\mathbf{E} \times \mathbf{B}\), if you curl your fingers from Y to Z, your thumb points along the X-axis. This is the direction of wave propagation and energy transfer.