Admin and Founder of The Secrets of the Universe and former intern at Indian Institute of Astrophysics, Bangalore, I am a science student pursuing Master’s in Physics from India. I love to study and write about Stellar Astrophysics, Relativity& Quantum Mechanics.
This article on Poynting Theorem is a guest article by Ariana Vlad, senior at the International Computers High School of Bucharest, Romania, where she focuses on studying Physics and Mathematics.
Meaning of Equation
"The transfer of energy by an electromagnetic wave is at right angles to both electric and magnetic components of the wave vibration and its rate is proportional to the vector product of their amplitudes”
The theorem was first formulated by John Henry Poynting while he was a professor at the University of Birmingham and published in late 1884. It links physical quantities like u - the energy density stored in the electromagnetic field -, S - the energy flux through an orthogonal surface (the Poynting vector) -, and J - which measures the rate at which charges go through a closed surface. Poynting introduced another groundbreaking theorem prior to this one: the eponym vector.
The Poynting Theorem is the equivalent of the work-energy balance - first applied in mechanics -used in electromagnetism. It also shows similarities to the continuity equation in hydrodynamics because it relates terms with energy flux dimension.
Individual Terms In Poynting Theorem
The first term is due to the change in the total energy stored in the given volume. It comes from the fluctuations in energy due to time variations of electric and magnetic fields and symbolizes the energy carried away by the electromagnetic wave. The minus sign links the value of the derivative to the sign convention: negative energy flows out of the system, while positive energy flows into the system. It should be noted that since this is the local form of the equation, partial derivatives are used. “Local” conservation refers to the idea that a charge can only move from one place to another if some conditions changed between the two places.
The second term is the divergence of the Poynting vector. This physical quantity represents the directional energy flux of an electromagnetic field (the energy transfer per unit area per unit time). For an electromagnetic wave, it points to the direction of propagation, representing the mean to express the radiative pressure. Divergence is a vector operator that gives the scalar measure of a vector field source in a given point.
Using vector calculus, one can find the formula for the Poynting vector:
The vector product indicates a direction perpendicular to both the electric and magnetic fields. An interesting exercise to try out is to find the Poynting vector in a simple electric circuit. The result is an explanation for a well-known fact: the energy flows from bigger to smaller electric potential.
The first two terms represent the field’s energy, which is not conserved. The easiest way to explain this belongs to Feynmann. When someone turns on the light switch in a dark room, it fills with light, and therefore with energy that wasn’t there before. In this case, the energy of matter balances the apparent paradox. One has to take into account that the electromagnetic field interacts with matter and does work on its surroundings. Therefore, the last part of the theorem represents the work done by the Lorentz force. Since work is a scalar product between force and displacement and the magnetic force is perpendicular to the velocity at any point, only the electric force gives a non-zero component.