The Unapologetic Mathematician

Mathematics for the interested outsider

Conservation of Electromagnetic Energy

Let’s start with Ampère’s law, including Maxwell’s correction:

\displaystyle\nabla\times B=\mu_0J+\epsilon_0\mu_0\frac{\partial E}{\partial t}

Now let’s take the dot product of this with the electric field:

\displaystyle E\cdot(\nabla\times B)=\mu_0E\cdot J+\epsilon_0\mu_0E\cdot\frac{\partial E}{\partial t}

On the left, we can run a product rule in reverse:

\displaystyle B\cdot(\nabla\times E)-\nabla\cdot(E\times B)=\mu_0E\cdot J+\epsilon_0\mu_0E\cdot\frac{\partial E}{\partial t}

Now, Faraday’s law tells us that

\displaystyle\nabla\times E=-\frac{\partial B}{\partial t}

so we can write:

\displaystyle-B\cdot\frac{\partial B}{\partial t}-\nabla\cdot(E\times B)=\mu_0E\cdot J+\epsilon_0\mu_0E\cdot\frac{\partial E}{\partial t}

Let’s rearrange this a bit:

\displaystyle-\frac{1}{\mu_0}B\cdot\frac{\partial B}{\partial t}-\epsilon_0E\cdot\frac{\partial E}{\partial t}=\nabla\cdot\left(\frac{E\times B}{\mu_0}\right)+E\cdot J

The dot product of a vector field with its own derivative should look familiar; we can rewrite:

\displaystyle-\frac{\partial}{\partial t}\left(\frac{1}{2\mu_0}B\cdot B-\frac{\epsilon_0}{2}E\cdot E\right)=\nabla\cdot\left(\frac{E\times B}{\mu_0}\right)+E\cdot J

But now we should recognize almost all the terms in sight! On the left, we’re taking the derivative of the combined energy densities of the electric and magnetic fields:

\displaystyle U=\frac{\epsilon_0}{2}\lvert E\rvert^2+\frac{1}{2\mu_0}\lvert B\rvert^2

The second term on the right is the energy density lost to Joule heating per unit time. The only thing left is this vector field:

\displaystyle u=\frac{E\times B}{\mu_0}

which we call the “Poynting vector”. It’s really named after British physicist John Henry Poynting, but generations of students remember it because it “points” in the direction electromagnetic energy flows.

To see this, look at the final form of our equation:

\displaystyle-\frac{\partial U}{\partial t}=\nabla\cdot u+E\cdot J

On the left we have the rate at which the electromagnetic energy is going down at any given point. On the right, we have two terms; the second is the rate electromagnetic energy density is being lost to heat energy at the point, while the first is the rate electromagnetic energy is “flowing away from” the point.

Compare this with the conservation of charge:

\displaystyle-\frac{\partial\rho}{\partial t}=\nabla\cdot J

where the rate at which charge density decreases is equal to the rate that charge is “flowing away” through currents. The only difference is that there is no dissipation term for charge like there is for energy.

One other important thing to notice is what this tells us about our plane wave solutions. If we take such an electromagnetic wave propagating in the direction k and with the electric field polarized in some particular direction, then we can determine that

\displaystyle u=\frac{E\times B}{\mu_0}=\frac{\lvert E\rvert^2}{\mu_0c}k=\epsilon_0c\lvert E\rvert^2k

showing that electromagnetic waves carry electromagnetic energy in the direction that they propagate.

February 17, 2012 Posted by | Electromagnetism, Mathematical Physics | 3 Comments

   

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