Different expressions for energy stored in an electromagnetic field

Question

This question is motivated by my readings in Griffiths. I've tried to be thorough in my background, but my final question is at the bottom.

In the electrostatics section, Griffiths gives the first description of energy in an electric field as:

$$ W_{E1} = \frac{\epsilon_0}2 \int{|\vec{E}|^2dV} $$ Later he also defines: $$ W_{E2} = \frac{1}2 \int{\vec{D} \cdot \vec{E}dV} $$ But he is quick to specify that the expression by $W_{E2}$ is the energy only of free charges, whereas $W_{E1}$ contains the energy of total charges, including both free and bound. This is illustrated in an example from the book here, where the energy of a uniformly polarized sphere is found to be zero from $W_{E2}$, but non-zero from $W_{E1}$.

Yet when he discusses magnetic fields in material, he emphasizes the equivalent expression for magnetic energy is given by: $$ W_{M1} = \frac{1}{2\mu_0} \int{|\vec{B}|^2dV} $$ Which I assume, by analog, should take into account all free and bound currents of a system. This would imply that the expression $$ W_{M2} = \frac{1}{2} \int{\vec{H} \cdot \vec{B}dV} $$ only takes into account energy by free currents. But Jackson describes this equation as the "most general". Finally, I have also commonly seen the expression $$ W_{M3} = \frac{\mu_0}{2} \int{|\vec{H}|^2dV} $$ which I do not understand how to interpret.

I have compared these equations by analyzing the energy stored in a magnetized or polarized sphere. Even Griffiths notices that it is "curious" that the fields inside the uniformly polarized sphere are given by $E_z =\frac{-1}{3\epsilon_0}P_0$ and inside a magnetized sphere $B_z = \frac{2\mu_0}3 M_0$. This leads to the following discrepancies. For the polarized sphere

• $W_{E1} = \frac{2\pi}{9\epsilon_0}R^3P_0^2$
• $W_{E2} =0$

For a magnetized sphere:

• $W_{M1} = \frac{4\pi\mu_0}{9}R^3M_0^2$
• $W_{M2} =0$
• $W_{M3} = \frac{2\pi\mu_0}{9}R^3M_0^2$

Can someone please help me make sense of the differences between these energy expressions? Or is there a fundamental difference between polarization and magnetization I am missing? Why is $W_{M1}$ a factor of 2 greater than $W_{E1}$? I am interested in the most general form of the expression, which will account for the total energy of the system in the presence of dielectrics or permeable materials.

Answer 1

So I found out the answer to this from Stratton's Electromagnetic Theory from 1941:

First let's clearly define what we mean by all these different fields even though it's a rather standard convention:

• $\vec{D} = \epsilon \vec{E}$
• $\vec{H} = \frac{1}{\mu} \vec{B}$
• $\vec{P} = \vec{D} - \epsilon \vec{E}$
• $\vec{M} = \frac{1}{\mu} \vec{B} - \vec{H}$

Stratton spends a great deal of time deriving expressions for electromagnetic energy from field quantities. He specifically tackles in section 2.10 the energy of a dielectric body in an electrostatic field, and covers the magnetic case in section 2.17. He derives the energy of a dielectric body placed in an external field to be calculated by:

$$W_E=-\frac{1}{2}\int_V{\vec{P} \cdot \vec{E}} \mathrm{d}v$$

This is none of the expressions I put originally! The total energy in the system is given by $W_E=\frac{1}{2} \int{\vec{E} \cdot \vec{D}}$ (i.e. $W_{E2}$ in my original post). But the energy of the dielectric body is only from the field due to the polarization of the body. This means the field we care about is (external + polarization) - (external). This treatment results in the expression with $P$. Similarly for magnetic fields:

$$W_M=\frac{1}{2}\int_V{\vec{M} \cdot \vec{B}} \mathrm{d}v$$

Stratton points out that the sign is different in the two expressions. He says this is because the work done by external forces is accompanied by a decrease of potential energy $W_E$, whereas mechanical forces exerted on a magnetized body are associated with an increase in $W_M$. This, he argues, is because magnetic energy is more similar to a kinetic energy and electric energy is more similar to potential energy. To me this is consistent with magnetic energy being associated with intertia forces and electric energy associated with reactive ones.

In conclusion:

• $W_{E1}$ is the electric energy of a field in a vaccuum

• $W_{E2}$ is the most general form of electric energy in a system with media

• $W_E=-\frac{1}{2}\int_V{\vec{P} \cdot \vec{E}} \mathrm{d}v$ is the energy associated with adding a dielectric body into an external electric field