This question is motivated off 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 for energy in of 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 Griffths 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.

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.