# Divergence of the Poynting vector with static fields

The Poynting vector is an energy flux density, and applying the divergence theorem shows that: $$\int_V (\nabla \cdot \vec{S}) dV = \oint_{\partial V} \vec{S} \cdot d\vec{a} = -\frac{\partial U}{\partial t} - \vec{J}_{free} \cdot \vec{E}$$

I'm fine with this in various dynamic cases where there should be energy flows, but there are several instances I've come across where the energy density should not be changing but the divergence of $\vec{S}$ is non-zero.

For instance, consider a very small toroid between the plates of a very large capacitor. Let's assume that the capacitor is large enough that the electric field is constant, and the turn density on the toroid is large enough that the magnetic field due to the (physical) circumferential current is negligible. Essentially, an ideal toroid and ideal capacitor.

Assume the toroid is centered on the origin, oriented along the z-axis, and the electric field is in the x-direction (to the right). The magnetic field inside the toroid is in the phi-direction. Then the Poynting vector on the positive x (right-hand) side of the toroid points in the z-direction, and it points down on the negative x (left-hand) side of the toroid. Thus the divergence of is positive on the lower-right and upper-left sides of the toroid, and negative on the upper-right and lower-left sides.

However I see no way for the energy in these regions of space to be changing. The energy density in the field is not changing, and though there is Joule heating occurring throughout the wire of the toroid it should uniformly produce a negative divergence at the surface. (And if we make the toroid superconducting then there should be no energy changes at all). If the surface for the surface integral above encloses the entire toroid there is no problem since $\vec{S}$ is zero outside the toroid. However a small surface centered on one of the surfaces apparently has energy flowing into it.

The end of Wikipedia's article on the Poynting vector has a brief discussion about angular momentum stored in the field, but that was with a circulating $\vec{S}$ with no divergence. Similarly there is a StackExchange question also dealing with static fields, but again without divergence.

How can I interpret the divergence of the Poynting vector in cases like this?

• Placing the time derivative of total energy on the right-hand side is tempting, but it is not used in any substantial way in macroscopic theory, because can't really transform all the other terms of the Poynting theorem such as $\int\mathbf j\cdot\mathbf E\,dV$ into time derivative of a function. We do not even have general equations for matter to use for $\mathbf j$. Commented Feb 18, 2015 at 19:27
• "Thus the divergence of is positive on the lower-right and upper-left sides of the toroid, and negative on the upper-right and lower-left sides." How do you mean? If EM field is constant in time, in vacuum $\mathbf S$ has zero divergence. It can only have non-zero divergence if $\mathbf j$ is non-zero - in current-conducting wire, for example. Commented Feb 18, 2015 at 19:29
• @ján-lalinský To your first comment: yes, I was lazy. Formally the right-hand side should be the time derivative of the stored energy in the fields plus the Joule heating term. However you can interpret heating as power flowing from the fields into heat, so I included both terms. Separating them out still doesn't help the issue, though, since there is no change in stored energy in the field, and Joule heating should be constant throughout the wire (or zero for a superconductor). Commented Feb 18, 2015 at 20:13
• @ján-lalinský To your second comment: E is constant everywhere in the region of interest, but B is not - in particular it's zero outside the toroid and nonzero inside. Thus on the right half of the toroid S points up inside the toroid, and is zero outside. Hence the divergence of S is positive on the lower-right and negative on the upper-right. Essentially S inherits the surface discontinuity of B due to J, and the cross product turns the discontinuity in parallel components of B into a discontinuity in perpendicular components of S. Commented Feb 18, 2015 at 20:20
• What I meant is you assume fields are constant in time. This already implies divergence of $\mathbf S$ vanishes at all points of vacuum. In your example, the divergence vanishes both inside and outside the torus, but it does necessarily vanish in the region where the wires are - the product $\mathbf j\cdot\mathbf E$ is not zero anymore there, since the wires carry current. Commented Feb 18, 2015 at 20:29