# Where does the energy go to for a charged sphere with pulsing radius?

In the question A charged sphere with pulsing radius the answer says that the charged sphere does not radiate. However, compressing a sphere of charge to a smaller radius requires work, so where does the electromagnetic energy needed to compress the charged sphere periodically go to?

The energy required to compress the sphere of charges goes into the electric field which has an associated energy density of $$\frac{1}{2}\epsilon_0 \mathbf E^2$$. As the radius of the sphere decreases, the integral of this quantity increases. Per Poynting's theorem the amount of this increase is equal to the electrical work done compressing the sphere.
• Thanks, are there sources for potential wave solutions for $\phi$ and $\overrightarrow{A}$ for which $\overrightarrow{E}=\overrightarrow{B}=0$ globally? So that $$0=\overrightarrow{E}=-\nabla\phi-\partial_{t}\overrightarrow{A}$$ and $$0=\overrightarrow{B}=\nabla\times\overrightarrow{A}$$ – v217 Aug 5 at 13:44
• Yes, you can just start with $\phi = 0$ and $A=0$ and then do any gauge transformation to get a solution for which $E=B=0$ globally. For instance, $\phi = -\sin(t-x-y-z)$ and $A=\left(\sin(t-x-y-z),\sin(t-x-y-z),\sin(t-x-y-z)\right)$ is a potential wave with no E or B fields. – Dale Aug 5 at 17:48