Rate of work done by fields in a finite volume

Question

In Jackson's Classical Electrodynamics, the rate of work done by fields in a finite volume is defined as $$\int _{v}\vec{J}\cdot\vec{E}\,d^{3}x^{'}$$ How?

Answer 1

Jackson states why this is the case beforehand:

For a single charge $q$ the rate of doing work by external electromagnetic fields $\mathbf E$ and $\mathbf B$ is $q\mathbf v\cdot\mathbf E$, where $\mathbf v$ is the velocity of the charge.... If there exists a continuous distribution of charges and current, the total rate of doing work by the fields in a finite volume $V$ is $$\int_V\mathbf J\cdot\mathbf E\,d^3x$$

For continuous distributions of charges. $$ q=\int\rho\,d^3x $$ which is used to change the charge $q$ into the volume charge density $\rho$. The fact that magnetic forces of point charges do no work allows us to ignore the magnetic component. Otherwise, this is an application of power $$ \frac{dW}{dt}=\mathbf F\cdot \mathbf v $$ for the Lorentz force.