Joule's law

Joule’s law is a statement of the dissipated Ohmic power $P$ inside a volume $v$:

$$\frac{dP}{dv}=\vec{E}\cdot \vec{J}=\sigma \vec{E}\cdot \vec{E}=\sigma |\vec{E}{|}^2$$ $$P={\iiint }_v\vec{E}\cdot \vec{J}dv={\iiint }_v\sigma |\vec{E}{|}^{2}dv$$

Derivation

The differential power is given by:

$$dP=\frac{dW}{dt}$$

where $dW$ is the differential work done by the field force to move $dq$ over $d\vec{l}$. We say that $dW$ is given by:

$$dW=\vec{F}\cdot d\vec{l}=dq\vec{E}\cdot d\vec{l}$$

The equivalence is from the statement of electric potential. Substituting that into $dP$:

$$dP=\frac{dW}{dt}=\frac{dq}{dt}\vec{E}\cdot d\vec{l}$$ $$=(\vec{J}\cdot d\vec{s})(\vec{E}\cdot d\vec{l})=(\vec{E}\cdot \vec{J})(d\vec{s}\cdot d\vec{l})$$

where $dq/dt=di=\vec{J}\cdot d\vec{s}$. Technically speaking this is inconsistent with how we view current. Prof Stickel says that we’re discussing an infinitesimal current so this is fine.

$$d\vec{s}\cdot d\vec{l}=dv$$

If we “divide” and take both sides by $dv$:

$$\frac{dP}{dv}=\vec{E}\cdot \vec{J}$$

As above.