Pre-Quantum Electrodynamics

• PM 7.10
• Gr 7.2.4

Work per unit time in current loop: \[ \frac{dW}{dt} = -{\cal E} I = L I \frac{dI}{dt} \] Start from zero current, integrate in time: \[ W = \frac{1}{2} L I^2 \label{Gr(7.29)} \] Nicer way (generalizable to surface and volume currents): from PLI, flux through loop is \(\Phi = L I\). But \[ \Phi = \int_{\cal S} {\bf B} \cdot d{\bf a} = \int_{\cal S} ({\boldsymbol \nabla} \times {\bf A}) \cdot d{\bf a} = \oint_{\cal P} {\bf A} \cdot d{\bf l}, \] so \[ LI = \oint {\bf A} \cdot d{\bf l} \] and \[ W = \frac{1}{2} I \oint {\bf A} \cdot d{\bf l} = \frac{1}{2} \oint ({\bf A} \cdot {\bf I}) dl \label{Gr(7.30)} \] Generalization to volume currents:

\[ W = \frac{1}{2} \int_{\cal V} d\tau ~({\bf A} \cdot {\bf J}) \tag{W_intAJ}\label{W_intAJ} \] Even better: use Ampère, \({\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J}\): \[ W = \frac{1}{2\mu_0} \int_{\cal V} d\tau ~{\bf A} \cdot ({\boldsymbol \nabla} \times {\bf B}) \label{Gr(7.32)} \] Integrate by parts using product rule 6: \[ {\boldsymbol \nabla} \cdot ({\bf A} \times {\bf B}) = {\bf B} \cdot ({\boldsymbol \nabla} \times {\bf A}) - {\bf A} \cdot ({\boldsymbol \nabla} \times {\bf B}), \] so \[ {\bf A} \cdot ({\boldsymbol \nabla} \times {\bf B}) = {\bf B} \cdot {\bf B} - {\boldsymbol \nabla} \cdot ({\bf A} \times {\bf B}). \] Then, \[ W = \frac{1}{2\mu_0} \left[ \int_{\cal V} d\tau B^2 - \int_{\cal V} d\tau {\boldsymbol \nabla} \cdot ({\bf A} \times {\bf B}) \right] = \frac{1}{2\mu_0} \left[ \int_{\cal V} d\tau B^2 - \oint_{\cal S} d{\bf a} \cdot ({\bf A} \times {\bf B}) \right] \label{Gr(7.33)} \] We can integrate over all space: after neglecting boundary terms (assuming fields fall to zero at infinity), we are left with

Summary: energy in electric and magnetic fields:

\begin{align} W_{elec} &= \frac{1}{2} \int d\tau ~V\rho &= \frac{\varepsilon_0}{2} \int d\tau ~E^2, \\ W_{mag} &= \frac{1}{2} \int d\tau ~({\bf A} \cdot {\bf J}) &= \frac{1}{2\mu_0} \int d\tau ~B^2, \end{align}

which are equations W_vcd, W_intEsq, W_intAJ and W_intBsq.

Note: gives easy way to find inductance, since \(W = \frac{1}{2} L I^2\).