Potentials

We begin our discussion of potentials by considering the two homogeneous equations. For example, if we wish to associate a potential with $\mbox{\boldmath$B$}$ such that $\mbox{\boldmath$B$}$ is the result of differentiating the potential, we observe that we can satisfy GLM by construction if we suppose a vector potential $\mbox{\boldmath$A$}$ such that:

$$\mbox{\boldmath$B$}= \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$A$}$$ (8.28)

In that case:

$$\mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$B$}= \mbox{\bo... ...cdot (\mbox{\boldmath$\nabla$}\times \mbox{\boldmath$A$}) = 0$$ (8.29)

as an identity.

Now consider FL. If we substitute in our expression for $\mbox{\boldmath$B$}$:

$$\mbox{\boldmath$\nabla$}\times \mbox{\boldmath$E$}+ \frac{\partial \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$A$}}{\partial t} = 0$$

$$\mbox{\boldmath$\nabla$}\times (\mbox{\boldmath$E$}+ \frac{\partial \mbox{\boldmath$A$}}{\partial t}) = 0$$ (8.30)

We can see that if we define:

$$\mbox{\boldmath$E$}+ \frac{\partial \mbox{\boldmath$A$}}{\partial t} = -\mbox{\boldmath$\nabla$}\phi$$ (8.31)

then

$$\mbox{\boldmath$\nabla$}\times (\mbox{\boldmath$E$}+ \frac{... ...box{\boldmath$\nabla$}\times \mbox{\boldmath$\nabla$}\phi = 0$$ (8.32)

is also an identity. This leads to:

$$\mbox{\boldmath$E$}= -\mbox{\boldmath$\nabla$}\phi - \frac{\partial \mbox{\boldmath$A$}}{\partial t}$$ (8.33)

Our next chore is to transform the inhomogeneous MEs into equations of motion for these potentials - motion because MEs (and indeed the potentials themselves) are now potentially dynamical equations and not just static. We do this by substituting in the equation for $\mbox{\boldmath$E$}$ into GLE, and the equation for $\mbox{\boldmath$B$}$ into AL. We will work (for the moment) in free space and hence will use the vacuum values for the permittivity and permeability.

The first (GLE) yields:

$$\mbox{\boldmath$\nabla$}\cdot (-\mbox{\boldmath$\nabla$}\phi - \frac{\partial \mbox{\boldmath$A$}}{\partial t}) = \frac{\rho}{\epsilon_0 }$$

$$\nabla^2\phi + \frac{\partial (\mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$A$})}{\partial t} = - \frac{\rho}{\epsilon_0 }$$ (8.34)

The second (AL) is a bit more work. We start by writing it in terms of $\mbox{\boldmath$B$}$ instead of $\mbox{\boldmath$H$}$ by multiplying out the $\mu_0$:

$$\mbox{\boldmath$\nabla$}\times \mbox{\boldmath$B$} = \mu_0 \mbox{\boldmath$J$}+ \mu_0 \epsilon_0 \frac{\partial \mbox{\boldmath$E$}}{\partial t}$$

$$\mbox{\boldmath$\nabla$}\times (\mbox{\boldmath$\nabla$}\times \mbox{\boldmath$A$}) = \mu_0 \mbox{\boldmath$J$}+ \mu_0 \epsilon_0 \frac{\partial }{\par... ...\mbox{\boldmath$\nabla$}\phi - \frac{\partial \mbox{\boldmath$A$}}{\partial t})$$

$$-\nabla^2\mbox{\boldmath$A$}+ \mbox{\boldmath$\nabla$}(\mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$A$}) = \mu_0 \mbox{\boldmath$J$}- \frac{1}{c^2}\mbox{\boldmath$\nabla$} ... ...{\partial t} - \frac{1}{c^2}\frac{\partial^2 \mbox{\boldmath$A$}}{\partial t^2}$$

$$\nabla^2\mbox{\boldmath$A$}+ - \frac{1}{c^2}\frac{\partial^2 \mbox{\boldmath$A$}}{\partial t^2} = - \mu_0 \mbox{\boldmath$J$} + \mbox{\boldmath$\nabla$}(\mbox{\bo... ...ath$A$})+ \mbox{\boldmath$\nabla$}\frac{1}{c^2}\frac{\partial \phi}{\partial t}$$

$$\nabla^2\mbox{\boldmath$A$}+ - \frac{1}{c^2}\frac{\partial^2 \mbox{\boldmath$A$}}{\partial t^2} = - \mu_0 \mbox{\boldmath$J$} + \mbox{\boldmath$\nabla$}\left (\mb... ...\cdot \mbox{\boldmath$A$}+ \frac{1}{c^2}\frac{\partial \phi}{\partial t}\right)$$ (8.35)