Potential Equations $\leftrightarrow$ Maxwell's Equation.

In Lorenz's paper [1] he begins with scaler and vector potentials (in retarded from) and derives Maxwell's equations from these equations. Typically, texts start with Maxwell's equations and develop the Lorenz Gauge [2,3] which has the benefit of seeming less ad hoc. Here we present a graphical representation of the development.

We begin with the Maxwell equations in general form,

$$\nabla \cdot E = \frac{\rho}{\epsilon_0} \qquad \nabla \times E = -\frac{\partial B}{\partial t}$$

$$\nabla \cdot B = 0 \qquad \nabla \times B = \mu_0J+\mu_0\epsilon_0\frac{\partial E}{\partial t}$$

We note that

$$\nabla \cdot B = 0 \ \Rightarrow \ B \equiv \nabla \times A \ni \nabla \times E = -\frac{\partial}{\partial t}(\nabla \times A)$$

$$\therefore \nabla \times \left(E+\fra... ...}{\partial t} = -\nabla V \ \ni \ E = -\nabla V - \frac{\partial A}{\partial t}$$

$$\ni \ \nabla \cdot E = \frac{\rho}{\epsilon_0} \ \rightarrow$$

$$\nabla^2V + \frac{\partial}{\partial t}(\nabla \cdot A)=-\frac{\rho}{\epsilon_0}$$ (1.1)

Finally we note that

$$\nabla \times B = \mu_0 J + \mu_0 \epsilon_0 \frac{\partial E}{\p... ...artial V}{\partial t}\right) - \mu_0\epsilon_0 \frac{\partial^2A}{\partial t^2}$$

Since

$$\nabla \times (\nabla \times A) = \partial_j\epsilon_{ijk}(\parti... ...\partial_j\epsilon_{ijk}(\partial_j A_i) = \nabla(\nabla \cdot A) - \nabla^2 A,$$

we have

$$\left(\nabla^2A -\mu_0\epsilon_0 \frac{\partial^2A}{\partial t^2}... ...abla \cdot A + \mu_0 \epsilon_0 \frac{\partial V}{\partial t}\right) = -\mu_0 J$$ (1.2)

Figure 1.1: Maxwell's Equations $\leftrightarrow$ Potential Equations