7.6: Appendix - Review of Free Electromagnetic Field

Here we review the derivation of the vector potential for the plane wave in free space. We begin with Maxwell’s equations (SI):

\[\begin{align} \overline {\nabla} \cdot \overline {B} &= 0 \label{6.78} \\[4pt] \overline {\nabla} \cdot \overline {E} &= \rho / \varepsilon _ {0} \label{6.79} \\[4pt] \overline {\nabla} \times \overline {E} &= - \dfrac {\partial \overline {B}} {\partial t} \label{6.80} \\[4pt] \overline {\nabla} \times \overline {B} &= \mu _ {0} \overline {J} + \varepsilon _ {0} \mu _ {0} \dfrac {\partial \overline {E}} {\partial t} \label{6.81} \end{align}\]

Here the variables are: \(\overline {E}\), electric field; \(\overline {B}\), magnetic field; \(\overline {J}\), current density; \(\rho\), charge density; \(\mathcal {E} _ {0}\), electrical permittivity; \(\mu _ {0}\), magnetic permittivity. We are interested in describing \(\overline {E}\) and \(\overline {B}\) in terms of a vector and scalar potential, \(\overline {A}\) and \(\varphi\).

Next, let’s review some basic properties of vectors and scalars. Generally, vector field \(\overline {F}\) assigns a vector to each point in space. The divergence of the field

\[\overline {\nabla} \cdot \overline {F} = \dfrac {\partial F _ {x}} {\partial x} + \dfrac {\partial F _ {y}} {\partial y} + \dfrac {\partial F _ {z}} {\partial z} \label{6.82}\]

is a scalar. For a scalar field \(\phi\), the gradient

\[\nabla \phi = \dfrac {\partial \phi} {\partial x} \hat {x} + \dfrac {\partial \phi} {\partial y} \hat {y} + \dfrac {\partial \phi} {\partial z} \hat {z} \label{6.83}\]

is a vector for the rate of change at one point in space. Here

\[\hat {x}^{2} + \hat {y}^{2} + \hat {z}^{2} = \hat {r}^{2}\]

are unit vectors. Also, the curl

\[\overline {\nabla} \times \overline {F} = \left| \begin{array} {l l l} {\hat {x}} & {\hat {y}} & {\hat {z}} \\ {\dfrac {\partial} {\partial x}} & {\dfrac {\partial} {\partial y}} & {\dfrac {\partial} {\partial z}} \\ {F _ {x}} & {F _ {y}} & {F _ {z}} \end{array} \right|\]

is a vector whose \(x\), \(y\), and \(z\) components are the circulation of the field about that component. Some useful identities from vector calculus that we will use are

\[\begin{align} \overline {\nabla} \cdot ( \overline {\nabla} \times \overline {F} ) &= 0 \label{6.85} \\[4pt] \nabla \times ( \nabla \phi ) &= 0 \label{6.86} \\[4pt] \nabla \times ( \overline {\nabla} \times \overline {F} ) &= \overline {\nabla} ( \overline {\nabla} \cdot \overline {F} ) - \overline {\nabla}^{2} \overline {F} \label{6.87} \end{align}\]

Gauge Transforms

We now introduce a vector potential \(\overline {A} ( \overline {r} , t )\) and a scalar potential \(\varphi ( \overline {r} , t )\), which we will relate to \(\overline {E}\) and \(\overline {B}\). Since

\[\overline {\nabla} \cdot \overline {B} = 0\]

and

\[\overline {\nabla} ( \overline {\nabla} \times \overline {A} ) = 0,\]

we can immediately relate the vector potential and magnetic field

\[\overline {B} = \overline {\nabla} \times \overline {A} \label{6.88}\]

Inserting this into Equation \ref{6.80} and rewriting, we can relate the electric field and vector potential:

\[\overline {\nabla} \times \left[ \overline {E} + \dfrac {\partial \overline {A}} {\partial t} \right] = 0 \label{6.89}\]

Comparing Equations \ref{6.89} and \ref{6.86} allows us to state that a scalar product exists with

\[\overline {E} = \dfrac {\partial \overline {A}} {\partial t} - \nabla \varphi \label{6.90}\]

So summarizing our results, we see that the potentials \(\overline {A}\) and \(\varphi\) determine the fields \(\overline {B}\) and \(\overline {E}\):

\[\begin{align} \overline {B} ( \overline {r} , t ) &= \overline {\nabla} \times \overline {A} ( \overline {r} , t ) \label{6.91} \\[4pt] \overline {E} ( \overline {r} , t ) &= - \overline {\nabla} \varphi ( \overline {r} , t ) - \dfrac {\partial} {\partial t} \overline {A} ( \overline {r} , t ) \label{6.92} \end{align}\]

We are interested in determining the classical wave equation for \(\overline {A}\) and \(\varphi\). Using Equation \ref{6.91}, differentiating Equation \ref{6.92}, and substituting into Equation \ref{6.81}, we obtain

\[\overline {\nabla} \times ( \overline {\nabla} \times \overline {A} ) + \varepsilon _ {0} \mu _ {0} \left( \dfrac {\partial^{2} \overline {A}} {\partial t^{2}} + \overline {\nabla} \dfrac {\partial \varphi} {\partial t} \right) = \mu _ {0} \overline {J} \label{6.93}\]

Using Equation \ref{6.87},

\[\left[ - \overline {\nabla}^{2} \overline {A} + \varepsilon _ {0} \mu _ {0} \dfrac {\partial^{2} \overline {A}} {\partial t^{2}} \right] + \overline {\nabla} \left( \overline {\nabla} \cdot \overline {A} + \varepsilon _ {0} \mu _ {0} \dfrac {\partial \varphi} {\partial t} \right) = \overline {\mu} _ {0} \overline {J} \label{6.94}\]

From Equation \ref{6.90}, we have

\[\overline {\nabla} \cdot \overline {E} = - \dfrac {\partial \overline {\nabla} \cdot \overline {A}} {\partial t} - \overline {\nabla}^{2} \varphi \label{6.95}\]

and using Equation \ref{6.79},

\[\dfrac {- \partial \overline {V} \cdot \overline {A}} {\partial t} - \overline {\nabla}^{2} \varphi = \rho / \varepsilon _ {0} \label{6.96}\]

Notice from Equations \ref{6.91} and \ref{6.92} that we only need to specify four field components (\(A_{x}, A_{y}, A_{z}, \varphi\) to determine all six \(\bar{E}\) and \(\bar{B}\) components. But \(\bar{E}\) and \(\bar{B}\) do not uniquely determine \(\bar{A}\) and \(\varphi\). So we can construct \(\bar{A}\) and \(\varphi\) in any number of ways without changing \(\bar{E}\) and \(\bar{B}\). Notice that if we change \(\bar{A}\) by adding \(\bar{\nabla} \chi \) where \(\chi\) is any function of \(\bar{r}\) and \(t\) this will not change \(\bar{B} \quad(\nabla \times(\nabla \cdot B)=0)\). It will change \(E\) by \(\left(-\frac{\partial}{\partial t} \bar{\nabla} \chi\right)\), but we can change \(\varphi\) to \(\varphi^{\prime}=\varphi-(\partial \chi / \partial t)\). Then \(\bar{E}\) and \(\bar{B}\) will both be unchanged. This property of changing representation (gauge) without changing \(\bar{E}\) and \(\bar{B}\) is gauge invariance. We can define a gauge transformation with

\[\bar{A}^{\prime}(\bar{r}, t)=\bar{A}(\bar{r}, t)+\bar{\nabla} \cdot \chi(\bar{r}, t) \label{6.97}\]

\[\varphi^{\prime}(\bar{r}, t)=\varphi(\bar{r}, t)-\frac{\partial}{\partial t} \chi(\bar{r}, t) \label{6.98}\]

Up to this point, \(A^{\prime} \text {and} \varphi^{\prime}\) are undetermined. Let’s choose a \(\chi\) such that:

\[\overline {\nabla} \cdot \overline {A} + \varepsilon _ {0} \mu _ {0} \dfrac {\partial \varphi} {\partial t} = 0 \label{6.99}\]

which is known as the Lorentz condition. Then from Equation \ref{6.93}:

\[- \nabla^{2} \overline {A} + \varepsilon _ {0} \mu _ {0} \dfrac {\partial^{2} \overline {A}} {\partial t^{2}} = \mu _ {0} \overline {J} \label{6.100}\]

The right hand side of this equation can be set to zero when no currents are present. From Equation \ref{6.96}, we have:

\[\varepsilon _ {0} \mu _ {0} \dfrac {\partial^{2} \varphi} {\partial t^{2}} - \nabla^{2} \varphi = \dfrac {\rho} {\varepsilon _ {0}} \label{6.101}\]

Equations \ref{6.100} and \ref{6.101} are wave equations for \(\overline {A}\) and \(\varphi\). Within the Lorentz gauge, we can still arbitrarily add another \(\chi\); it must only satisfy Equation \ref{6.99}. If we substitute Equations \ref{6.97} and \ref{6.98} into Equation \ref{6.101}, we see

\[\nabla^{2} \chi - \varepsilon _ {0} \mu _ {0} \dfrac {\partial^{2} \chi} {\partial t^{2}} = 0 \label{6.102}\]

So we can make further choices/constraints on \(\bar{A} \text {and} \varphi\) as long as it obeys Equation \ref{6.102}. We now choose \(\varphi=0\), the Coulomb gauge, and from Equation \ref{6.99} we see

\[\overline {\nabla} \cdot \overline {A} = 0 \label{6.103}\]

So the wave equation for our vector potential when the field is far currents (\(J= 0\)) is

\[- \overline {\nabla}^{2} \overline {A} + \varepsilon _ {0} \mu _ {0} \dfrac {\partial^{2} \overline {A}} {\partial t^{2}} = 0 \label{6.104}\]

The solutions to this equation are plane waves:

\[\overline {A} = \overline {A} _ {0} \sin ( \omega t - \overline {k} \cdot \overline {r} + \alpha ) \label{6.105}\]

where \(\alpha\) is a phase. \(k\) is the wave vector which points along the direction of propagation and has a magnitude

\[k^{2} = \omega^{2} \mu _ {0} \varepsilon _ {0} = \omega^{2} / c^{2} \label{6.106}\]

Since \(\overline {\nabla} \cdot \overline {A} = 0\) (Equation \ref{6.103}), then

\[- \overline {k} \cdot \overline {A} _ {0} \cos ( \omega t - \overline {k} \cdot \overline {r} + \alpha ) = 0\]

and

\[\overline {k} \cdot \overline {A} _ {0} = 0 \label{6.107}\]

So the direction of the vector potential is perpendicular to the direction of wave propagation (\(\overline {k} \perp \overline {A _ {0}}\)). From Equations \ref{6.91} and \ref{6.92}, we see that for \(\varphi = 0\):

\[\begin{align} \overline {E} &= - \dfrac {\partial \overline {A}} {\partial t} \\[4pt] &= - \omega \overline {A} _ {0} \cos ( \omega t - \overline {k} \cdot \overline {r} + \alpha ) \label{6.108} \\[4pt] \overline {B} &= \overline {\nabla} \times \overline {A} \\[4pt] &= - \left( \overline {k} \times \overline {A} _ {0} \right) \cos ( \omega t - \overline {k} \cdot \overline {r} + \alpha ) \label{6.109} \end{align}\]

Here the electric field is parallel with the vector potential, and the magnetic field is perpendicular to the electric field and the direction of propagation (\(\overline {k} \perp \overline {E} \perp \overline {B}\)). The Poynting vector describing the direction of energy propagation is

\[\overline {S} = \varepsilon _ {0} c^{2} ( \overline {E} \times \overline {B} )\]

and its average value, the intensity, is

\[I = \langle S \rangle = \dfrac {1} {2} \varepsilon _ {0} c E _ {0}^{2}.\]