Maxwell's equations of classical electrodynamics are, in...

Maxwell's equations of classical electrodynamics are, in vacuo, $$ \begin{array}{ll} \nabla \cdot \mathbf{E}=\rho, & \nabla \times \mathbf{E}+\frac{\partial \mathbf{B}}{\partial t}=0 \\ \nabla \cdot \mathbf{B}=0, & \nabla \times \mathbf{B}-\frac{\partial \mathbf{E}}{\partial t}=\mathbf{j} \end{array} $$ (where we are using Heaviside-Lorentz rationalized units, see Appendix C of Aitchison and Hey). Show that these equations are equivalent to the following covariant equation for $A^{\mu}$ : $$ \square^{2} A^{\mu}-\partial^{\mu}\left(\partial_{\nu} A^{\nu}\right)=j^{\mu}, $$ with $j^{\mu}=(\rho, \mathbf{j})$, and where $A^{\mu}=(\phi, \mathbf{A})$, the four-vector potential, is related to the electric and magnetic fields by $$ \mathbf{E}=-\frac{\partial \mathbf{A}}{\partial t}-\nabla \phi, \quad \mathbf{B}=\nabla \times \mathbf{A} $$

Maxwell's equations of classical electrodynamics are, in vacuo,
$$
\begin{array}{ll}
\nabla \cdot \mathbf{E}=\rho, & \nabla \times \mathbf{E}+\frac{\partial \mathbf{B}}{\partial t}=0 \\
\nabla \cdot \mathbf{B}=0, & \nabla \times \mathbf{B}-\frac{\partial \mathbf{E}}{\partial t}=\mathbf{j}
\end{array}
$$
(where we are using Heaviside-Lorentz rationalized units, see Appendix C of Aitchison and Hey). Show that these equations are equivalent to the following covariant equation for $A^{\mu}$ :
$$
\square^{2} A^{\mu}-\partial^{\mu}\left(\partial_{\nu} A^{\nu}\right)=j^{\mu},
$$
with $j^{\mu}=(\rho, \mathbf{j})$, and where $A^{\mu}=(\phi, \mathbf{A})$, the four-vector potential, is related to the electric and magnetic fields by
$$
\mathbf{E}=-\frac{\partial \mathbf{A}}{\partial t}-\nabla \phi, \quad \mathbf{B}=\nabla \times \mathbf{A}
$$

Key Concepts

Gauge Conditions

Gauge conditions are constraints imposed on the electromagnetic potentials to remove ambiguities that arise due to gauge freedom—the fact that different potentials can yield the same physical fields. In the covariant formulation, a common choice is the Lorenz gauge, which simplifies the equations and makes the underlying symmetries of the theory more manifest.

d'Alembertian Operator

Also known as the wave operator, the d'Alembertian operator is a second-order differential operator in space and time. It generalizes the Laplace operator to a relativistic setting and is essential in formulating the wave equations that describe the propagation of fields in space-time.

Covariant Formulation

This concept involves rewriting physical laws such that they take the same form in all inertial frames, highlighting the connection with special relativity. By expressing Maxwell's equations using four-vectors and tensors, one achieves a formulation that is explicitly Lorentz invariant, thereby unifying electric and magnetic fields into a single electromagnetic field tensor.

Maxwell's Equations

These equations form the cornerstone of classical electrodynamics, describing how electric and magnetic fields are generated and altered by charges and currents. They encompass the laws of Gauss for electricity and magnetism, Faraday’s law of induction, and Ampère’s law with Maxwell’s correction, providing a complete description of the dynamics of electromagnetic fields in space and time.

Electromagnetic Four-Potential

The four-potential is a four-dimensional vector that combines the scalar potential and the vector potential into one entity, facilitating the reformulation of electrodynamics in a relativistically invariant manner. It provides a convenient way to express both the electric and magnetic fields through derivatives, and its use is fundamental in transitioning from the classical to the covariant perspective.

Transcript

00:01 So for this, we're going to start off with the maxwell equation, which tells us the dell cross e is equal to negative 1 over c.

00:24 Then we want to apply the curl on both sides.

00:30 So we'll get something like this.

00:45 Since we know that c is constant, this is going to be the same thing as the negative 1 over c.

01:01 Then since we're assuming the magnetic field vector h has continuous derivatives, we can change the order of the curl operator.

01:14 So ultimately what this is going to look like is we get this.

01:35 So now what we end up having is that this is equal to negative 1 over c times.

02:08 Move that right here.

02:13 So then with this in mind, we now see as a result this right here, which is what we wanted to show.

02:47 Then for part b, since that was a, for part b, we're going to start with the maxwell equation like this.

03:12 And then what we're going to do is we're going to take the cross product on both sides of dell.

03:28 So then once we do that, we know that c is constant once again.

03:36 So assuming all these things we know, we can then get it down to what we did before, which is this will be equal to 1 over c, delta t or d t um and then like this so then with that in mind um we will just have actually this right here but this is an h and this is negative um and that can be multiplied by 1 over c, delta delta t.

04:43 So therefore, what we are able to show is that this right here, we can just duplicate it, is going to be equal to a negative 1 over c squared times that.

05:11 Then for part c, we're going to use exercise 29 to help us...

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