# Field equations

### Charged particles and fields

Last time we discussed the Lorentz force, which describes how electromagnetic fields affect charged particles, by exerting forces of the form, $$F^\mu = \frac{\mathrm{d} P^\mu}{\mathrm{d} \tau} = q F^{\mu \nu} U_\nu$$ on the particles, where $F^{\mu \nu}$ is the electromagnetic field tensor, and $q$ is the charge of the particle which is the property of the particle that allows fields to interact with it.

But we can also ask what creates and influences the fields themselves and in which way are they affected? If charged particles are affected by fields, for example, by transferring energy to and from the particle, then Newton's third law - action/reaction - requires that the particle must be able to transfer energy to and from the wave.

The equations that describe electromagnetic fields are called Maxwell's equations. We shall convert the classical version of the equations to their covariant form, making them the same in all reference frames.

Maxwell's equations also predict the speed of light, so making the same in all reference frames will also show that the speed of light is the same in all reference frames.

### Qualitative effects of particles on fields

Figure 23.1 - Fields due to charges and currents

A charged particle is a source of electric fields that point radially away from the particle (outward/inward depending on the polarity of the charge). Currents - charges moving along paths - are sources for magnetic fields which circulate about the path.

What happens if the position of the particle, or the path of the current, changes? The field will have to re-arrange itself, but won't be able to do this instantaneously. Effects will take time to propagate throughout space (and time).

For example, a charge moving back and forth will generate an electromagnetic wave. Similary, if you somehow instantaneously changed the direction of a current, then the associated magnetic field would reverse its direction of circulation, but this effect would take time to reach a far-off observer.

We can think of fields over all space and time being modified by a waves of re-arrangement, which as Einstein determined, could not travel faster than the speed of light. It is these waves that are described by Maxwell's equations.

### Divergence and curl of fields in space

We can think of the three partial derivatives $$\frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z}$$ as the components of a vector operator called the del operator, denoted by $$\nabla = \left(\frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z}\right)$$

Nb. This is analagous to the covariant derviative in tensor analysis. If we used component notation then $\nabla \rightarrow \partial_m$ for $m=1,2,3$.

Then, we define the gradient of a field, $\phi$, to be the del operator acting on the field, $$\mathbf{grad}\; \phi = \nabla \phi = \left(\frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y}, \frac{\partial \phi}{\partial z}\right)$$

This measures the rate of change of the field across space.

We define the divergence of a three-vector, $\vec{a}$, to be the dot product of the del operator with the vector, $$\mathbf{div}\; \vec{a} = \nabla \cdot \vec{a} = \frac{\partial a_x}{\partial x} + \frac{\partial a_y}{\partial y} + \frac{\partial a_z}{\partial z}$$

Qualitively, a field has positive divergence when it ‘emanates’ from a source, and negative divergence when it ‘drains’ into a sink. Examples of fields with divergence are electric fields and Newtonian gravitational fields.

We can write the wave equation in terms of the gradient and divergence operators, $$\mathbf{div}\; \mathbf{grad}\; \phi = \nabla \cdot \nabla \phi = \nabla^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2}$$ and so the wave equation becomes $$\frac{\partial^2 \phi}{\partial t^2} - \nabla^2 \phi = 0$$

We define the curl of a three-vector, $\vec{a}$, to be the cross product of the del operator with the vector, $$\mathbf{curl}\; \vec{a} = \nabla \times \vec{a} = \begin{bmatrix}\frac{\partial a_z}{\partial y} - \frac{\partial a_y}{\partial z}\\\frac{\partial a_x}{\partial z} - \frac{\partial a_z}{\partial x}\\\frac{\partial a_y}{\partial x} - \frac{\partial a_x}{\partial y}\end{bmatrix}$$

Qualitively, a field has curl it seems to ‘circulate’ about some location. Main example is the magnetic field.