Dr Phill’s Science Made Simple

Mathematics is key to understanding physics. Many physics articles have the unforunate problem that they dive deep into mathematics without establishing a context. This make many such articles incomprehensible.

Another issue is that some mathematical constructs can be written in different ways. Each construct is valid provided that it is used consistently.

Also, the mathematics describing a physical phenomenon can often the derived in several different ways. An equation can sometimes be written in different forms. There are also shorthand notations used to simplify equations. It is also common to get rid of constants by setting their value to $1$. These simplifications can make the understanding of the mathematics difficult to understand.

Field Operators

If a scalar field, which is a function of position, is $\varphi$ then the gradient is:

$$\mathrm{\nabla }\varphi =(\frac{\partial {\varphi }_1}{\partial x_1},\frac{\partial {\varphi }_2}{\partial x_2},\frac{\partial {\varphi }_3}{\partial x_3})$$

This is a vector field that points in the direction of greatest change in the scalar field.

The divergence of a vector field $\mathrm{\nabla }\cdot \overrightarrow{a}$ is a scalar field. It is the flux generation per unit volume at each point of the field.

$$\mathrm{\nabla }\cdot \overrightarrow{a}=\frac{\partial a_1}{\partial x_1}+\frac{\partial a_2}{\partial x_2}+\frac{\partial a_3}{\partial x_3}$$

The Laplacian is the divergence of a gradient. It is a scalar second deriviative of a scalar field that represents the curvature of the field.

$${\mathrm{\nabla }}^2\varphi =\mathrm{\nabla }\cdot \mathrm{\nabla }\varphi =\frac{{\partial }^2{\varphi }_1}{\partial x_1^2}+\frac{{\partial }^2{\varphi }_2}{\partial x_2^2}+\frac{{\partial }^2{\varphi }_3}{\partial x_3^2}$$

There is also a vector Laplacian.

$${\mathrm{\nabla }}^2\overrightarrow{a}=\mathrm{\nabla }\cdot \mathrm{\nabla }\overrightarrow{a}=(\frac{{\partial }^2{a}_1}{\partial x_1^2},\frac{{\partial }^2{a}_2}{\partial x_2^2},\frac{{\partial }^2{a}_3}{\partial x_3^2})$$

The curl of a vector field $\mathrm{\nabla }\times \overrightarrow{a}$ is a vector field. The curl of a vector field at point a measures the tendency of particles at a to rotate about the axis that points in the direction of the curl at P.

$$\mathrm{\nabla }\times \overrightarrow{a}=(\frac{\partial a_3}{\partial x_2}-\frac{\partial a_2}{\partial x_3},\frac{\partial a_1}{\partial x_3}-\frac{\partial a_3}{\partial x_1},\frac{\partial a_2}{\partial x_1}-\frac{\partial a_1}{\partial x_2})$$

As a curl is a cross product it is perpendicular to both vectors. The dot product of thwo perpendicular vectors is always zero. Hence, the divergence of a curl is always identically zero.

$$\mathrm{\nabla }\cdot \mathrm{\nabla }\times \overrightarrow{a}\equiv 0$$

The curl of a curl of a vector vector field is another vector field.

$$\mathrm{\nabla }\times \mathrm{\nabla }\times \overrightarrow{a}=\mathrm{\nabla }\mathrm{\nabla }\cdot \overrightarrow{a}-{\mathrm{\nabla }}^2\overrightarrow{a}$$

Spacetime Vectors

The spacetime vector has four components, where the time dimension is multiplied by the speed of light $c$ to make it a distance.

In Minkowski spacetime, the vector is:

$$\overrightarrow{x}=\left[\begin{array}{c}ct\\ x\\ y\\ z\end{array}\right]$$

The metric tensor ${\eta }_{ij}$ is required to calculate the distance between differential offsets in each dimension $d{x}_i$. Where ${\eta }_{00}=1$, ${\eta }_{ii}=-1i>0$, ${\eta }_{ij}=0i\ne j$.

$$d{s}^2={\eta }_{ij}d{x}^{i}d{x}^j=c^{2}d{t}^2-d{x}^2-d{y}^2-d{z}^2$$

The $ct$ coordinate is sometimes the fourth element rather than the first. The metric tensor $\eta$ can also be negated. All of the options are valid as long as they are used consistently.

If $d{s}^2>0$ then events are time-like and can see each other after a period of time.

If $d{s}^2=0$ then events are light-like and can see each other if travelling at the speed of light.

If $d{s}^2<0$ then events are space-like and can’t see each as it would require faster than light travel.

Another, more convenient approach is to make the time dimension complex.

$$\overrightarrow{x}=\left[\begin{array}{c}ict\\ x\\ y\\ z\end{array}\right]$$

In this case the metric tensor is not required as it is the identity matrix.

$$d{s}^2=d{x}^{i}d{x}^j=-c^{2}d{t}^2+d{x}^2+d{y}^2+d{z}^2$$

Note that the sign is negated and $d{s}^2<0$ is now time-like.

Relativistic Energy Equation

The total energy of a relativistic body is derived from the rest mass and momentum.

$$E^2=p^2{c}^2+(m{c}^2{)}^2$$

Heaviside-Lorentz (HL) Units

Electromagnetic equations in the SI system contain the constants ${ϵ}_0$ and ${\mu }_0$. The units of related parameters also differ. Oliver Heaviside and Hendrik Lorentz proposed changes to units to simplify equations. These units are in common usage.

The vacuum permittivity ${ϵ}_0$ has SI units $C^2{s}^2{m}^{-3}k{g}^{-1}$. In HL this constant is merged into other units.

The vacuum permeability ${\mu }_0$ has SI units $C^{-2}mkg$. In HL this constant is merged into other units.

The electric field vector $\overrightarrow{E}$ has SI units $m{s}^{-2}{C}^{-1}kg$. In HL $\overrightarrow{E}={ϵ}_0{\overrightarrow{E}}^{SI}$ has units $C{m}^{-2}$.

The polarisation density vector $\overrightarrow{P}$ has SI units $C{m}^{-2}$. It is the same in HL.

The electric displacement field vector $\overrightarrow{D}$ has SI units $C{m}^{-2}$. It is the same in HL.

The magnetic flux density vector $ has SI units $s^{-1}{C}^{-1}kg$. In HL $\overrightarrow{B}=\frac{1}{{\mu }_0}{\overrightarrow{B}}^{SI}$, $\frac{1}{c}\overrightarrow{B}$ has units $C{m}^{-2}$.

The magnetic field strength vector $\overrightarrow{H}$ has SI units $C{m}^{-1}{s}^{-1}$. In HL $\frac{1}{c}{\overrightarrow{H}}^{SI}$ has units $C{m}^{-2}$.

The magnetisation vector $\overrightarrow{M}$ has SI units $C{m}^{-1}{s}^{-1}$. In HL $\frac{1}{c}\overrightarrow{M}$ has units $C{m}^{-2}$.

The current density $\overrightarrow{J}$ has SI units $C{m}^{-2}{s}^{-1}$. In HL $\frac{1}{c}\overrightarrow{J}$ has units $C{m}^{-3}$.

The electric potential $\varphi$, or voltage $v$ has SI units $m{s}^{-2}{C}^{-1}kg$. In HL $\varphi ={ϵ}_0{\varphi }^{SI}$ has units $C{m}^{-2}$.

We can now compare SI equations with HL equations.

Maxwell’s Equations Revisted

Maxwell’s equations in HL become:

$$\mathrm{\nabla }\cdot \overrightarrow{D}=\rho$$ $$\mathrm{\nabla }\cdot \overrightarrow{B}=0$$ $$\mathrm{\nabla }\times \overrightarrow{E}=-\frac{1}{c}\frac{\partial \overrightarrow{B}}{\partial t}$$ $$\mathrm{\nabla }\times \overrightarrow{H}=\frac{1}{c}\frac{\partial \overrightarrow{D}}{\partial t}+\frac{1}{c}\overrightarrow{J}$$

The Lorentz force defines the effect of electric and magnetic fields on a moving charged particle with charge $q$, velocity $\overrightarrow{v}$, and momentum $\overrightarrow{p}$.

$$\overrightarrow{F}=\frac{d\overrightarrow{p}}{t}=q(\overrightarrow{E}+\frac{1}{c}\overrightarrow{v}\times \overrightarrow{B})$$

Using Potentials

Equations can be simplified using potentials.

The scalar electric potential, or voltage, is $\varphi$.

The magnetic vector potential $\overrightarrow{A}$ is a vector satisfying:

$$\mathrm{\nabla }\times \overrightarrow{A}=\overrightarrow{B}$$

The curl operator doesn’t have an exact inverse. So, any vector $\overrightarrow{A}$ that gives the desired value of $\overrightarrow{B}$ can be used.

The electic field vector can also be defined in terms of potentials.

$$\overrightarrow{E}=-\mathrm{\nabla }\cdot \varphi -\frac{\partial \overrightarrow{A}}{\partial t}$$