Supplement 1.1: Maxwell's Equations

We consider the vectors of the electric field $\vec{E}$ and the magnetic field $\vec{B}$ in the presence of an electric charge density ρ and current density $\vec{j}$ . Electric and magnetic matter properties are given by the electrical permittivity ε and the magnetic permeability μ. It is ε=ε_oε_r and μ=μ_oμ_r, with the relative permittivity ε_r and relative permeability μ_r of the material, and where ε_o and μ_o are the vacuum permittivity and vacuum permeability, respectively.

Units of quantities, values of constants ↓

We note the units of these quantities as well as the values of the dielectric constant and the magnetic field constant. This will be needed more often later on.

[\vec{E}] = \frac{N}{C} = \frac{V}{m}

[\vec{B}] = \frac{N}{C m/s} = Tesla

[ρ] = \frac{C}{m^{3}}

[\vec{j}] = \frac{A}{m^{2}}

ε_{o} = 8.854 \cdot 10^{- 12} \frac{A s}{V m}

μ_{o} = 1.26 \cdot 10^{- 6} \frac{V s}{A m}

The material properties $ε_{r}$ and $μ_{r}$ are dimensionless; in a vacuum, their values are equal to 1.

Maxwell's equations combine all these quantities into a system of four integral equations or differential equations. An intuitive understanding is often easier with the integral equations, and this is why the integral forms are more often used in physics class. The differential equations are given here:

\nabla \cdot \vec{E} = \frac{ρ}{ε}

\nabla \cdot \vec{B} = 0

i.e., the electric charge is the source of an electric field, while magnetic charges do not exist; and

\nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}

\nabla \times \vec{B} = μ \vec{j} + ε μ \frac{\partial \vec{E}}{\partial t}

a time-varying magnetic field causes a curl of the electric field, and a time-varying electric field or an electric current causes a curl of the magnetic field.

Maxwell's equations in integral form ↓

Equations ↓

The terms ∇⋅ and ∇× denote the divergence and the curl of the vector which follows. These are spatial derivatives of vectors, using the nabla (∇) operator which is also a vector. The symbols "⋅" and "×" denote the dot product and the cross product of two vectors. For example, in Cartesian (x,y,z) coordinates, the nabla operator and the electric field E read:

\nabla = (\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z})

\vec{E} = (E_{x}, E_{y}, E_{z})

Then the divergence and the curl of $\vec{E}$ are:

\nabla \cdot \vec{E} = \frac{\partial E_{x}}{\partial x} + \frac{\partial E_{y}}{\partial y} + \frac{\partial E_{z}}{\partial z}

\nabla \times \vec{E} = (\frac{\partial E_{z}}{\partial y} - \frac{\partial E_{y}}{\partial z}, \frac{\partial E_{x}}{\partial z} - \frac{\partial E_{z}}{\partial x}, \frac{\partial E_{x}}{\partial y} - \frac{\partial E_{y}}{\partial x})

We refer to textbooks on vector calculus if you are not familiar with these operations, e.g., Murray R. Spiegel, 1959: Schaum's Outline of Vector Analysis (McGraw Hill) 225 pp.

Question 1: Divergence and curl of a vector ↓

Please show that the equations given above for the divergence $\nabla \cdot \vec{E}$ and the curl $\nabla \times \vec{E}$ of the electric field vector are correct. Are these terms scalar quantities or vectors?

Check your results on this page!

Question 2: Gradient of a scalar quantity ↓

To be complete, the ∇ operator can further be applied with scalar quantities. Let $φ (x, y, z)$ be a scalar function in space, e.g. the electric potential. The term ∇φ then denotes the spatial derivative of φ, also called grad φ. ∇φ is a vector since the vector ∇ is multiplied with the scalar φ.

Please write ∇φ in components along the x, y, and z directions.

Check your result on this page!

Understanding Spectra from the Earth

Supplement 1.1: Maxwell's Equations