# What are Maxwell's equations

1. Jul 24, 2014

### Greg Bernhardt

Definition/Summary

Maxwell's equations are a set of four equations which must be satisfied by all electric and magnetic fields throughout all space at all times.

They comprise of Gauss' law, Gauss' law for magnetism, Maxwell's modification of Ampère's law, and Faraday's law.

Each equation has a differential version (involving div or curl), and an equivalent integral version. These equivalences come from the Divergence Theorem (for div), and from Stokes Theorem (for curl).

Equations

Gauss' Law:

$$\text{div}\left(\mathbf{D}\right) = \rho_f$$

$$\oint_S\mathbf{D\cdot n}\;\;dS = Q_f$$

Gauss' Law for Magnetism:

$$\text{div}\left(\mathbf{B}\right) = 0$$

$$\oint_S\mathbf{B\cdot n}\;\;dS = 0$$

Ampère-Maxwell Law:

$$\text{curl}\left(\mathbf{H}\right) = \mathbf{j_f}+\frac{\partial\mathbf{D}}{\partial t}$$

$$\oint_C\mathbf{H\cdot}d\mathbf{\ell} = I_f + \frac{\partial}{\partial t}\int_S\mathbf{D\cdot n}\;\;dS$$

$$\text{curl}\left(\mathbf{E}\right) = - \frac{\partial\mathbf{B}}{\partial t}$$

$$\oint_C\mathbf{E\cdot}d\mathbf{\ell} = -\frac{\partial}{\partial t}\int_S \mathbf{B\cdot n}\;\;dS$$

Extended explanation

1.0 Introduction

This article will briefly discuss each of Maxwell's equations, but detailed analysis of each equation will be left to separate articles. The special cases of Maxwell's equations in Linear, Homogeneous, Isotropic and non-Dispersive (LHIND) materials and in the absence of free charge, currents and dielectrics will also be discussed. We will also touch on the application of Maxwell's equations to electromagnetic radiation.

1.1 Notation

$\mathbf{D}$ $\mathbf{E}$ $\mathbf{B}$ and $\mathbf{H}$ are the electric displacement field (or flux density), the electric field, the magnetic field, and the magnetic field strength respectively.

$\rho_f$ represents the net free charge density and $Q_f$ represents the net free charge. $\mathbf{j_f}$ represents that free current density and $\mathbf{I_f}$ represents the net free current.

$S$ is any surface, oriented by the normal vector $\mathbf{n}$. In the case of both of Gauss' laws, the surface must be closed. In the case of Ampère's and Faraday's laws, the surface needn't be closed, but is bounded by a single, smooth, closed curve $C$.

$dS$ is the surface element of surface $S$ and $d\mathbf{\ell}$ is the differential path element whose direction is tangential to the closed curve $C$.

2.0 The general form of Maxwell's equations

2.1 Gauss' Law

The differential form simply states that the divergence of the electric displacement field is equal to the local free charge density.

The integral form states that the net flux of the electric displacement field through a closed surface $S$ is equal to the net free charge inside that surface.

2.2 Gauss' Law for Magnetism

The differential form states that the divergence of any magnetic field must be identically equal to zero. It therefore implies that there are no magnetic monopoles.

To a vector field whose divergence is identically zero ("is solenoidal"), there can be associated a (non-unique) vector field (the "vector potential") whose curl is that first field.

Therefore it is possible1 to associate a magnetic vector potential $\mathbf{A}$ with any magnetic field $\mathbf{B}$:

$$\mathbf{B} = \text{curl}\left(\mathbf{A}\right) = \nabla \times \mathbf{A}$$

The integral form states that the net flux of the magnetic field through any closed surface must be identically equal to zero. It therefore implies that magnetic field lines can have no sources or sinks, and must form closed loops (or extend to infinity).

2.3 Ampère-Maxwell Law

The differential form states that the curl of the magnetic field strength is the sum of the free current density and the time differential of the electric displacement field.

The integral form states that the integral of the magnetic field strength around a closed loop is equal to the sum of the net free current passing through any surface enclosed by the loop and the time derivative of the flux of the electric displacement field through that same surface.

The differential form of Faraday's law shown above is often called the Faraday-Maxwell equation since it was Maxwell that first developed the differential form. The differential form was later modified by Heaviside, but sadly his name doesn't appear in the title of the equation.

The differential form states that the curl of the electric field is equal to the negative [partial] derivative of the magnetic field with respect to time.

In other words: the circulation of the electric field is proportional to the rate of change of the magnetic field.

The integral form states that the integral of the electric field around a closed loop is equal to the negative [partial] time derivative of the magnetic flux through any surface bounded by that loop.

3.0 Maxwell's Equations in Linear Homogeneous Isotropic & non-Dispersive Media

In the case of LHIND materials one can relate the electric field to the electric displacement field and the magnetic field to the magnetic field strength by scalar constants (simple multiplicative factors):

$$\mathbf{D} = \varepsilon_0\varepsilon_r\mathbf{E}$$

$$\mathbf{B} = \mu_0\mu_r\mathbf{H}$$

where $\varepsilon_0$ and $\mu_0$ are the permittivity and permeability of free space and $\varepsilon_r$ and $\mu_r$ are the relative permittivity and relative permeability of the dielectric material.

In the case of non-LHIND materials it is still possible to relate the fields as shown above, but the coefficients $\varepsilon_r$ and $\mu_r$ are no longer scalar constants instead they become tensors, and/or functions of the fields themselves.

So Maxwell's equations in LHIND materials may be written in terms of only two fields, the electric and magnetic fields, $\mathbf{E}$ and $\mathbf{B}$:

3.1 Gauss' Law

$$\text{div}\left(\mathbf{E}\right) = \frac{\rho_f}{\varepsilon_0\varepsilon_r}$$

$$\oint_S\mathbf{E\cdot n}\;\;dS = \frac{Q_f}{\varepsilon_0\varepsilon_r}$$

3.2 Gauss' Law for Magnetism

$$\text{div}\left(\mathbf{B}\right) = 0$$

$$\oint_S\mathbf{B\cdot n}\;\;dS = 0$$

3.3 Ampère-Maxwell Law

$$\text{curl}\left(\mathbf{B}\right) = \mu_0\mu_r\mathbf{j_f}+\mu_0\mu_r\varepsilon_0 \varepsilon_r\frac{\partial\mathbf{E}}{\partial t}$$

$$\oint_C\mathbf{B\cdot}d\mathbf{\ell} = \mu_0\mu_rI_f + \mu_0\mu_r \varepsilon_0\varepsilon_r \frac{\partial}{\partial t}\int_S\mathbf{E\cdot n}\;\;dS$$

$$\text{curl}\left(\mathbf{E}\right) = - \frac{\partial\mathbf{B}}{\partial t}$$

$$\oint_C\mathbf{E\cdot}d\mathbf{\ell} = -\frac{\partial}{\partial t}\int_S \mathbf{B\cdot n}\;\;dS$$

4.0 Maxwell's Equations in the Absence of Dielectrics

In the absence of dielectric or magnetic media $\mu_r = \varepsilon_r = 1$ which further simplifies the equations shown in section three, but there is no need to explicitly re-write the equations in this case.

5.0 Maxwell's Equations in Free Space

Taking the case where there is no dielectric or magnetic media (i.e. when $\mu_r = \varepsilon_r = 1$) and further assuming that there are no free charges or currents (i.e. $\rho_f = \mathbf{j_f} = 0$), we can further simplify the general case of Maxwell's equations to obtain the so-called Maxwell's Equations in Free Space.

5.1 Gauss' Law

$$\text{div}\left(\mathbf{E}\right) = 0$$

$$\oint_S\mathbf{E\cdot n}\;\;dS = 0$$

5.2 Gauss' Law for Magnetism

$$\text{div}\left(\mathbf{B}\right) = 0$$

$$\oint_S\mathbf{B\cdot n}\;\;dS = 0$$

5.3 Ampère-Maxwell Law

$$\text{curl}\left(\mathbf{B}\right) = \mu_0\varepsilon_0 \frac{\partial\mathbf{E}}{\partial t}$$

$$\oint_C\mathbf{B\cdot}d\mathbf{\ell} = \mu_0\varepsilon_0 \frac{\partial}{\partial t} \int_S\mathbf{E\cdot n}\;\;dS$$

$$\text{curl}\left(\mathbf{E}\right) = - \frac{\partial\mathbf{B}}{\partial t}$$

$$\oint_C\mathbf{E\cdot}d\mathbf{\ell} = -\frac{\partial}{\partial t}\int_S \mathbf{B\cdot n}\;\;dS$$

6.0 Maxwell's Equations & the Speed of Light

Using Maxwell's equations one can quite easily determine the velocity of propagation of the electromagnetic fields through various media. First we shall consider the case where the electromagnetic fields propagate through dielectric (LHIND) media in the absence of free currents and charge and then consider the special case of free space.

6.1 In Dielectric Media

We start from Maxwell's equations in LHIND media but assume that $\mathbf{j_f} = \rho_f = 0$ and take the curl of the differential form of Faraday's law:

$$\text{curl}\left\{ \text{curl}\left(\mathbf{E}\right) \right\} = \text{curl} \left\{-\frac{\partial\mathbf{B}}{\partial t}\right\} = -\frac{\partial}{\partial t} \text{curl} \left(\mathbf{B}\right)$$

Substituting for the curl of the magnetic field using the Ampère-Maxwell law:

$$\text{curl}\left\{ \text{curl} \left(\mathbf{E}\right) \right\} = -\mu_0 \mu_r \varepsilon_0 \varepsilon_r \frac{\partial^2\mathbf{E}}{\partial t^2}$$

We have the identity $\text{curl} \left\{ \text{curl}\left( \mathbf{F} \right) \right\} = \text{grad} \left\{ \text{div}\left( \mathbf{F} \right) \right\}- \nabla^2 \mathbf{F}$ for any vector field $\mathbf{F}$. Applying this theorem to the LHS and noting that $\text{div}\left(\mathbf{E}\right) = 0$ by Gauss' law (assuming that $\rho_f=0$) we may write:

$$\nabla^2\mathbf{E} = \mu_0\mu_r\varepsilon_0\varepsilon_r\frac{\partial^2\mathbf{E}}{\partial t^2}$$

Comparing this to the standard wave equation:

$$\nabla^2\mathbf{\psi} = \frac{1}{v^2} \frac{\partial^2\mathbf{\psi}}{\partial t^2}$$

It can be deduced that:

$$\frac{1}{v^2} = \mu_0\mu_r\varepsilon_0\varepsilon_r$$

Hence, the speed of propagation of the electromagentic fields2 in dielectric media is:

$$v = \frac{1}{\sqrt{\mu_0\mu_r\varepsilon_0\varepsilon_r}}$$

6.2 In Free Space

As said previously, in free space $\mu_r=\varepsilon_r=1$. Hence, using the result from the previous section it is obvious that the speed of light in vacuum may be written:

$$c = \frac{1}{\sqrt{\mu_0\varepsilon_0}}$$

And as such we may re-write the speed of propagation in dielectric media thus:

$$v = \frac{c}{\sqrt{\mu_r\varepsilon_r}} = \frac{c}{n}$$

Where $n$ is the refractive index of the media.

6.3 Newtonian 4-vectors in Free Space

Changing to units in which $\varepsilon_0$ $\mu_0$ and $c$ are 1, we may combine the two 3-vectors $\mathbf{E}$ and $\mathbf{B}$ into the 6-component Faraday 2-form $(\mathbf{E};\mathbf{B})$, or its dual, the Maxwell 2-form $(\mathbf{E};\mathbf{B})^*$.

And we may define the current 4-vector J as $(Q_f,\mathbf{j}_f)$.

Then the differential versions of Gauss' Law and the Ampère-Maxwell Law can be combined as:

$$\nabla \times (\mathbf{E};\mathbf{B})^*\,=\,(\nabla \cdot \mathbf{E}\ ,\ \frac{\partial\mathbf{E}}{\partial t}\,+\,\nabla\times\mathbf{B})^*\,=\,J^*$$

and those of Gauss' Law for Magnetism and Faraday's Law can be combined as:

$$\nabla \times (\mathbf{E};\mathbf{B}) = (\nabla \cdot \mathbf{B}\ ,\ \frac{\partial\mathbf{B}}{\partial t}\,+\,\nabla\times\mathbf{E})^*\,=\,0$$

(1) This can be shown using the Fundamental theorem of vector calculus.
(2) Technically we have only shown the speed of propagation of the electric field, we should also determine the speed of propagation of the magnetic field and show that it is the same as the electric field. However, we leave this as an exercise for the reader.

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