Maxwell's Equations

Similar to Newton's laws of motion in mechanics and the laws of thermodynamics in thermodynamics, Maxwell's equations are the synthesis of electromagnetism! The electricity and magnetism are not separate but they are together. You'll know from the analysis of Maxwell's equation that one kind of field induces another kind and vice versa and they can exist together and propagate in space as electromagnetic wave. If we consider the Newton's laws of motion as the heart of Mechanics, laws of thermodynamics as the heart of thermodynamics, Maxwell's equations are the heart of electromagnetism.

Displacement Current and Ampere's Law

Before listing all Maxwell's equations, we need to fix the Ampere's law. What if a circuit has a capacitor that's charging. The capacitor plates are not connected to each other and the circuit is open. And if you make an amperian loop or closed path between the plates, there is no enclosed current but the circuit indeed has conduction current $i_C$ in the wires.

To address this caveat, Maxwell introduced a new current called displacement current $i_D$. In Figure 1, you can see the flat surface labelled $S_1$ and curved surface $S_2$, where there is conduction current $i_C$ through $S_1$ but there is no current through $S_2$ because there is only either vacuum or insulator between the capacitor plates. Note the surface $S_2$ is not the flat surface like $S_1$. Since the current is the same, surface $S_2$ is curved or extended to include the space between the plates; both surfaces are part of the same amperian loop.

Figure 1 Displacement current; the flat surface encloses conduction current but the curved surface encloses displacement current equal to conduction current.

From the above figure, for the same amperian loop, there is current for one surface and no current for other surface. The new Maxwell's displacement current $i_D$ through the capacitor plates equal to the conduction current $i_C$ in the wires solves the problem of discontinuity of current in the circuit. The word "displacement" for current is historic and it has nothing to do with the displacement of motion. If you include this current in Ampere's law, it has the form,

\[\oint \vec B \cdot d\vec l = \mu_0(i_C + i_D)_\text{encl} \tag{1} \label{1}\]

In the wires, the displacement current is zero, and within the plates the conduction current is zero and hence we have a fully functional Ampere's law. Is the Maxwell's displacement current $i_D$ is really imaginary? Did we just add a factious current to solve our problem? Does this current really exist? To answer these questions, let's dig a little bit more about this current.

As the capacitor is charging (or discharging), the electric flux changes within the plates, that is the changing electric field is considered to be equivalent to the current between the plates. You know from capacitance that, if $q$ is the charge on the plate at any instant and $v$ is the voltage between the plates at that instant,

\[q = \epsilon_0 \frac{A}{d}v = \epsilon_0 \frac{A}{d}E\,d = \epsilon_0 \Phi_E\]

where $\Phi_E$ is the electric flux. We know the conduction current $i_C$ is $dq/dt$ and therefore,

\[i_C = \epsilon_0 \frac{d\Phi_E}{dt}\]

Since this flux change takes place within the plates, we can regard this change in electric flux as equivalent to the displacement current within the plates, so

\[i_D = \epsilon_0 \frac{d\Phi_E}{dt}\]

Now we have the displacement current and the corresponding displacement current density $j_D$ is $j_D = i_D/A$. If you want to restate the Ampere's law mathematically including displacement current, you can rewrite the expression of Ampere's law as,

\[\oint \vec B \cdot d\vec l = \mu_0(i_C + \epsilon_0\frac{d\Phi_E}{dt})_\text{encl} \tag{2} \label{2}\]

Now we have the full form of Ampere's law without any problems. This corrected law is sometimes called Ampere-Maxwell law. Now we are ready to list all four equations Maxwell put together. He did these equations, but introduced displacement current. What about the magnetic field of displacement current if it is really a current?

If there is really a kind of current within the plates, there must be corresponding magnetic field similar to that of a straight wire within the region between the plates. In Figure 2, the we make a circular closed path of radius $r$, and we have circular capacitor plates of radius $R$. The magnetic field is tangent at each point on our integration path. Therefore, Ampere's law gives $\oint \vec B \cdot d\vec l = 2\pi \, r \,B$.

Figure 2 Magnetic field of displacement current.

The total current enclosed by our closed path is the current density $j_D$ times the area of the path, that is $j_D\, \pi\,r^2$. Note that $j_D = i_D / \pi\,R^2$ and the current enclosed $I_\text{encl} = i_D\,r^2 / R^2$. The Ampere's law tells us that the line integral over the closed path is the $\mu_0$ times the total current enclosed by the path, so $2\pi\,r\,B = \mu_0\,i_D\,r^2 / R^2$ and the magnetic field is

\[B = \frac{\mu_0}{2\pi} \frac{r}{R^2}i_D = \frac{\mu_0}{2\pi} \frac{r}{R^2}i_C\]

Note that $i_D = i_C$ for charging or discharging capacitor. The experimental measurements of magnetic field within the plates reveal that there is indeed the magnetic field given by the above equation. This remarkable discovery of changing electric field being equivalent to current and the changing electric field is inducing the magnetic field has great importance to understand the electromagnetic phenomena. In other words, it is clear from the above discussion that both current and changing electric field produce magnetic field.

Four Maxwell's Equations

Now we list four equations Maxwell put forward which allow us to understand all electromagnetic phenomena. The four equations are Gauss's law for electric fields, Gauss's law for magnetic fields, Faraday's law and Ampere's law(Ampere-Maxwell law) and they are explained below.

Gauss's law for electric fields is

\[\int \vec E \cdot d \vec A = \frac{q_\text{encl}}{\epsilon_0} \tag{3} \label{3}\]

Gauss's law for magnetic fields is

\[\int \vec B \cdot d\vec A = 0 \tag{4} \label{4}\]

Faraday's law is

\[\oint \vec E \cdot d\vec l = -\frac{d\Phi_B}{dt} \tag{5} \label{5}\]

Ampere's law (Ampere-Maxwell law) is

\[\oint \vec B \cdot d\vec l = \mu_0(i_C + \epsilon_0 \frac{d\Phi_E}{dt})_\text{encl} \tag{6} \label{6} \]

These four Equations \eqref{3}, \eqref{4}, \eqref{5}, \eqref{6} are called Maxwell's four equations and here we expressed these equations for vacuum for simplicity; all of these equations above are applicable to vacuum where there is no dielectric or magnetic material.

The Gauss's law of electric field tells us that, the net electric flux through any surface is equal to the charge enclosed by the surface divided by $\epsilon_0$. This law describes the electric field of a charge distribution that created it.

Since no magnetic monopole has ever been discovered, the Gauss's law for magnetic fields states that the net magnetic flux through any closed surface is always zero. What happens is that, the magnetic field lines entering the closed surface must come out of the surface and there is no net flow of lines out of the surface. This law is the confirmation that no magnetic monopole has ever existed despite they are part of continuous search.

Next is the Faraday's law of electromagnetic induction. This law states that the line integral of electric field over a closed path is the negative of the rate of change of magnetic flux for the surface bounded by the path. The line integral of the electric field over the closed path is the induced emf due to the change in magnetic flux. This law demonstrates that the changing magnetic flux or field creates the electric field.

Our final equation is the corrected form of Ampere's law sometimes called Ampere-Maxwell law. The ampere law basically tells us that the line integral of magnetic field over a closed path is $\mu_0$ times the total current enclosed by the path. Now the displacement current completes this law for any discontinuity of current in the circuit such as in the case within the capacitor plates providing the fact that the changing electric field is equivalent to the current and produces its own magnetic field. In other words a changing electric flux or field creates the magnetic field.

With all those equations in place, we need one more to explain all electromagnetic interactions. If there are both electric and magnetic fields, the total force on a charge $q$ moving with velocity $v$ is

\[\vec F = q\vec E + \vec v \times \vec B \]

This total force is called Lorentz's force. If the charge is stationary, it's velocity is zero and there is no magnetic force. Using Maxwell's equations and this force, we can explain all relationships in electromagnetism!

Maxwell's equations are the heart of electromagnetism! These equations predict the relationships between electric and magnetic fields and their dependence on each other. These equations explain how these two fields are two sides of a coin, that is they are combined to form one entity, that is one field induces another field and can co-exist (example is electromagnetic wave). Maxwell's equations contain all basic relationships of electromagnetism and therefore they form the basis of electromagnetism. For example, you can derive Coulomb's law from Gauss's law and Biot-Savart law from Ampere's law etc.

Symmetry in Maxwell's Equations

The Gauss's law for electric and magnetic fields are straightforward. Here we look at the Faraday's law and the Ampere-Maxwell's law. The remarkable symmetry between these two equations is that (you may be already familier with if you read above text) is that, a changing (time-varying) electric field produces magnetic field and a changing (time-varying) magnetic field produces electric in the neighboring region of space.

The equations that lead us to conclude the field of one type can induce the field of another type are the equation of Faraday's law of electromagnetic induction and the equation of Ampere-Maxwell's law. This discovery led Maxwell understand that there is something called electromagnetic wave. From the Faraday's law of electromagnetic induction, you can conclude that a changing magnetic field can induce electric field, and from the Ampere-Maxwell law, you can conclude that a changing electric field can induce magnetic field. This is the most amazing symmetry in nature.

This symmetry of one field can induce another field led Maxwell to predict that both fields can exist together and can propagate in space without any medium. And such propagation of both fields in the form of a wave is called electromagnetic wave. We can obtain the wave equation for electromagnetic wave using Equations \eqref{5} and \eqref{6} and can determine the speed of the electromagnetic wave.

Hertz (Heinrich Rudolf Hertz) performed experiments that directly confirm the theoretical predictions of Maxwell's equations. Hertz found from his experiments that the speed of the radiation was close to $3 \times 10^8 \text{m/s}$. It is a known fact that the speed of electromagnetic wave is equal to the speed of light and this led to understand that light waves are electromagnetic waves.

Electromagnetism
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