Maxwell Equations

Introduction

If you have spent some time around the concepts of electrodynamics, chances are that you have already come across Maxwell’s equations in one form or another. Even if you didn’t know what they were called at the time, they still must have found their way into your concepts.

Maxwell’s equations form the fundamentals of electrodynamics. Along with the Lorentz Force Law, these four equations form the basis of electrodynamics, and every single problem can be tackled using these five equations alone. In this tutorial, we will discuss displacement current and Maxwell’s equations.

What Is Displacement Current?

You must have studied Gauss’s Law, Ampere’s Law, Faraday’s law, etc. You must have also studied that the divergence of magnetic flux density $\mathrm{\overrightarrow{B}}$ is a vanishing quantity. That is $\mathrm{\overrightarrow{\nabla}.\overrightarrow{B}\:=\:0}$ These equations are well and good for the static case. However, when the current is not constant in time, then Ampere’s law starts to fall apart.

To correct the problems in Ampere’s law, Maxwell proposed the addition of a term known as the displacement current density, represented as the time derivative of the displacement field, 𝐷⃗→. Displacement current density physically represents the contribution of a time-varying electric field to Ampere’s law.

Note that displacement current density is not an actual current, per se. Instead, it is the time derivative of the electric displacement field. Mathematically,

$$\mathrm{\overrightarrow{D}\:=\:\varepsilon_{0}\overrightarrow{E}\:+\:\overrightarrow{P}}$$

Here, 𝜖0 is the permittivity of free space, $\mathrm{\overrightarrow{E}}$ is the applied electric field, and $\mathrm{\overrightarrow{P}}$ is polarization. In such a case,

$$\mathrm{Displacement\:current\:Density\:=\:\frac{\partial\overrightarrow{D}}{\partial\:t}}$$

Maxwell First Equation

While there is no particular order in which Maxwell’s equations must be discussed, one can consider Gauss’ law to be the first one owing to its simplicity.

According to Gauss’ law, the amount of electric field that flows out of a surface is proportional to the charge enclosed within it. Note that we are talking about a static electric field in this case. The choice of the surface is completely arbitrary while discussing Gauss’ law, and we choose the most convenient one.

Mathematically,

$$\mathrm{\overrightarrow{\nabla}.\overrightarrow{E}\:=\:\frac{Q}{\varepsilon_{0}}}$$

𝑄 represents the total charge enclosed within the surface of our choice.

Maxwell Second Equation

For the second equation, we turn our attention to the relationship that is often referred to as the Gauss law for magnetism, though it isn’t truly Gauss’ law. This relation holds universally true and there is no situation in which it is violated.

The second equation implies the vanishing divergence of magnetic flux density. Mathematically,

$$\mathrm{\overrightarrow{\nabla}.\overrightarrow{B}\:=\:0}$$

The physical significance of this equation is that magnetic monopoles cannot exist. Thus, you can have positive and negative charges that exist independently of each other. However, you cannot somehow have a situation where there is a north pole and a south pole that are independent of each other. Even if you cut a bar magnet in half, it would lead to two pairs of south and north poles, not one isolated north and one isolated south pole.

Maxwell Third Equation

The concept behind the third equation is quite interesting and suggests that a changing magnetic field gives rise to an electric field. Thus, this relation is the one that establishes a connection between electricity and magnetism.

This concept was given by Faraday and is also referred to as Faraday’s law of electromagnetic induction. It reads like so −

$$\mathrm{\overrightarrow{\nabla}.\overrightarrow{E}\:=\:-\:\frac{\partial\overrightarrow{B}}{\partial\:t}}$$

The minus sign represents the opposite direction. This is Faraday’s law of electromagnetic induction represented in differential form and terms of electric and magnetic fields. You must have studied it in an alternate form as:

$$\mathrm{\varepsilon\:=\:-\:\frac{\;d\phi}{dt}}$$

Here, 𝜖 is the EMF induced, while 𝜙 is the magnetic flux passing through the area of the coil.

Maxwell Fourth Equation

The fourth Maxwell equation is the most elegant and ground-breaking discovery in the history of electrodynamics. The third equation establishes a relation between electricity and magnetism, while this equation strengthens that connection and makes it two-way instead of one.

As per this equation, a changing electric field gives rise to a magnetic field, just as a changing magnetic field gives rise to an electric field.

Originally, this was Ampere’s law and only connected magnetic flux density with the current enclosed. However, Maxwell discovered that in the dynamic case, it falls apart and thus, “fixes” it by including an additional term. Now, the fourth Maxwell equation is known as Ampere-Maxwell law and reads as below −

$$\mathrm{\overrightarrow{\nabla}\times\overrightarrow{B}\:=\:\mu_{0}(\overrightarrow{J}\:+\:\varepsilon_{0}\frac{\partial\overrightarrow{E}}{\partial\:t})}$$

𝐽→ is the electric current density.

Significance of Maxwell Equations

Maxwell’s equations are extremely important and fundamental relations in electrodynamics within which lies the entirety of our knowledge of electrodynamics. You must be aware that the speed of light in free space is a universal and fundamental constant. It can be shown that electromagnetic waves travel at this velocity by combining Maxwell’s equations and comparing them with the general wave equation.

Maxwell’s equations are also used in various forms to solve boundary value problems and analyse different scenarios with considerable ease. These equations also play quite well with quantum mechanics and Einstein’s general theory of relativity.

Conclusion

Displacement current density represents the contribution of a time-varying electric field to the magnetic flux density and appears in Maxwell’s equations to compensate for the problems encountered in Ampere’s circuital law. It holds meaning in the case of varying electric fields only and vanishes when the electric field is constant in time. If $\mathrm{\overrightarrow{D}}$represents the electric displacement field, then displacement current density is given by $\mathrm{\frac{\partial\overrightarrow{D}}{\partial\:t}}$

Maxwell’s equations form the fundamentals of electrodynamics and along with Lorentz Force Law, can completely explain the concepts encompassed therein. These equations are extremely important for understanding the concepts of electrodynamics and can be easily adjusted for the static case as well.

FAQs

1. Is displacement current different from displacement current density?

The nomenclature is quite confusing. The term displacement current in itself does not represent a physical quantity that is used in electrodynamics. Instead, we either deal with the electric displacement vector 𝐷⃗→, or its time derivative, which is the displacement current density.

2. What would happen to the second Maxwell equation if magnetic monopoles could exist?

In such (as of yet, not realized case), the divergence of magnetic flux density would be non-zero. Its sign would depend on the exact scenario in question.

3. How do we prove that an electromagnetic wave travels at the speed of light?

This is achieved by representing Maxwell’s equations in terms of scalar and vector potentials. We essentially arrive at an inhomogeneous wave equation for the electric and magnetic fields, which gives us the value of the speed of the wave

4. I have seen an integral representation of Maxwell’s equations. Are they correct?

Yes. Maxwell’s equations can be represented in both differential and integral forms. Converting from one form to another is quite easy by applying the concepts of mathematical physics.

Q5. Where does the name displacement current density come from?

The correction in Ampere’s law that Maxwell performed included adding the term $\mathrm{\mu_{0}\epsilon_{0}\:=\:\frac{\partial\overrightarrow{E}}{\partial\:t}}$ . This can be connected to the electric displacement vector,$\mathrm{\overrightarrow{D}}$ , Since, $\mathrm{\overrightarrow{D}\:=\:\varepsilon_{0}\overrightarrow{E}\:+\:\overrightarrow{P}}$