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Maxwell’s Equation With Their Laws

Aug 10, 2022

Maxwell Equations

This article is all about Maxwell Equations. Let’s begin. 

Science is a subject based on applications, and it helps us learn about our environment differently. It has three disciplines— Physics, which involves the study of how the universe behaves; chemistry, which is concerned with the study of substances, their properties, and chemical reactions; and Biology, which involves the study of living organisms in the environment and how they function.


As discussed earlier, Physics deals with the study of the different processes that explain how nature behaves. Its applications are widely used in daily life, and its laws govern the universe. Mobile phones, car seat belts, camera lenses, earphones, steam iron, and ballpoint pens are some of the innumerable examples of the application of Physics in the day-to-day life of human beings.

The first person to calculate the speed of electromagnetic wave propagation was Maxwell. Maxwell explained that the speed was found to be the same as that of light, which made him conclude that electromagnetic waves and visible light are actually quite similar. 


Maxwell Equations

The Maxwell Equations are the foundation of electromagnetism, which includes the electromagnetic induction law given by Faraday, the Law of electricity given by Gauss, and the Law of current given by Ampere in a conductor. 

These Maxwell Equations give way to a mathematical model for electricity, static electricity, radio technologies, optics, power generation, radar, electric motor, lenses, etc. These equations describe the working nature of electric and magnetic fields and how they are produced by charges, currents and due to changes in electric or magnetic fields.


These Maxwell Equations are named after Scottish mathematical physicist James Clerk Maxwell, who formulated the classical theory of electromagnetic radiation. He published these questions by including the Lorentz Force law between the years 1861 and 1862. Maxwell’s first equation discusses the electromagnetic nature of light.

Maxwell First Equation

The basis of Maxwell’s first equation is the Gauss electrostatic law, according to which “the total electric flux of an enclosed surface is equal to the charge inside it divided by the permittivity.” In mathematical terms, Gauss law is expressed as “the product of the surface integral and the electric flux density vector over a closed surface equals the charge enclosed by the surface.” 


The Law of electricity by Gauss establishes the relationship between an electric field that is static and the electric charge produced by it. The direction in which a static electric field point is always away from the positive charge and towards the negative charge. The Law also explains that the total outflow of the electric field from a closed surface is directly proportional to the total amount of charge within that surface.

The lines of the electric field start at a positive charge and end at a negative charge. The net number of the lines of the electric field that cross a closed surface, divided by the free space dielectric constant, gives the net amount of charge on that surface. 


The integral form of Maxwell’s first equation:

e = q/e0 ——– (i)


Additionally, e = ∫E⃗ .dA⃗ ——- (ii)

Comparing equations (i) and (ii), we have:


∫E⃗ .dA⃗ = q/∈₀   —- (iii)

Differential form of Maxwell’s first equation:

The amount of total charge density in terms of the density of volume charge is q = ∫pdv

So, the equation (iii) becomes:

∫E⃗ .dA⃗ =1e0∫Pdv

If we apply the divergence theorem to the left-hand side of the equation, we get,

∫(▿⃗ .E⃗ )d.V=1ϵ0∫pdv

∫(▿⃗ .E⃗ )d.V−1ϵ0∫pdv=0

∫[(▿⃗ .E⃗ )−Pϵ0]d.V=0

(▿⃗ .E⃗ )−Pϵ0

Maxwell Second Equation

Maxwell’s second differential equation is based on Gauss’s Law of Magnetism, which states that the total magnetic flux of a magnetic field crossing a closed surface is zero. This is due to the fact that magnets always exist in dipoles. There are no magnetic monopoles.

The magnetic field is created because of the dipole nature of the magnetic field. The total outflow from a magnetic field through a closed system is zero. Magnetic dipoles act as current loops with negative and positive magnetic charges that cannot be separated from one another. 

Gauss’s Law of magnetism says that loops are created by magnetic field lines that originate from the magnet and extend to infinity and vice versa. In other words, if the magnetic field lines enter a body, they will also come out of it. On a Gaussian surface, the total magnetic field is zero. Also, it is a solenoidal vector field.

Above is a representation of the magnetic field lines that neither start nor end; rather, they form loops.

Maxwell’s Third Equation

Maxwell’s third equation is based on Faraday’s Law of Electromagnetic Induction. Faraday’s Law of Induction was modified by Maxwell, and it describes the electric field produced by a magnetic field that is time-varying. According to this law, the work required to move a unit charge around a closed structure such as a loop is equal to the magnetic field transforming around that loop. 

The electric field lines induced are similar to the magnetic field lines if they are not superimposed by a static electric field. This electromagnetic induction concept is the basis of the operating principle behind several electric devices, such as rotating bar magnets used to create varying magnetic fields. This further leads to the production of electric fields in a conducting wire lying nearby.

The magnetic field of the Earth is altered in a geomagnetic storm because of an increase in the flux of charged particles. This further leads to the induction of an electric field in the atmosphere of the Earth. 

Integral Formula for Maxwell’s Third Equation:

∈ = -Ndm/dt- ————– (1)

As the electromotive force is related to the electric field by the relation ∈ =∫E .dA, if we put these values in equation 1, we get:

∫E.dA=−N∫E .d A∫B⃗ .d A

If N = 1, we get

∫E .dA=−ddt∫B .d A

Differential Formula for Maxwell’s Third Equation:

Applying stoke’s theorem, we get:

∫(▿⃗ .E )dA =−ddt∫B .dA

∫(▿⃗ .E )dA +ddt∫B .dA =0

(▿⃗ .E )+dB dt=0

(▿⃗ .E )=−dB dt

Maxwell’s Addition to Ampere’s Law

According to this law, the magnetic field can be created either by altering the electric field or by electric current. The second part of the statement is as per Ampere’s Law, while the first part is as per Maxwell’s addition. The magnetic field that is induced around a closed loop is related to the current displaced through that closed-loop or the electric current in it directly. 

This Law establishes a relationship to form an equation set mathematically aligned with the non-static fields without altering Gauss’s Law for static fields and Ampere’s Law. However, a magnetic field is produced by a changing electric field and vice versa. Therefore, a possibility for electromagnetic waves that can sustain themselves is created by these equations so that they can travel through a vacuum.

According to the observations and calculations, the electromagnetic wave speed equals light speed. Like radio waves and X-rays, light is also a form of electromagnetic radiation. Maxwell established the relationship between light and electromagnetic waves in 1861. From there, the theories of electromagnetism and optics were unified. 


These equations given by Maxwell, along with the Lorentz law of force, are fundamental to classical optics, electric circuits, and classical electrodynamics. The integral form of Maxwell’s equation best explains how electric currents and electric charges form electric and magnetic fields. It describes how a magnetic field is created by an electric field and vice versa. Hope this article was able to clear all your doubts about  the Maxwell Equations. 

Frequently Asked Questions

1. What is the significance of the maxwell equations?

Maxwell’s equation best explained how electric and magnetic fields are created by varying currents and charges. The navigation of varying magnetic and electric fields at the speed of light is described by these equations. They are fundamental to the functioning of most modern devices and appliances like computers, mobile phones, and electricity.

2.What is the significance of Maxwell’s equations in thermodynamics?

The changing of thermodynamic variables from one set to another is aided by Maxwell’s equations. Suppose you are required to calculate the entropy change of a system at a constant enthalpy and with regard to a given pressure. Although the temperature, volume, and pressure of a system can be measured very easily, there is no device for measuring the entropy of a system. 

3.What did Maxwell alter Ampere’s Law?

Maxwell made some modifications to Ampere’s Law as he found some shortcomings in it. For Ampere’s circuital Law to be accurate, he assumed that some current must exist between the capacitor’s plates. The current outside the capacitor was due to electron flow.

4. Who are maxwell equations named after?

Maxwell gave three equations. The Maxwell addition to Ampere’s Law is also an important equation in this respect. Maxwell equations are named after the Scottish physicist James Maxwell, who gave the classical theory of electromagnetic radiation.

Maxwell Equations


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