Maxwell's Equations And Their Meaning

Today, the flagship equation of physics is the mass energy equivalence, the E = mc^2. This equation is so famous that even a layman has come across it or has seen it somewhere at least once in his life. The mass energy equivalence belongs to the theory of special relativity, devised by Albert Einstein in 1905. But, what is the origin of special relativity? The answer is Electricity and Magnetism. Yes! You read it correct. The roots of theory of relativity are embedded in the four famous equations of electricity and magnetism, the Maxwell's Equations.

The Four Maxwell's Equations

Maxwell's Equations
The Maxwell's Equations

James Maxwell was a theoretical physicist who proved Faraday right. Faraday hypothesised that the light is an electromagnetic wave. Faraday wasn't a rigorous theoretical scientist, but Maxwell was. These 4 equations are not formulated by Maxwell. The first two equations are Gauss's laws in electricity and magnetism respectively, the third equation is the Faraday's law and the fourth one is the Ampere's circuital law with a correction term that was introduced by Maxwell. So why are they named after him? The reason is that Maxwell brought these independent equations of electricity and magnetism together and showed that the two are deeply connected.

Maxwell's equations were the turning point in the history of mankind. As Carl Sagan said," They have had a greater impact on human history than any ten presidents." Let us now look at these game changing equations and what do they really mean.

1. Gauss' Law of Electrostatics

Maxwell's Equations
Gauss' law in electrostatics. Remember, if the electric field did not follow the inverse square law, Gauss' law would not have been possible.

Meaning of equation

Gauss' law in electrostatics tells us three important things. First: There can exist an independent electric charge in space. This means a single positive charge or a single negative charge can have its own existence. They do not have to exist in pairs. Second: The number of electric field lines that comes out of a volume is directly proportional to the charge that is contained in it. Third: The electric field, like the gravitational field, follows the inverse square law. Thus the Gauss' law is just another version of Coulomb's law in electrostatics. To read about Gauss' law in detail, click here.

2. Gauss' Law In Magnetostatics

Maxwell's Equations
Gauss' law in magnetostatics.

Meaning of Equation

This is the easiest of the four Maxwell's equations. The one line meaning of this equation is: Magnetic monopoles do not exist in nature. So if you cut a bar magnet into two pieces, you won't have a separate north and south poles. Instead, you'll have two new bar magnets, each having its own pair of north and south poles. If magnetic monopoles are discovered, the right hand side of the equation will change. To read this law in detail, click here.

3. Faraday's Law

Maxwell's Equations

Meaning of Equation

If you are using anything that runs on electricity, thank this equation. Nothing would have been possible without this third equation, known as the Faraday's law. This equation is the architect of the modern world. It says, A changing magnetic flux through a closed circuit will produce an electromotive force in that circuit. A remarkable statement in itself. Thus you can produce an electric field from the magnet itself. No other equation captures the beautiful link between the two diverse fields than this one. The Faraday's equation made it possible to construct solenoids, electric motors, generators and transformers. To read about this law in detail, click here.

4. Ampere's Circuital Law

Maxwell's Equations - Ampere's Circuital Law
Ampere's Ciruital Law

Meaning of Equation

The Ampere's law tells us that a current carrying wire produces a magnetic field that circles around the wire and is proportional to the amount of current flowing through the conducting wire. Well, this equation was a landmark in itself. It tells that all the magnetic effects are actually caused by the currents, which are nothing but charged particles in motion. To read about this law in detail, click here.

Another important thing about this law is that it is mathematically incorrect in the form given above. This was noticed by Maxwell and he added a new term to the law. So the modified Ampere's law is:


The second term on the right hand side is the correction added. This not only corrects the mathematics of the equation but also adds a beautiful symmetry to the Maxwell's equations. The correction actually means that we can have a magnetic field out of a changing electric field, just opposite to the third equation, which tells that electric field can be generated out of a changing magnetic field.

So now you can see how important these equations are in physics. These 4 equations along with the continuity equation are the building blocks of whole of electrodynamics.

Also Read: Basics of Astrophysics Series

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