Gauss' Law In Magnetism | Month of Equations

Today is the ninth day of Month of Equations series and so far we have discussed only one equation on electrodynamics: The Poisson's equation. I'm going to shed light on yet another famous equation: the Gauss' law in magnetism. This equation is one of the four Maxwell's equations. Let us see what it says.

The Gauss' law in magnetism

Gauss' Law in Magnetism
The Gauss' in Magnetism

The meaning of this equation is simple. Gauss' law in magnetism means that there exist no magnetic monopoles in nature. But how does mathematical equation comes to this conclusion? To understand this, we need to learn about the "inverted triangle" shown in the equation above.

The Divergence of a vector

The inverted triangle in the above equation is known as the del operator. It reads as ( i ∂/∂x + j ∂/∂y + k ∂/∂z ) where ij and k are unit vectors along x, y and z direction respectively. This operator acts on some function, in this case the magnetic field. If we take the dot product of this operator with a vector, it is known as divergence. Getting complicated? Let us make it easier.

Divergence, as the same suggests, answers the question: Does any point in space act as a source or sink of some quantity? Suppose you take two reference points A and B along the course of a river separated by some distance. Suppose x units of water is crossing point A. If the same amount of water crosses B after travelling some distance, then there is no divergence in the flow of water (here our vector function is the water flow).

If lesser water crosses B than A, then the field (water flow) has negative divergence, i.e. it has a sink and if more water flows through point B, then there is positive divergence, i.e. there is a source in the field. All we want is the net flow, meaning there can be any number of sources or sinks between the points A and B, but if the total flow through B remains same as that of A, then also net divergence is zero. Simple enough to understand!

Gauss' law in Magnetism
This integral equation is equivalent to the equation written above. This is the integral form of Gauss' law in magnetostatics while the above equation is the differential form of the same.

The divergence of magnetic field

Now this equation says that divergence of magnetic field B is zero. From our above understanding of divergence, this means there is no source or sink of magnetic field anywhere in the universe. But we know that magnetic field has a source, the magnet. Here is the point! This equation tells that the net divergence of B is zero.

So there are equal number of sources and sinks of magnetic field in the universe. Here source is the north pole of the magnet from where field lines originate and the sink is the south pole of the magnet where the field lines end. So even if you cut the magnet into two pieces, there is a new magnet with new north pole and new south pole to keep the net divergence zero. So magnetic monopoles (isolated north or south pole) do not exist in nature. If they exist, the R.H.S. of above equation will have to change.

Previous in series: The Bethe-Weizsaecker formula in Nuclear Physics

Author's Message

At first I thought of writing all the 4 Maxwell's equations as one single article in this series. But then I split the four into different articles because I wanted to explain them in detail. Through the Gauss' law in magnetism, I explained the concept of divergence too. This will make it easier to understand the Gauss' law in electrostatics. The subject of electrodynamics is of fundamental importance in theoretical and experimental physics. If you want to deeply study this subject, the book "An introduction to electrodynamics" by D. Griffiths is one of the best books (and of course the one by J.D. Jackson).

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