Month of Equations: What Does Gauss’ Law in Magnetism Really Mean?

                                         October 2: Gauss’ Law in Magnetism

2. Div B

Meaning of Equation: “Magnetic monopoles do not exist in nature

But how do we infer this one line result from the above equation. Let us find out.

The inverted triangle in the above equation is known as the del operator. It reads as ( i ∂/∂x + j ∂/∂y + k ∂/∂z ) where i, j 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!

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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.

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. This equation is one of the 4 Maxwell’s equations of electrodynamics.

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