This is Part four(Differential form of Gauss’s Law for Magnetic Field) of my eight part series on Maxwell Equations.

I had already covered Gauss’s law for electric field in two articles and links to both these articles are

- Maxwell’s Equations – Integral form of Gauss’s Law (Part 1)
- Maxwell’s Equations – Differential form of Gauss’s Law (Part 2)

Previous Article in this series was about Integral form of Gauss’s Law for magnetic fields and its link is

Now this article would be about Differential form of Gauss’s Law for Magnetic Field

The differential form of Gauss’s Law for magnetic fields is written as

$\vec{\nabla} \cdot \vec{B}=0$

The left side of this equation is simply a mathematical description of the divergence of the magnetic field which is the tendency of the magnetic field to flow more strongly away from a point than toward it and the right side is simply zero. So Gauss’s Law in differential form states that

The divergence of magnetic field at any point is zero

Now in case of differential form of Gauss’s law for electric fields divergence of electric field is proportional to electric charge density but here in case of magnetic fields divergence of field at any point is zero because here it is not possible to have isolated magnetic poles as magnetic poles always appear in pair of north and south poles. So there is no such thing as magnetic charge density and this means that divergence of magnetic field is zero.

Now I shall write about the meaning of each symbol used in the mathematical form of differential form of Gauss’s law for magnetic fields

- $\vec{\nabla}$ : $\nabla$ is the differential operator known as ‘del’ or ‘nabla’ operator and overhead arrow reminds that this del operator is a vector.
- $\cdot$ : The dot product turns the del operator into the divergence .
- $\vec{B}$ : $\vec{B}$ is the magnetic field measured in Tesla and it is a vector quantity.
- $\vec{\nabla} \cdot \vec{B}$ of the magnetic field : This expression gives the divergence of the magnetic field. Since divergence is by definition the tendency of a field to ‘‘flow’’ away from a point and since no point sources or sinks of the magnetic field are present so the amount of ‘‘incoming’’ field is exactly the same as the amount of ‘‘outgoing’’ field at every point. Vector fields with zero divergence are called ‘‘solenoidal’’ fields, and all magnetic fields are solenoidal.