Gauss's Law for Magnetism: Uses, Derivation and Forms

Gauss's Law for Magnetism states that the total magnetic flux through a closed surface is zero. This one is a discrete form that is used in approximated or computational contexts whenever the surface S is broken into finite segments instead of infinitesimal parts. Mathematically, in the summation form, it is expressed as:

${\phi }_B=\sum \mathbf{B}\cdot \Delta \mathbf{S}=0$ 

• ${\phi }_B$refers to total magnetic flux that is passing through a closed surface. A positive flux indicates that magnetic field lines are leaving the surface, whereas a negative flux indicates that those magnetic field lines are entering the closed surface.
• ∑ indicates the summation over each surface element
• B represents the magnetic flux density vector
• ΔS refers to the outward-pointing area vector of surface patch (m²)
• B · ΔS indicates the dot product that provides flux through that patch

This equation states that there is no net magnetic flux b that passes through a closed arbitrary surface S. In simple words, it means that the number of magnetic lines entering and leaving the closed surface S is equal/same.

It also means that magnetic monopoles cannot exist. For every positive magnetic pole, there must be an equal number of negative magnetic poles.

This is a calculus-based/ continuous form of Gauss's law for Magnetism. IIT JAM entrance exam and JEE Main exam covers questions related to this form of Gauss law. This Gauss's law formula is used whenever there is a smooth surface and we need to calcuate for infinitely small elements of area.

$$\underset{S}{\overset{}{\oint }}\mathbf{B}\cdot \mathrm{dA\; =\; 0}$$

Here:

• ∮ is the closed surface integral over 'S' surface
• B is the magnetic field vector
• dA is the infinitesimal area vector on 'S' surface 
• B⋅dA produces the component of B that is passing through 'S' surface

Divergence Theorem relates the flux of a vector field through the closed surface S to the divergence of the field over volume enclosed by the surface. Through this theorem, the integral form can be converted to the differential form. The Divergence Theorem is stated as:

$$\underset{S}{\overset{}{\oint }}\mathbf{B}\cdot \mathrm{dA}=\underset{V}{\overset{}{\int }}(\nabla \cdot \mathbf{B})dV$$

Here:

∇ · B is the divergence of magnetic field B,

and,

dV is an infinitesimal volume element.

The integral form of Gauss’s Law for Magnetism states that total magnetic flux through any closed surface is zero, it follows that:

$$\underset{V}{\overset{}{\int }}(\nabla \cdot \mathbf{B})dV=0$$

NEET exam and CUET entrance exam aspirants must take a look at this derivation of Gauss's law formula. For this equation to be true for any arbitrary volume V, the integrand must be zero everywhere. As we obtain the differential form of Gauss Law Formula for Magnetism:

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

Through the following points, you will be able to understand the applications of Gauss's law for magnetism:

1. The Gauss's law for magnetism indicates that magnetic lines are continous, thus, they form closed loops. It also indicates that magnetic monopoles cannot exist.
2. Gauss's law for magnetism is used in design and analysis of magnetic materials and devices. It is used in the design of electric motors, transformers and generators.
3. Principles derived from this law of magnetism are used for designing the magnetic shields. IISER exam aspirants must know about this application since it is often asked in exams.
4. Gauss law in magnetism helps in understanding phenomena like magnetic anamolies.
5. This maxwell's equation describes how electric and magnetic fields are generated and how they are altered by each other.
6. Gauss's law for magnetism is one of the fundamental law used in the operation of MRI machines.

Let us take a look at some of the important questions related to Gauss law for Magnetism: