Question

if magnetic monopoles existed , how would the gauss' s law of magnetidm be modified? ????

Answer 1

Gauss’ law for magnetism says that the net magnetic flux through any closed surface is zero. Mathematically, it is written in the integral form as:

∬B⃗ ⋅dA→=0

An immediate consequence of this law is the non-existence of magnetic monopoles in nature. Extensive research has been done on the existence of magnetic monopoles and we dont have any evidence of their existence.

Suppose, if on some day in future we discover magnetic monopoles, that integral above will not remain zero anymore. A correction has to be made to the existing Gauss’ Law as follows:

Cosider a magnetic monopole of magnetic charge qm. Let us construct a spherical gaussian surface around the magnetic monopole, of radius R, with the magnetic monopole at the centre of the sphere as shown.

At every point on the sphere, the magnetic field due to the monopole and the Area vector are in the same direction.

Also, the magnetic field due to a magnetic monopole at a distance R is given by:

B=(μ04π)(qmR2)

Now, let us find the net magnetic flux through the sphere using the above relation. If dA→ is a small Area element on the sphere, then the total Magnetic Flux through the sphere can be found by integrating B⃗  over the surface of the sphere:

∬B⃗ ⋅dA→

=∬BdAcos(0°)

=∬(μ04π)(qmR2)dA

=(μ04π)(qmR2)∬dA

(The B⃗  field is the same for all points on the sphere, since every point on the sphere is equidistant from qm. This is why, (μ04π)(qmR2) has been brought outside the integral.)

∬dA is just the surface area of the sphere which is 4πR2.

Hence,

∬B⃗ ⋅dA→

=(μ04π)(qmR2)×4πR2

=μ0qm

Thus, if magnetic monopoles exist, the corrected version of Gauss’ law will be:

∬B⃗ ⋅dA→=μ0qm

Note that the Gauss’ Law for magnetism takes different forms depending upon the system of units we choose. While deriving the above expression, I have chosen the SI system. Similar approach works for other sytems of units.

Answer 2

Hey !!

Gauss law of magnetism describes that divergence of magnetic field will be zero while divergence of electric field is not zero which shows the non existence of magnetic monopole. As per Gauss law of magnetism

                                  ∫B.ds = 0

If monopole exists, then the right side will be equal to the monopole which is multiplied by μ₀.

Good luck !!