Gauss’ Law Of Magnetism

What is Magnetism? 

It is the property of any object that can attract a piece of iron or steel, such as natural magnets and artificial magnets.

Bar magnet: It has two poles, the north pole and the south pole. Magnetic field lines produced from the bar magnet create a closed-loop.

If the bar magnet is cut into two equal pieces transversely, then there is no change in the pole strength of the bar magnet. But if it is cut longitudinally into two equal pieces, the pole strength of each piece becomes half of the original value.

What is Gauss’ law?

Gauss’ law includes two statements, which describe electric and magnetic flux. Gauss’ law of magnetism states that magnetic flux across any closed surface is zero. This law observed that isolated magnetic poles (monopoles) do not exist.

What is Magnetostatics?

It is the study of magnetic fields where the current is not changing with time, i.e. it is steady. The steady current produces the magnetic field.

 dB = [(μ)(I) (dL) (x)(Ir)] /4πr2 

 Where dL = infinite small length of a conductor carrying electric current.

 Ir = unit vector to specify the direction of the vector distance r from the current to the field point.

And this is known as Biot-Savart’s law.

What is Biot-Savart’s Law?

This law states that the magnetic field due to the current-carrying conductor is directly proportional to a) the magnitude of the current. b) length of element dL. c) sine of the angle between r and dL and inversely proportional to the square of the distance between source and the field point.

B = ഽdB = ഽμ⦁IdL sinθ/4πr2

This law is also used in the derivation of Biot-Savart’s law.

Magnetic Flux

It is defined as the number of magnetic lines of force passing through any surface.

  𝛟 = B.dS 

Gauss’ law of magnetism states that the net magnetic flux linked with any closed surface in a magnetic field is always zero. The integral equation for this is,

∮B.dS = 0           …….(1)

 Where, 𝛟 = B.dS 

Therefore, 𝛟 = 0

Magnetic field lines always create closed loops, so the number of magnetic field lines entering the surface will be equal to the number of magnetic field lines existing for any given Gaussian surface. This is explained by the concept of a magnet with a north and a south pole, where the strength of the north pole is equal to the strength of the north pole. This law reflects the fact that magnetic monopoles do not exist.

Ampere’s circuital law states that the line integration of the magnetic field along the closed path is μ⦁ times the current threading through.

∮B.dl =μ⦁I 

Here, there is some value because of line integration, but in Gauss’ law of magnetism, there is a surface integration. That is why it is zero.

This also tells us about the nature of magnetic fields, that they do not have any starting and terminating point.

Differential Equation

Gauss’ law of magnetism has a differential equation that can be derived using the divergence theorem. The divergence theorem states:

∫(▽.f)dv=∮f.dS           …….(2)

Where f is a vector.

Now, by using divergence theorem equation (1) rewritten as follows:

0=∮B.dS=∫(▽.B)dv

Because the equation is set to zero, the integrand(▽.B) must be zero. Thus the differential form of Gauss’ law becomes:

▽.B=0

Derivation using Biot-Savart’s law:

Gauss law can be derived using Biot-Savart’s law.

Here, Biot-Savart’s law can also be written as:

Φ(r) = [(μ) (∫J(r’)dv (r)]/ 4π|r-r’|2

whereΦ(r) is the magnetic flux at any point r.

J(r’) is the current density at r’.

Now, taking the divergence of both sides of this equation:

▽.Φ(r) = [(μ) ∫▽.J(r’)dv (r)]4π|r-r’|2

To carry the divergence of the integrand in this equation, the following vector identity is used:

▽.(A ✕ B) = B. (▽ ✕ A) – A.(▽ ✕ B)

Thus, the integral becomes:

[J(r’).(▽ ✕ (r|r-r’|2))]-[(r|r-r’|2).(▽ ✕ J(r’))]

The first part of this equation is zero as the curl of r|r-r’|2 is zero. And the second part of this equation becomes zero because J depends on r’ and ▽ only on r. Thus we see that:

▽.Φ(r) = 0

Which is known as Gauss’ law of magnetism in differential form

Difference between Gauss’ Law in Electrostatic and Gauss’ Law in Magnetism

Gauss’ law in electrostatic extends to magnetism because Gauss’ law in electrostatic states that the electric flux coming out of the close surface is 1/𝜖͒ times the charge(q) inside the closed surface, that is:

   𝛟 = ∮E.dS = q/𝜖͒

But in magnetism, Gauss law is always zero because we cannot isolate the North and South poles.

Conclusion

We learned about the basic and main concept of Gauss’ law of magnetism, which states that the net magnetic flux linked with any closed surface in the magnetic field is always zero. Magnetism is a property of any object which can attract a piece of iron and steel and produce magnetic field lines.