Magnetic Flux: Definition, Magnetic Field, Gauss’s Law

Magnetic flux is a very important physical quantity in the field of electromagnetism. It is a measurement of the total magnetic field that passes through an area. The measurement of magnetic flux depends on the particular chosen area. The given article will discuss the magnetic flux in detail and learn other quantities like the magnetic field lines and magnetic field. We will learn about one of the four maxwell’s equations which is analogous to Gauss’s law which we studied in electrostatics. We will also discuss various phenomena related to magnetic flux, such as electromagnetic induction.

Magnetic Field

We all have seen magnets attract iron nails kept near them, but how do they attract it without coming in contact. This phenomenon can be explained using field, so we also have a magnetic field just as we had an electric and gravitational field.

How are magnetic fields produced?
Some materials have an inherent magnetic property, and they produce a magnetic field and are known as magnets.
A static charge produces an electric field only, but it produces both the electric field and magnetic field when the charge is moving. We can say that a moving charge produces a magnetic field.
When an electric current is flowing inside a conductor, the electrons in the conductor are in motion. Due to the motion of these-electrons, a magnetic field is generated around the conductor.
The magnetic field is a vector quantity and is denoted by \(\overrightarrow B \). Its SI unit is Tesla.

Magnetic flux can be defined as the number of magnetic field lines passing through an area. It signifies both the direction and the magnitude of the magnetic field passing through the surface. It is denoted by using the Greek letter Phi\((\phi)\). Its SI unit is Weber \((\rm{Wb})\).
Magnetic flux is given by the dot product of the magnetic field and the area vector,

\(\phi = \overrightarrow B \cdot \overrightarrow S \)
Where,
\(\overrightarrow B\) is the magnetic field.
\(\overrightarrow S\) is the area vector.

Magnetic Field Lines

Magnetic field lines or magnetic lines of force is a way to represent the magnetic field pictorially. The density and direction of the lines in a space signify the magnitude and direction of the magnetic field.
Properties of Magnetic field are as follows;

1. Two magnetic field lines never intersect.
   This is because magnetic field lines give the direction of the net magnetic field, and if they intersect, there will be two directions for the net magnetic field, which is impossible.
2. Magnetic field lines form a closed loop.
3. Magnetic field lines emerge from the north pole and terminate at the south pole.

Gauss’s Law for Magnetics

Gauss’s law for magnetic is one of maxwell’s four equations for electromagnetism.
According to this law, the magnetic flux through any closed surface will be zero.
\({\phi _{{\text{closed}}\,{\text{surface}}}} = \oint \overrightarrow B \cdot \overrightarrow {dA} = 0\)
Where,
\(\overrightarrow B\) is the magnetic field.
\(\overrightarrow {dA}\) is a small area vector.

Electromagnetic Induction

Electromagnetic induction was discovered by the famous scientist Michael faraday in 1931. Electromagnetic induction is a phenomenon in which the electric current is generated in a conductor due to a change in magnetic flux.
It can be considered as the reverse of the magnetic effect of current as in the magnetic effect of current, the magnetic field is produced due to current, and in electromagnetic induction, electric current is produced due to magnetic field.

Faraday’s Law of Electromagnetic Induction

According to Faraday’s law of electromagnetic induction, the emf induced in a loop due to electromagnetic induction is directly proportional to the rate of flux change with respect to time.
\(E = – \frac{{d\phi }}{{dt}}\)

Lenz’s Law

Lenz’s law states that an induced electric current in a loop flows in a direction such that the current opposes the change in flux that induced it.
Lenz’s law only gives us the direction of the current flow. In simple words, if the flux through a loop decreases, then the induced current will increase the flux through the loop, and if the current is produced due to an increase in flux, then the current induced will decrease the flux.

Self-Induction and Mutual Induction

Self-induction is a phenomenon in which the emf is induced in the coil or loop is due to variation in the flux through it due to change in the current in the circuit. Due to this, a coil or a loop resists any change in the current through it.
Mutual induction is a phenomenon in which the current is induced in a loop or coil due to the current flowing in another loop kept close to it. This phenomenon is used in transformers.

Induced Electric Field

This induced electric field is due to a change in flux through a loop due to changing magnetic field. It is important to note that this induced electric field differs from the electric field due to a static charge. The induced electric field is a non-conservative field. The electric potential and potential difference are not defined.
\(\oint {\overrightarrow E } \cdot \overrightarrow {dl} = \frac{{d\phi }}{{dt}}\)
The only similarity of the induced field with the electric field due to static charge is that the force on a charge due to the field is the magnitude of the charge multiplied by the strength of the field.
\(\overrightarrow {{F_E}} = \overrightarrow {q{E_E}} \)
And,
\(\overrightarrow {{F_{IE}}} = \overrightarrow {q{E_I}} \).

Sample Problems on Magnetic Flux

Q.1. If the conducting rod \(AB\) moves with the velocity ‘\(v\)’ in the given figure, find the emf induced in the circuit. The length of the conductor is ‘\(l\)’, and a constant magnetic field \(\overrightarrow B \) is coming out of the plane.

Ans: Given,
The length of the conductor is ‘\(l\)’.
The velocity of the conductor is ‘\(v\)’
The magnitude of the magnetic field is \(\overrightarrow B \).
The flux of the circuit is given by,
\(\phi = \oint \overrightarrow B \cdot \overrightarrow {dA} \)
\( = \phi = Blx\)
From Faraday’s law of electromagnetic induction, we have,
\(E = – \frac{{d\phi }}{{dt}}\)
\( \Rightarrow E = Bl\frac{{dx}}{{dt}}\)
\( = E = Blv\).

Q.2. Find the flux through the given area when the magnetic field \(\overrightarrow B \) makes an angle \(60\) degrees with the area vector \(\overrightarrow S \).

Ans: Given,
The magnetic field is equal to \(\overrightarrow B \).
The area is equal to \(\overrightarrow S \).
The angle between the magnetic field and the Area vector is \(60^\circ\).
The flux due to the magnetic field is given by,
\(\phi = \oint {\overrightarrow B } \cdot \overrightarrow {dS} \)
\( \Rightarrow \phi = BS\,\cos \left( {60} \right) = \frac{1}{2}BS\).

Summary

Magnetic flux is the amount of magnetic field passing through an area. The flux through a closed surface is always zero. This is because the magnetic monopole does not exist. Magnetic field lines always form a closed loop. Magnetism and electric current are related to each other. Change in the flux through a conducting loop induces emf inside the loop. Faraday explained this. The emf induced is equal to the rate of change of flux. Lenz’s law gives the direction of the induced current. Self-induction resists any change in current in a loop, whereas mutual induction induces a current in another loop kept near a current-carrying loop.

FAQs on Magnetic Flux

Q.1. What is magnetic flux?
Ans: Magnetic flux is the amount of the magnetic field passing through an area. It can also be defined as the number of magnetic field lines passing through an area. Its SI unit is Tesla. It is calculated by the dot product of the magnetic field and area vector.
\(\phi = \overrightarrow B \cdot \overrightarrow A \).
Where,
\(\overrightarrow B\) is the magnetic field.
\(\overrightarrow {A}\) is the area vector.

Q.2. What is Gauss’s law for magnetic?
Ans: Gauss’s law for magnetics states that the magnetic flux through any closed surface will always be zero.
\({\phi _{{\text{closed}}\,{\text{surface}}}} = \oint {\overrightarrow B } \cdot \overrightarrow A = 0\).
Where,
\({\overrightarrow B }\) is the magnetic field.
\(\overrightarrow {A}\) is the area vector.

Q.3. Can magnetic monopoles be isolated?
Ans: No, the magnetic monopoles cannot be isolated. Magnetic poles always exist in pairs known as the north and the south pole.

Q.4. What is electromagnetic induction?
Ans: Electromagnetic induction is a phenomenon in which electromotive force is induced due to a change in magnetic flux. Faraday’s law of electromagnetic induction gives the value of induced emf. The emf induced in a loop due to electromagnetic induction is directly proportional to the rate of change of flux with respect to time.
\(E = – \frac{{d\phi }}{{dt}}\)
Lenz’s law gives the direction of the induced emf. The induced electric current in a loop flows in a direction such that the current opposes the change in flux that induced it. In simpler words, the effect opposes the change.

Q.5. What is an induced electric field?
Ans: An induced Electric field is a force field that applies force on a charge in that field as,
\(\overrightarrow {{F_{IE}}} = \overrightarrow {q{E_I}} \)
This field is different from the electric field, which is generated by a static charge. The induced electric field lines can form a closed loop. The electric potential and potential difference are not defined for points in an induced electric field.
The induced electric field is generated by changing magnetic flux through an area. It is related to change in flux as,
\(\oint {\overrightarrow E } \cdot \overrightarrow {dl} = \frac{{d\phi }}{{dt}}\).

We hope you find this article on ‘Magnetic Flux‘ helpful. In case of any queries, you can reach back to us in the comments section, and we will try to solve them.