Magnetic Flux

Magnetic flux is the number of magnetic field lines or magnetic lines of force passing through a surface. Thus, in simplified terms, the magnetic flux through some surface is proportional to the number of field lines passing through that surface. The magnetic flux across a surface is defined as the electric field flux used with Gauss Law.

Definition

The physical description of magnetic flux through a surface is the surface integral of the scalar product (dot product) of the magnetic field vector (B) and elementary area vector (dA). It is denoted by the symbol Φ or ΦB (B in the suffix represents magnetic). The SI unit of magnetic flux is the Weber (Wb) or Tesla metre squared (T m2).

Mathematical expression

Consider a surface divided into elemental area dA. For each element, we obtain the components of B normal and tangent to the surface at the position of that area element. These components will generally vary from point to point on the surface.

The normal component of a magnetic field is B⟂ = B cosϴ.

The elementary magnetic flux through this area is defined as

d Φ= B⟂dA=B cosϴ dA = B.dA

Hence the total magnetic flux through the surface is the sum of the contributions from the individual area elements and is given by

Φ=BdAcos =B.dA

Magnetic Flux through closed surface

For Magnetic flux through a closed surface, the surface integral used in the definition of magnetic flux is taken over a closed loop.

i.e., 

Φ=B.dA = 0

The magnetic flux through a closed loop is equal to zero. This is referred to as Gauss’ Law in Magnetism. This law results shows that magnetic monopoles have never been found in nature, i.e., magnetic monopoles do not exist.

How do we measure magnetic flux?

The SI unit of magnetic flux is the Weber, named after a German Physicist. The unit Weber has the symbol Wb. Because the magnetic flux is just a way of expressing the magnetic field in a given area, it can be measured with a magnetometer in the same way as the magnetic field.

A related term that you may come across is magnetic flux density. This is measured in Wb/m2. Because we are dividing flux by area, we could also directly state the flux density units in Tesla. 

Magnetic Flux and Faraday’s Law

With a series of experiments, Faraday discovered that a changing magnetic flux induces an electric current through a coil kept in the field. Thus Faraday’s law explains the principle of Electromagnetic Induction.

Faraday’s law is thus stated as an emf/voltage is induced in a circuit whenever there is a relative motion between a conductor and a magnetic field. The induced emf, in turn, produces induced current in the circuit. Also, the magnitude of induced emf / induced voltage is proportional to the rate of change of magnetic flux.

The Mathematical expression of Faraday’s law

Rate of change of Magnetic Flux = dΦ / dt

Induced emf (e) α time rate of change of magnetic flux (dΦ / dt)

“Induced emf (induced voltage) is proportional to the time rate of change of magnetic flux.”

Φ = B.A = BA cosϴ ( if B is uniform)

Thus for the time derivative of magnetic flux to be non-zero, the following condition needs to be satisfied.

1. B(t)

The magnetic field is changing with respect to time. The magnetic field is time-dependent.

1. A(t)

Area changes with respect to time. Suppose the area of the circuit or the coil loop kept in the magnetic field changes.

1. ϴ(t)

Theta changes with time. If the coil loop is rotating, the angle between the area vector and the magnetic field changes.

Applications

Magnetic flux is the scalar product (dot product) of the magnetic field vector (B) and elementary area vector (dA). Thus, it is a scalar quantity. It is observed that magnetic flux is a quantity of significance in the statement of Faraday’s Law. Also, it has applications in the discussion of objects like transformers and solenoids. A similar application is found in Electric motors and generators that apply Faraday’s law to coils that rotate in a magnetic field. This changing magnetic flux induces an electric current.

Conclusion

Magnetic flux, which is a scalar quantity, can be defined as the number of magnetic field lines passing through a given surface. The significance of magnetic flux can be inferred from Faraday’s law and its description of the phenomenon of Electromagnetic Induction. The time rate of change of magnetic flux induces an electric current. This principle of induced voltage and hence induced current due to changing magnetic flux has many applications. One of the applications is the Electric motor.

Thus magnetic flux is a significant quantity. A term related to magnetic flux is magnetic flux density, defined as magnetic flux per unit area.