Magnetic Flux

The magnetic flux is a scalar quantity and can be quantified by establishing an imaginary surface represented by an area vector in the vicinity of a magnetic field. The magnetic flux φ is given by the dot product of the magnetic field and the area vector.

If the angle between the magnetic field vector and the area vector is θ, then the magnetic flux is given by

Symbol: Greek letter phi, φ

SI Unit: Weber or Wb

CGS Unit: Maxwell

Dimensions: [M¹ L² T^-2 I^-1]

When θ = 0^o , φ = BA. Therefore, the flux is maximum when the area under consideration is perpendicular to the magnetic field.

Suppose the magnetic field and the area vector are in opposite directions, then the angle between the two is 180^o and the flux will be negative.

Problem 1: A rectangular loop is 0.50 m long and 0.40 m wide. It is placed in a constant magnetic field of 0.3 T such that its area vector A makes an angle of 30° with B. Determine the magnetic flux through the surface.

Solution:

Given,

B = 0.3 T

θ = 30  ͦ

l = 0.5 m

w = 0.4 m

A = l x w = 0.5 x 0.4 = 0.2 m²

Therefore,

φ = BA cos θ

or, φ = 0.3 T x 0.2 m² x cos 30  ͦ

or, φ = 0.052 Wb

Problem 2: A magnetic field of 1.5 T passes perpendicular to a disc of diameter 6 cm. Find the magnetic flux passing through the disc.

Solution:

Given,

B = 1.5 T

d = 6 cm

r = d/2 = 6/2 cm = 3 cm = 0.03 m

A = πr² = π x (0.03 m)² = 0.0028 m²

We have,

φ = BA

or, φ = 1.5 T x 0.0028 m²

or, φ = 0.0042 Wb

Magnetic Flux Density

In the above equation, when A = 1, φ = B. Therefore, the magnetic flux density is defined as the flux passing through a unit cross-sectional area.

Magnetic Field from Magnetic Flux

By rearranging the above equation, we get

Thus, the magnetic field is given by the flux per unit area perpendicular to the field.

SI Unit: Wb/m²

Faraday’s Law

Magnetic flux is significantly essential in Faraday’s law of electromagnetic induction. Suppose a conducting wire is wound into a loop or a coil, and a magnet is brought closer. According to Faraday’s law, a current will be induced in the wire whose electromotive force or emf is given by,

where,

ε: Induced emf

N: Number of turns of the coil

Δφ/Δt: Change in magnetic flux over time

Lenz’s Law

Suppose a loop of wire is placed in a changing magnetic field such that the magnetic flux passes through the wire. Then, Lenz’s law gives the direction of the induced current in the wire and the polarity of the induced emf. The induced current will generate an induced magnetic field, whose direction is given by the right-hand rule.

Gauss’s Law for Magnetism

Gauss’s law for magnetism states that the net magnetic flux coming out of a closed surface is zero. This statement gives an understanding of magnetic field sources. For example, all field lines in a bar magnet from the north pole must enter the south pole. A closed surface enclosing a bar magnet will have no flux coming out. Therefore, Gauss’s law for magnetism proves that a magnetic monopole does not exist.

	Electric Flux	Magnetic Flux
Definition	Number of electric field lines passing through a given surface	Number of magnetic field lines passing through a given surface
Density gives	Electric field	Magnetic field
Formula	φ = EA cos θ	φ = BA cos θ
SI Unit	Volt-meter (V-m)	Weber (Wb)