Magnetic Flux

Magnetic flux

(a) The number of lines of force passing through a given area is defined as magnetic flux.

(b) The magnetic flux passing through unit normal area is defined as magnetic induction (B).

(c) When the magnetic field is normal to the plane then $\varphi$= BA, when A = 1m² then $\varphi$= B.

(d) When magnetic field makes an angle $\theta$ with the normal to the plane :

(i) Magnetic flux linked with the plane = Area of the plane

(A) × Component of magnetic field normal to the plane (B cos $\theta$)

i.e. $\varphi$= AB cos $\theta$ If the number of turns in the coil is N. then $\varphi$= NAB cos $\theta$

(ii) $\varphi$= magnetic field normal to the plane (B) × component of A in the direction of magnetic field (A cos $\theta$)

i.e. $\varphi$= BA cos $\theta$

In both cases $\theta$ is the angle between $\vec{B}and\hat{n}$.

(iii) When $\vec{B}and\vec{A}$are mutually parallel then $\theta$ = 0^o and $\varphi$= BA.

(iv) When $\vec{B}and\vec{A}$are mutually perpendicular, $\theta$ = 90^o and $\varphi$= 0.

(v) When $\vec{B}and\vec{A}$are antiparallel, then $\theta$ = 180^o and $\varphi$= BA.

(e) When the angle between B and the plane of coil is $\theta$ then $\varphi$= BA sin $\theta$.

If the number of turns in the coil is N then $\varphi$= NBA sin $\theta$.

(i) When the plane of coil is parallel to $\vec{B}$ then $\theta$ = 0^o and $\varphi$= 0.

(ii) When $\vec{B}$and the plane of coil are mutually perpendicular i.e. $\theta$ = 90, then $\varphi$= BA.

(iii) When $\vec{B}$and the plane of coil are mutually antiparallel i.e. $\theta$ = 180^o, then $\varphi$= 0.

(f) Magnetic flux linked with a small surface element dA $d\varphi =\vec{B}.d\vec{A}=BdA\cos \theta$

where d $\vec{A}$= area of small element.

$\hat{n}$= unit normal vector.

(g) The flux linked with total area of the surface A

$\varphi =\int d\varphi =\int B.dA=\int \vec{B}\vec{d}A\cos \theta$

(h) Positive magnetic flux: When the magnetic induction $vecB$ and the unit normal vector $\hat{n}$ are in the same direction then $\varphi$ is called the positive magnetic flux.

$\to \vec{B}$

$\to \vec{A}$

$\varphi$ = BA

(i) Negative magnetic flux: When the magnetic induction $\vec{B}$ and unit normal vector are mutually in opposite directions then $\varphi$ is called negative magnetic flux

$\varphi$ = – BA

(j) The flux emanating out of a surface is positive and the flux entering the surface is negative.

(k) $\varphi$ is a scalar quantity.

(l) The net magnetic flux coming out of a closed surface is always zero, i.e.

$\varphi =\underset{A}{\oint }B.dA=0$

or $\vec{\nabla }.\vec{B}=0$

Magnetic flux density or magnetic induction

(a) The magnetic lines of force passing through unit normal area in a magnetic field is defined as magnetic induction.

(b) $B=\frac{\varphi }{A}whenA=1m{.}^{2}thenB=\varphi$

(c) Direction of magnetic induction: The direction in the magnetic field in which if a current carrying conductor is placed then no force acts on it, is known as the direction of magnetic induction.

(d) Magnetic induction is a vector quantity.

(e) The magnetic induction due to a bar magnet

(i) In axial position $B=\frac{2KM}{r^3}$

(ii) In equatorial position $B=\frac{KM}{r^3}$

Here $K=\frac{{\mu }_0}{4\pi }={10}^{-7}Weber/A.m$