Define magnetic flux. Give units and dimensions of magnetic flux. Give an expression for magnetic flux associated with variable magnetic field.
Magnetic flux ($\phi$) through any surface is defined as " the total number of magnetic lines passing through that surface."
Consider a small surface of area A. Let $\hat{n}$ be the unit vector which is drawn normal to the plane of the surface.
If $\theta$ is the angle between $\hat{n}$ and the uniform magnetic field $\vec{B}$ , then the magnetic flux ($\phi$) through the surface is given by,
$\phi_B = \vec{B} \cdot \vec{A} = BA \cos\theta \tag{i}$
or
$\phi_B = (B \cos\theta) A \tag{ii}$
Now $B \cos \theta$ is the component of the magnetic field normal to the plane of the surface
Then eqn. (ii) can be written as,
$\Phi_B = BA \tag{iii}$
Thus, magnetic flux through a given surface is defined as the product of the area of the surface and the component of the magnetic field ($B$) normal to the plane of the surface.
Special Cases :
(i) When $\theta = 0^\circ$, i.e. uniform magnetic field is acting perpendicular to the plane of the surface, then
$\Phi_B = BA \cos 0^\circ$
$(\because \cos 0^\circ = 1)$
$\Phi_B = BA = \text{maximum}$
Thus, magnetic flux through a given surface is maximum (i.e. maximum number of magnetic field lines pass through the given surface), when magnetic field is perpendicular to the plane of the surface.
(ii) When $\theta = 90^\circ$, i.e. uniform magnetic field is along the plane of the surface, then
$\Phi_B = BA \cos 90^\circ$
$(\because \cos 90^\circ = 0)$
$\Phi_B = 0$
Thus, the magnetic flux through a given surface is zero, when the magnetic field is along the plane of the surface.
Definition of Magnetic flux density or Strength of Magnetic field or Magnetic Induction (B) :
Since
$\Phi_B = BA$
$\therefore B = \frac{\Phi_B}{A}$
Thus, magnetic flux density (B) is defined as the magnetic flux (associated normally) per unit area.
Units of magnetic flux :
In SI, magnetic flux is measured in weber (Wb)
Since
$\Phi_B = BA$
$1 \text{weber} = 1 Tesla \times metre^{2}$
$1 \text{Wb} = 1 T m^{2}$
In C.G.S. system, unit of magnetic flux is maxwell
$1 \text{ maxwell} = 1 \text{ G} \times 1 \text{ cm}^2$
$=10^{-4} \text{ T} \times 10^{-4} \text{ m}^2$
$= 10^{-8} \text{ T m}^2$
$= 10^{-8} \text{ weber}$
Dimensional formula for magnetic flux :
We know
$\Phi_B = BA \cos \theta$
Since
$B = \frac{F}{qv}$
and $\cos \theta$ is dimensionless
So,
$[\Phi_B] = [FA/qv]$
$(\because q = It)$
$[\Phi_B] = \left[ \frac{FA}{Ivt} \right]$
or
$[\Phi_B] = \frac{[F][A]}{[I][v]}$
$[\Phi_B] = \frac{[MLT^{-2}][L^2]}{[A][LT^{-1}]}$
$[\Phi_B] = [ML^2 T^{-2} A^{-1}]$
Nature of Magnetic Flux : Since $\Phi = \vec{B} \cdot \vec{A}$ i.e. dot product of $\vec{B}$ and $\vec{A}$, so $\Phi$ is a scalar physical quantity.
Magnetic Flux associated with variable magnetic field :
We know, magnetic flux $\Phi_B = \vec{B} \cdot \vec{A} = BA \cos \theta$ holds good only if the magnetic field is uniform over the surface. If the magnetic field ($\vec{B}$) is variable, then the surface is divided into number of area elements, each of which has area small enough to have constant magnetic field through it.
Let $\vec{B}_1, \vec{B}_2, \vec{B}_3, \dots \vec{B}_n$ be the magnetic field at the area elements of area $\Delta \vec{A}_1, \Delta \vec{A}_2, \Delta \vec{A}_3, \dots \Delta \vec{A}_n$ respectively of a surface (Figure 6), then the total magnetic flux through the surface is the sum of the magnetic flux through all area elements.
That is,
$\Phi_B = \vec{B}_1 \cdot \Delta \vec{A}_1 + \vec{B}_2 \cdot \Delta \vec{A}_2 + \dots + \vec{B}_n \cdot \Delta \vec{A}_n$
$\Phi_B= \sum \vec{B} \cdot \Delta \vec{A} \qquad ...(iv)$
If $\Delta A \rightarrow 0$, then eqn. (iv) can be written as
$\Phi_B = \int_S \vec{B} \cdot d\vec{A} \qquad ...(v)$