10.2 Magnetic Flux and Flux Density

10.2 Magnetic Flux and Flux Density

We can represent the strength of a magnetic field $B$ by the lines of force in the same way as for electric field. Then, the population of these lines in the field per unit area passing through a surface perpendicular to the field will represent the magnetic flux. Thus:

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The magnetic flux through a patch of area $A$ is the number of magnetic lines passing through this area.

If $B$ represents the number of lines passing through unit area placed perpendicular to the field, then the total flux through area $A$ perpendicular to the field will be:

The surface $A$ may not be perpendicular to the field, that is the normal to the surface makes an angle $\theta$ with $B$ as shown in Fig. 10.5. Then, we will have to use component of $B$ ($B\cos\theta$) along the vector area $\mathbf{A}$ (Fig. 10.6). Therefore, the flux passing through the surface will be:

As $\mathbf{B}$ and $\mathbf{A}$ both are vectors, so we can write:

Equation (10.6) shows that $\phi_{B}$ is a scalar quantity. Therefore, we can define magnetic flux as:

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The magnetic flux $\phi_{B}$ through a plane element of area $A$ in a uniform magnetic field $B$ is given by dot product of $\mathbf{B}$ and $\mathbf{A}$.

Note that $\mathbf{A}$ is a vector whose magnitude is the area of the element and whose direction is along the normal to the surface of the element and $\theta$ is the angle between the directions of the vectors $\mathbf{B}$ and $\mathbf{A}$ (Fig. 10.5). In Fig. 10.7(a), the field is directed along the normal to the area, so $\theta$ is zero ($\cos 0^{\circ} = 1$) and the flux is maximum, equal to $BA$. When the field is parallel to the plane of the area (Fig. 10.7-b), the angle between the field and normal to area is $90^{\circ}$, i.e., $\theta = 90^{\circ}$ ($\cos 90^{\circ} = 0$), so the flux through the area in this orientation is zero.

In case of a curved surface placed in a uniform magnetic field, the curved surface is divided into a number of small surface elements, each element being assumed plane and the flux through the whole curved surface is calculated by the sum of the contributions from all the elements of the surface using Eq. (10.5).

From the definition of tesla, the unit of magnetic flux is $\mathrm{N\,m\,A^{-1}}$ which is called Weber $(\mathrm{Wb})$. According to Eq. 10.5, the magnetic induction $B$ is the flux per unit area of a surface perpendicular to $B$, hence, it is also called as magnetic flux density. Its unit is $\mathrm{Wb\,m^{-2}}$. Therefore, magnetic induction, i.e. the magnetic field strength is measured in terms of $\mathrm{Wb\,m^{-2}}$ or $\mathrm{N\,A^{-1}\,m^{-1}}$ (tesla).

Example 10.2

A rectangular loop of wire is placed in a uniform magnetic field of magnitude $1.2\,\mathrm{T}$. If the loop is $25\,\mathrm{cm}$ long and $20\,\mathrm{cm}$ wide, determine the magnetic flux through the loop for the three orientations as shown in Fig. 10.8.

Solution

For orientation (a), angle between $\mathbf{B}$ and area vector $\mathbf{A}$ is $\theta = 60^{\circ}$

Using $\phi_{B} = BA\cos\theta$

For orientation (b), angle $\theta = 45^{\circ}$

For orientation (c), angle $\theta = 30^{\circ}$