define magnetic flux as the product of the magnetic flux density and the cross-sectional area perpendicular to the direction of the magnetic flux density

Electromagnetic Induction

Objective

Define magnetic flux as the product of the magnetic flux density and the cross‑sectional area perpendicular to the direction of the magnetic flux density.

Key Concepts

• Magnetic flux density \$\mathbf{B}\$ – measured in tesla (T).
• Cross‑sectional area \$A\$ – the area through which the field lines pass, measured in square metres (m²).
• Angle \$\theta\$ – the angle between the field direction and the normal to the area.

Definition of Magnetic Flux

The magnetic flux \$\Phi\$ through a surface is given by

\$\Phi = \int_S \mathbf{B}\cdot d\mathbf{A} = B A \cos\theta\$

When the magnetic field is uniform and the surface is flat, the integral reduces to the simple product of the magnetic flux density, the area, and the cosine of the angle between them.

Special Cases

1. Field perpendicular to the area (\$\theta = 0^\circ\$): \$\Phi = B A\$.
2. Field parallel to the area (\$\theta = 90^\circ\$): \$\Phi = 0\$ (no flux passes through).

Symbols and Units

Worked Example

Calculate the magnetic flux through a circular coil of radius \$r = 0.10\ \text{m}\$ placed in a uniform magnetic field of \$B = 0.5\ \text{T}\$, with the field perpendicular to the plane of the coil.

1. Find the area: \$A = \pi r^{2} = \pi (0.10\ \text{m})^{2} = 3.14 \times 10^{-2}\ \text{m}^{2}\$.
2. Since the field is perpendicular, \$\theta = 0^\circ\$ and \$\cos\theta = 1\$.
3. Apply \$\Phi = B A\$: \$\Phi = 0.5\ \text{T} \times 3.14 \times 10^{-2}\ \text{m}^{2} = 1.57 \times 10^{-2}\ \text{Wb}\$.

Thus, the magnetic flux through the coil is \$1.57 \times 10^{-2}\ \text{Wb}\$.