recall and use Φ = BA

Electromagnetic Induction – Magnetic Flux

In this section we recall and apply the fundamental relationship between magnetic flux, magnetic field and area: $$\Phi = B A \cos\theta$$ where θ is the angle between the field and the normal to the area.

Key Definitions

• Magnetic Flux (Φ) – measure of the quantity of magnetic field passing through a given area.
• Magnetic Field (B) – vector field produced by moving charges, measured in teslas (T).
• Area (A) – cross‑sectional area perpendicular to the field, measured in square metres (m²).
• θ – angle between the field direction and the normal to the surface.

Formula Derivation

For a uniform field perpendicular to the surface (θ = 0°), the flux simplifies to $$\Phi = B A$$. If the field is at an angle, the effective component of B normal to the surface is \(B \cos\theta\), giving the general form.

Units and Conversion

Practical Example

Calculate the magnetic flux through a square loop of side 0.10 m placed in a uniform magnetic field of 0.50 T directed perpendicular to the loop.

1. Area: \(A = (0.10\,\text{m})^2 = 0.01\,\text{m}^2\).
2. Flux: \(\Phi = B A = 0.50\,\text{T} \times 0.01\,\text{m}^2 = 0.005\,\text{Wb}\).

Common Mistakes

• Using the wrong unit for B (e.g., gauss instead of tesla).
• Ignoring the angle θ; assuming θ = 0° when the field is not perpendicular.
• Calculating area incorrectly for non‑rectangular shapes.

Practice Problems

1. A rectangular coil of dimensions 0.20 m × 0.30 m is placed in a magnetic field of 0.80 T that makes an angle of 30° with the normal to the coil. Find the magnetic flux.
2. What is the flux through a circular loop of radius 0.05 m in a field of 0.25 T perpendicular to the loop?
3. If the flux through a coil is 0.02 Wb and the area is 0.04 m², what is the magnitude of the magnetic field?

Summary

The magnetic flux through a surface is given by \(\Phi = B A \cos\theta\). For perpendicular fields, this reduces to \(\Phi = B A\). Remember to use consistent SI units and account for the angle between the field and the surface normal.