understand and use the concept of magnetic flux linkage

Electromagnetic Induction – Magnetic Flux Linkage

1. What is Magnetic Flux?

The magnetic flux, $ \Phi $, through a surface of area $A$ is the product of the magnetic field strength $B$ and the component of the area perpendicular to the field:

$$\Phi = B A \cos\theta$$

where $ \theta $ is the angle between the magnetic field direction and the normal to the surface.

2. Flux Linkage

When a coil of $N$ turns is considered, the total flux linked with the coil is called the flux linkage, denoted $ \Lambda $:

$$\Lambda = N\Phi = N B A \cos\theta$$

Flux linkage is measured in weber‑turns (Wb·turn). It is the quantity that appears in Faraday’s law for a coil.

3. Faraday’s Law of Electromagnetic Induction

Faraday’s law states that the induced emf $ \mathcal{E} $ in a coil equals the negative rate of change of its flux linkage:

$$\mathcal{E} = -\frac{d\Lambda}{dt}= -N\frac{d\Phi}{dt}$$

The negative sign represents Lenz’s law – the induced emf always opposes the change producing it.

4. Lenz’s Law

Lenz’s law can be expressed qualitatively:

• If the magnetic flux through a coil increases, the induced current creates a magnetic field that opposes the increase.
• If the flux decreases, the induced current creates a field that tries to maintain the original flux.

5. Calculating Induced emf – Worked Example

Consider a single‑turn rectangular loop of width $w = 0.10\ \text{m}$ and height $h = 0.20\ \text{m}$ rotating at a constant angular speed $ \omega = 50\ \text{rad s}^{-1}$ in a uniform magnetic field $B = 0.30\ \text{T}$. Find the maximum induced emf.

1. Flux through the loop at any time $t$: $$\Phi(t) = B A \cos(\omega t)$$ where $A = w h = 0.10 \times 0.20 = 0.020\ \text{m}^2$.
2. Differentiate to obtain emf: $$\mathcal{E}(t) = -\frac{d\Phi}{dt}= B A \omega \sin(\omega t)$$
3. Maximum emf occurs when $\sin(\omega t)=1$: $$\mathcal{E}_{\max}= B A \omega = 0.30 \times 0.020 \times 50 = 0.30\ \text{V}$$

6. Factors Affecting Flux Linkage

Factor	How it changes $ \Lambda $	Effect on Induced emf
Number of turns $N$	$\Lambda$ proportional to $N$	Induced emf increases linearly with $N$
Magnetic field $B$	$\Lambda$ proportional to $B$	Stronger $B$ → larger emf for a given rate of change
Area $A$ of coil	$\Lambda$ proportional to $A$	Larger coil → greater emf
Orientation $ \theta $	$\Lambda = N B A \cos\theta$	Maximum when $\theta =0^\circ$, zero when $\theta =90^\circ$
Rate of change of any factor	Appears in $d\Lambda/dt$	Faster change → larger induced emf

7. Common Applications

• AC generators – rotating coils in a magnetic field produce a sinusoidal emf.
• Transformers – changing flux linkage in the primary coil induces emf in the secondary.
• Induction cookers – rapidly varying flux in a coil induces currents in metal cookware.

8. Summary Checklist

• Flux linkage $ \Lambda = N B A \cos\theta $.
• Faraday’s law: $ \mathcal{E} = -d\Lambda/dt $.
• Lenz’s law determines the direction of the induced current.
• Increasing any of $N$, $B$, $A$, or the rate of change of $\theta$ increases the induced emf.

9. Practice Questions

1. A coil of 200 turns, each of area $5.0\times10^{-3}\ \text{m}^2$, is placed in a magnetic field that increases uniformly from $0$ to $0.8\ \text{T}$ in $0.25\ \text{s}$. Calculate the average induced emf.
2. A rectangular loop $0.15\ \text{m}\times0.30\ \text{m}$ rotates at $60\ \text{rev s}^{-1}$ in a $0.25\ \text{T}$ field. Determine the peak emf.
3. Explain qualitatively how the direction of the induced current changes when the loop is rotated past the position where the flux is maximum.