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

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.