State the factors affecting the magnitude of an induced e.m.f.

Published by Patrick Mutisya · 14 days ago

Electromagnetic Induction – 4.5.1

Objective

State the factors that affect the magnitude of an induced e.m.f.

Key Principle

Faraday’s law of electromagnetic induction states that the induced e.m.f. (\$\mathcal{E}\$) in a coil is proportional to the rate of change of magnetic flux (\$\Phi\$) linking the coil:

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

where

  • \$N\$ = number of turns of the coil

  • \$\Phi = BA\cos\theta\$ is the magnetic flux through one turn

  • \$B\$ = magnetic field strength (tesla, T)

  • \$A\$ = area of the coil (m²)

  • \$\theta\$ = angle between the magnetic field direction and the normal to the coil surface

Factors Affecting the Magnitude of the Induced e.m.f.

The magnitude of the induced e.m.f. depends on how quickly the magnetic flux through the coil changes. The following factors influence this rate of change:

  1. Number of turns of the coil (\$N\$)

    More turns increase the total change in flux, giving a larger e.m.f.

  2. Strength of the magnetic field (\$B\$)

    A stronger field produces a greater flux for a given area, so a change in \$B\$ yields a larger e.m.f.

  3. Area of the coil (\$A\$)

    Larger area intercepts more field lines; a change in area (e.g., by moving a rectangular loop) leads to a larger change in flux.

  4. Rate of change of the magnetic field or area (\$\frac{dB}{dt}\$, \$\frac{dA}{dt}\$)

    Faster changes give a greater \$\frac{d\Phi}{dt}\$ and thus a larger e.m.f.

  5. Relative motion between a conductor and magnetic field

    When a straight conductor of length \$l\$ moves with velocity \$v\$ perpendicular to a uniform field \$B\$, the induced e.m.f. is \$\mathcal{E}=Blv\$. Faster motion or longer conductor increases the e.m.f.

  6. Angle between the field and the coil (\$\theta\$)

    Since \$\Phi = BA\cos\theta\$, changing \$\theta\$ changes the flux. Rotating the coil more rapidly (larger \$\frac{d\theta}{dt}\$) increases the induced e.m.f.

Summary Table

FactorEffect on Induced e.m.f.
Number of turns (\$N\$)Proportional – double \$N\$, double \$\mathcal{E}\$
Magnetic field strength (\$B\$)Proportional – stronger \$B\$ gives larger \$\mathcal{E}\$
Coil area (\$A\$)Proportional – larger \$A\$ increases flux change
Rate of change of \$B\$ or \$A\$Directly proportional – faster change → larger \$\mathcal{E}\$
Length of moving conductor (\$l\$) in a uniform fieldProportional – \$\mathcal{E}=Blv\$
Velocity of motion (\$v\$)Proportional – faster motion → larger \$\mathcal{E}\$
Angle \$\theta\$ between \$B\$ and coil normalFlux varies as \$\cos\theta\$; rapid change of \$\theta\$ increases \$\mathcal{E}\$

Typical Example

A rectangular coil of 20 turns has an area of \$0.04\ \text{m}^2\$ and is placed in a uniform magnetic field of \$0.5\ \text{T}\$. The coil is rotated from \$\theta = 0^\circ\$ to \$\theta = 90^\circ\$ in \$0.2\ \text{s}\$. Calculate the average induced e.m.f.

Solution:

\$\Phi_i = BA\cos0^\circ = (0.5)(0.04)(1)=0.02\ \text{Wb}\$

\$\Phi_f = BA\cos90^\circ = (0.5)(0.04)(0)=0\ \text{Wb}\$

\$\Delta\Phi = \Phif-\Phii = -0.02\ \text{Wb}\$

\$\mathcal{E}_{\text{avg}} = -N\frac{\Delta\Phi}{\Delta t}= -20\frac{-0.02}{0.2}=2\ \text{V}\$

Suggested Diagram

Suggested diagram: (a) coil rotating in a uniform magnetic field, showing change of angle \$\theta\$; (b) straight conductor moving perpendicular to magnetic field lines, illustrating \$\mathcal{E}=Blv\$.