Electromagnetic Induction

AS UNIT 4

Fields and Options

4.5 Electromagnetic Induction

Learners should be able to demonstrate and apply their knowledge and understanding of:
a) The definition of magnetic flux as [math]Φ = AB cosθ[/math] and flux linkage [math] = NΦ[/math]
b) The laws of Faraday and Lenz
c) How to apply the laws of Faraday and Lenz (i.e. emf = – rate of change of flux linkage)
d) The idea that an emf is induced in a linear conductor moving at right angles to a uniform magnetic field
e) Qualitatively, how the instantaneous emf induced in a coil rotating at right angles to a magnetic field is related to the position of the coil, flux density, coil area and angular velocity

  • a)   Definition of Magnetic Flux

  • Magnetic flux [math]Φ[/math] represents the total number of magnetic field lines passing through a surface. It is given by:
  • [math]Φ = BAcosθ[/math]
  • Where:
  • – Φ = Magnetic flux (Weber, Wb)
  • – B = Magnetic field strength (Tesla, T)
  • – A = Area through which the field passes ([math]m^2[/math] )
  • – θ = Angle between the magnetic field and the normal to the surface
  • – Maximum flux ( [math]Φ = BA[/math]) occurs when the field is perpendicular to the surface
  • [math]θ = 0^0, cos 0^0 = 1[/math]
  • Zero flux ([math]Φ = 0[/math] ) occurs when the field is parallel to the surface
  • [math]θ = 90^0, cos⁡90 = 0[/math]
  • Figure 1 Magnetic flux
  • ⇒ Definition of Flux Linkage
  • Flux linkage ([math] NΦ[/math] ) is the total magnetic flux linked with a coil having N turns:
  • [math]\text{Flux Linkage} = NΦ[/math]
  • Where:
  • – N = Number of turns of the coil
  • – Φ = Magnetic flux linked with one loop

  • b)   Faraday’s Law of Electromagnetic Induction

  • ⇒ Faraday’s First Law:
  • A changing magnetic flux induces an electromotive force (emf) in a conductor or coil.
  • ⇒ Faraday’s Second Law:
  • The induced emf (E) is proportional to the rate of change of flux linkage:
  • [math]E = -\frac{d(N\Phi)}{dt}[/math]
  • A faster change in flux results in a higher emf.
  • The negative sign follows Lenz’s Law, which ensures the direction of the induced current opposes the change in flux.
  • Figure 2 Faraday’s law of Electromagnetic Induction
  • ⇒  Lenz’s Law
  • Lenz’s Law states:
  • “The direction of the induced current is such that it opposes the change in magnetic flux that caused it.”
  • This is represented in Faraday’s equation by the negative sign.
  • It is a direct consequence of energy conservation—the induced current tries to counteract any external influence causing the flux change.
  • Figure 3 Lenz’s Law

  • c)    Application of Faraday’s and Lenz’s Laws

  • The induced emf can be calculated using:
  • [math]E = -\frac{d(N\Phi)}{dt}[/math]
  • ⇒ Examples of applications:
  • 1. Moving a magnet through a coil:
  • Moving a north pole into a coil increases flux; the induced current produces a field opposing this motion.
  • Pulling the magnet out decreases flux, and the induced current reverses direction.
  • 2. Rotating a coil in a magnetic field (AC Generator):
  • A coil rotating in a magnetic field experiences a changing flux over time, inducing an alternating emf.
  • Figure 4 AC Generator rotating move
  • 3. Transformers:
  • A changing current in the primary coil creates a changing flux, inducing an emf in the secondary coil (used in power transmission).

  • d)   Induced EMF in a Linear Conductor Moving at Right Angles to a Uniform Magnetic Field

  • ⇒  Basic Concept:
  • When a conductor moves perpendicularly to a uniform magnetic field, an electromotive force (emf) is induced across it due to the motion of free electrons in the conductor. This is an application of Faraday’s Law of Electromagnetic Induction.
  • ⇒  Formula for Induced EMF:
  • The magnitude of the induced emf ([math]ε[/math] ) is given by:
  • [math]ε = Blv[/math]
  • Where:
  • – B = Magnetic flux density (Tesla, T)
  • – l = Length of the conductor moving through the field (meters, m)
  • – v = Velocity of the conductor (m/s)
  • Figure 5 Induced EMF
  • ⇒  Explanation Using Lorentz Force:
  • When a conductor moves perpendicularly to a magnetic field, the free electrons inside experience a force due to the Lorentz Force Law:
  • [math]F = qvB[/math]
  • This force causes charge separation within the conductor, leading to an accumulation of charges at the ends of the conductor.
  • A potential difference (emf) is established between the ends of the conductor.
  • If the circuit is closed, current will flow, following Lenz’s Law, which opposes the motion.
  • ⇒  Real-Life Applications:
  • Railguns: A moving conductor in a magnetic field accelerates a projectile.
  • Power Generation: Motion of wires through Earth’s magnetic field generates a small emf in power lines.
  • DC Generators: A rotating coil can be thought of as moving conductors generating emf.

  • e)    Induced EMF in a Rotating Coil (AC Generator Principle)

  • When a coil rotates in a magnetic field, the magnetic flux passing through it changes with time, inducing an alternating emf.
  • ⇒  Instantaneous EMF Formula:
  • The emf induced in a coil rotating at a uniform angular velocity (ω) is given by:
  • [math]ε = ε_0 sinωt[/math]
  • Where:
  • [math]ε_0 = NBAω[/math](Maximum emf)
  • N = Number of turns of the coil
  • B = Magnetic flux density (Tesla)
  • A = Area of the coil (m²)
  • ω = Angular velocity of the coil (rad/s)
  • t = Time (s)
  • Figure 6 AC Generator principle
  • ⇒  Explanation of Coil Position and EMF Relationship:
Coil Position Magnetic Flux Induced EMF

Perpendicular to Field

( [math]0^0 \text{or} 180^0[/math] )

Maximum

(Φ=BA)

Zero

(E=0)
Angled to Field Reduced Increasing

Parallel to Field

( [math]90^0 \text{or} 270^0[/math] )

 Zero Maximum
  • When the coil is perpendicular to the field, maximum flux passes through it, but the rate of change of flux is zero, so no emf is induced.
  • When the coil is parallel to the field, the flux through it is minimum, but the rate of change of flux is maximum, so the induced emf is maximum.
  • ⇒  Applications:
  • – AC Generators (Alternators): Convert mechanical rotation into electrical energy.
  • – Wind Turbines: Rotating blades move a coil inside a magnetic field, generating electricity.
  • ⇒  Conclusion:
  • A moving conductor in a uniform magnetic field experiences induced emf due to the motion of charges.
  • A rotating coil in a magnetic field experiences an alternating emf depending on its position and angular velocity.
  • The induced emf follows Faraday’s Law, and its direction follows Lenz’s Law.
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