D.4.2 Applications of electromagnetic induction (HL only)

Rotating Coils: Sinusoidal EMF Generation in Electric Generators

How Does a Rotating Coil Generate EMF?

Imagine a coil rotating in a magnetic field.
As it rotates, the magnetic flux through the coil changes, inducing an emf according to Faraday’s law.

Hint

The induced emf is proportional to the rate of change of magnetic flux.

Mathematical Description of Sinusoidal EMF

If a coil with $N$ turns and area $A$ rotates in a uniform magnetic field $B$ with angular velocity $v$ , the magnetic flux $Φ$ through the coil is given by:

$Φ = N B A \cos (ω t)$

The induced emf is the negative rate of change of this flux:

$ε = - \frac{d Φ}{d t} = N B A ω \sin (ω t)$

This equation describes a sinusoidal emf, which oscillates between positive and negative values.

Hint

The peak emf is given by $ε_{0} = N B A ω$ , and the emf varies as $ε = ε_{0} \sin (w t)$ .

Why Sinusoidal?

The sinusoidal nature arises because the flux changes cyclically as the coil rotates.
When the coil is perpendicular to the field, the flux is maximum (but the rate of change is zero), resulting in zero emf.
When the coil is parallel to the field, the flux is zero, but the rate of change is maximum, producing maximum emf.

Analogy

Think of the coil’s rotation like a pendulum swinging back and forth.
At the extremes, the pendulum is momentarily still (zero emf), but at the midpoint, it moves fastest (maximum emf).

Time-Varying Magnetic Fields: Induction by Moving Magnets or Coils

How Does Motion Induce EMF?

EMF can be induced by:

Moving a magnet near a coil.
Moving a coil in a magnetic field.
Changing the strength of the magnetic field.

Example

When a magnet is pushed into a coil, the galvanometer needle deflects, indicating a current.
Pulling the magnet out causes the needle to deflect in the opposite direction.

Factors Affecting Induced EMF

Speed of Motion: Faster movement increases the rate of change of flux, inducing a larger emf.
Strength of the Magnetic Field: A stronger field induces a larger emf.
Number of Turns in the Coil: More turns increase the flux linkage, enhancing the induced emf.
Orientation of Motion: Motion perpendicular to the field lines induces the maximum emf.

Common Mistake

A common mistake is to assume that a stationary magnet near a coil will induce emf. Only relative motion or a changing magnetic field induces emf.

Eddy Currents: Applications in Braking Systems

Definition

Eddy currents

Eddy currents are loops of induced current that form in conductors exposed to a changing magnetic field.

While they can cause energy losses, they are also harnessed in eddy current braking systems.

How Eddy Current Brakes Work

Magnetic Field Interaction:
- When a conductor (such as a metal disc) moves through a magnetic field, eddy currents are induced in the disc.
Opposing Motion:
- According to Lenz’s law, these currents create a magnetic field that opposes the motion of the disc, producing a braking force.
Heat Dissipation:
- The kinetic energy of the moving disc is converted into thermal energy, slowing down the system.

Example

Eddy current brakes are commonly used in high-speed trains and roller coasters because they provide smooth, contactless braking without wear and tear.

Advantages of Eddy Current Brakes

No Physical Contact:
- Reduces maintenance and wear.
Smooth Operation:
- Provides consistent braking force.
Adjustable:
- The braking force can be controlled by varying the strength of the magnetic field.

Common Mistake

A common mistake is to assume that eddy currents always cause energy losses. While they do dissipate energy as heat, this property is intentionally used in applications like braking systems.

Reflection and Key Takeaways

Theory of Knowledge

How does Lenz’s Law illustrate the broader principle of conservation of energy?
Can you think of other systems where opposing forces maintain balance?

Self review

How does the rotation of a coil in a magnetic field produce a sinusoidal emf?
What factors affect the magnitude of the induced emf when a magnet moves near a coil?
How do transformers use electromagnetic induction to step up or step down voltage?
Why is Lenz’s Law essential for energy conservation in electromagnetic systems?

Electromagnetic induction is the foundation of modern electrical technology. By understanding how rotating coils, time-varying fields, and transformers work, you can appreciate the principles that power our world.

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