12.10 Faraday's Law

Faraday's Law of Induction is a fundamental law of electromagnetism that describes how a changing magnetic field can generate an electric current in a conductor. Discovered by Michael Faraday in 1831, this principle of electromagnetic induction is the foundation for electric generators, transformers, and many other modern technologies.

Magnetic Flux

Before understanding Faraday's Law, one must understand magnetic flux. Magnetic flux is a measure of the total number of magnetic field lines passing through a given area. A change in this flux is the key to inducing a current. For more on how these quantities are measured, see Derived Units→.

Faraday's Law of Induction

Faraday's Law can be summarized in two parts:

1. A changing magnetic flux through a loop of wire induces an electromotive force (EMF), which is essentially a voltage.
2. The magnitude of the induced EMF is directly proportional to the rate of change of the magnetic flux.

Mathematical Formulation:

For a single loop of wire, the induced EMF ($\epsilon$) is given by: $\epsilon =-\frac{\Delta \varphi }{\Delta t}$

For a coil with $N$ turns, the total EMF is multiplied by the number of turns: $\epsilon =-N\frac{\Delta \varphi }{\Delta t}$

where:

• $\epsilon$: Induced EMF (in Volts)
• $N$: Number of turns in the coil
• $\Delta \varphi$: Change in magnetic flux (in Webers, Wb)
• $\Delta t$: Change in time (in seconds, s)

Lenz's Law and the Negative Sign

The negative sign in Faraday's Law is a representation of Lenz's Law.

Lenz's Law states: The direction of the induced current is such that it creates a magnetic field that opposes the change in magnetic flux that produced it.

• This is a statement of the conservation of energy. If the induced current created a field that assisted the change, it would lead to a runaway, self-perpetuating current, which is physically impossible.

Faraday's Key Experiments

Faraday discovered induction through a series of brilliant experiments.

Experiment 1: A Moving Magnet and a Coil

• Setup: A coil of wire is connected to a galvanometer (a sensitive current meter).
• Observations:
  1. When a bar magnet is held stationary near the coil, no current is induced.
  2. When the magnet is moved towards the coil, the galvanometer deflects, indicating an induced current.
  3. When the magnet is moved away from the coil, the current is induced in the opposite direction.
• Conclusion: It is the relative motion between the magnet and the coil that causes a change in magnetic flux and induces a current.

Experiment 2: The Transformer Effect

• Setup: Two coils are placed near each other. The primary coil (P) is connected to a battery and a switch, and the secondary coil (S) is connected to a galvanometer.
• Observations:
  1. When the switch in the primary circuit is closed, the current in P goes from zero to a maximum. This changing current creates a changing magnetic field, which induces a brief current in S.
  2. When the current in P is steady, the magnetic field is constant, and no current is induced in S.
  3. When the switch is opened, the current in P drops to zero. This change again induces a brief current in S, but in the opposite direction.
• Conclusion: A changing current in one coil can induce a current in a nearby coil. This is the basic principle of a transformer.

Factors Affecting Induced EMF

The magnitude of the induced EMF (and thus the induced current) depends on:

1. The Number of Turns ($N$): More turns in the coil lead to a larger induced EMF.
2. The Rate of Change of Flux ($\Delta \varphi /\Delta t$):
   • Speed of Motion: Moving the magnet or coil faster increases the rate of change.
   • Strength of the Magnetic Field: A stronger magnet produces a larger flux change.
   • Area of the Coil: A larger coil area can capture more flux.

Significance: Faraday's Law is the cornerstone of electrical technology, explaining how mechanical energy can be converted into electrical energy and how electrical energy can be efficiently transferred and utilized.