recall and use Faraday’s and Lenz’s laws of electromagnetic induction

Electromagnetic Induction

Learning Objective

Recall and use Faraday’s and Lenz’s laws to analyse situations involving a changing magnetic flux.

Key Concepts

• Magnetic flux, $\Phi = \int \mathbf{B}\cdot d\mathbf{A}$
• Induced emf, $ \mathcal{E} $
• Direction of induced current
• Conservation of energy in electromagnetic systems

Faraday’s Law of Electromagnetic Induction

When the magnetic flux through a closed conducting loop changes, an emf is induced in the loop. The magnitude of the induced emf is given by

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

where

• $\mathcal{E}$ is the induced emf (V)
• $\Phi$ is the magnetic flux (Wb)
• $t$ is time (s)
• The negative sign expresses Lenz’s law (see below).

Lenz’s Law

Lenz’s law states that the direction of the induced emf (and thus the induced current) is such that it opposes the change in magnetic flux that produced it. This is a consequence of the conservation of energy.

In practice, the right‑hand rule is used to determine the direction:

1. Identify the change in flux (increase or decrease).
2. Imagine a magnetic field produced by the induced current that would oppose that change.
3. Use the right‑hand rule for a solenoid or loop to find the current direction.

Factors Affecting the Induced emf

• Speed of motion of the conductor relative to the magnetic field.
• Strength of the magnetic field, $B$.
• Number of turns in the coil, $N$ (for a coil, $\mathcal{E} = -N\,d\Phi/dt$).
• Area of the loop or the rate of change of area.

Common Situations

1. Moving conductor in a uniform field – $\mathcal{E}=Blv$ for a straight rod of length $l$ moving at speed $v$ perpendicular to $\mathbf{B}$.
2. Rotating coil (AC generator) – $\Phi = NBA\cos\theta$, $\theta = \omega t$, giving $\mathcal{E}=NBA\omega\sin\omega t$.
3. Changing magnetic field – If $B$ varies with time, $\mathcal{E}= -A\,\frac{dB}{dt}$ for a fixed area $A$.

Worked Example

Problem: A rectangular loop of width $0.10\,$m and height $0.20\,$m moves at $2.0\,$m s⁻¹ into a region where a uniform magnetic field of $0.50\,$T points into the page. The loop has $N=5$ turns. Find the maximum induced emf.

Solution:

1. Maximum emf occurs when the rate of change of flux is greatest – when the leading edge of the loop just enters the field.
2. Rate of change of area entering the field: $dA/dt = \text{width}\times v = 0.10\;\text{m}\times2.0\;\text{m s}^{-1}=0.20\;\text{m}^2\text{s}^{-1}$.
3. Change in flux per turn: $d\Phi/dt = B\,dA/dt = 0.50\;\text{T}\times0.20\;\text{m}^2\text{s}^{-1}=0.10\;\text{Wb s}^{-1}$.
4. Induced emf for $N$ turns: $\displaystyle \mathcal{E}_{\max}=N\left|\frac{d\Phi}{dt}\right|=5\times0.10=0.50\;\text{V}$.

Key Equations Summary

Phenomenon	Equation	Notes
Magnetic flux	$\displaystyle \Phi = \int \mathbf{B}\cdot d\mathbf{A}$	Units: weber (Wb)
Faraday’s law (single loop)	$\displaystyle \mathcal{E} = -\frac{d\Phi}{dt}$	Negative sign → Lenz’s law
Faraday’s law (coil of $N$ turns)	$\displaystyle \mathcal{E} = -N\frac{d\Phi}{dt}$	Flux is per turn
Moving straight conductor	$\displaystyle \mathcal{E}=Blv$	$\mathbf{B}\perp\mathbf{v}$ and $\mathbf{B}\perp$ length $l$
Rotating coil (generator)	$\displaystyle \mathcal{E}=NBA\omega\sin\omega t$	Peak emf $=NBA\omega$

Common Misconceptions

• Thinking the induced emf “creates” energy – it always opposes the cause of the change, preserving energy.
• Confusing the direction of the magnetic field with the direction of induced current; Lenz’s law determines current direction.
• Assuming a static magnetic field can induce an emf without motion or change; a change in flux is essential.

Practice Questions

1. A circular coil of radius $0.15\,$m and $N=10$ turns rotates at $50\,$rad s⁻¹ in a uniform magnetic field of $0.30\,$T. Write the expression for the instantaneous emf and calculate its maximum value.
2. A solenoid of length $0.40\,$m and $200$ turns carries a current that is increasing at $3.0\,$A s⁻¹. The solenoid’s cross‑sectional area is $2.5\times10^{-3}\,$m². Determine the magnitude of the induced emf in a single loop placed around the solenoid.
3. Explain qualitatively why a metal ring placed on a magnetic core that is being switched on experiences a repulsive force.

Summary

Faraday’s law quantifies how a changing magnetic flux induces an emf, while Lenz’s law provides the direction of the induced current, ensuring that the induced effect always opposes the cause. Mastery of these principles enables analysis of generators, transformers, inductive sensors, and many other devices encountered in A‑Level physics.