LEC 5: Electromagnetic Induction II: Faraday’s Law, Motional EMF, and Eddy Currents

Electromagnetic Induction II

Overview and Historical Context

Electromagnetic induction is a fundamental concept in physics, describing how a changing magnetic field can induce an electromotive force (emf) in a conductor. This principle was discovered through experiments by scientists such as Faraday and Lenz, and is central to the operation of electric generators and motors.

• Field Concept: Faraday introduced the idea of electric and magnetic fields, explaining how changes in these fields produce effects in other objects (electromagnetic induction).

• Unification by Maxwell: Maxwell’s equations describe the behavior of electric and magnetic fields and their interactions, including electromagnetic wave propagation.

Faraday’s Law and Lenz’s Law

Fundamental Principles

Faraday’s Law and Lenz’s Law govern the induction of emf in a circuit due to changing magnetic flux.

• Faraday’s Law: The induced emf ($\varepsilon$) is the negative rate of change of magnetic flux ($\Phi_m$) through a circuit: $\Phi_m = \int \vec{B} \cdot d\vec{A}$ $\varepsilon = -\frac{d\Phi_m}{dt}$ For a coil with N turns: $\varepsilon = -N\frac{d\Phi_m}{dt}$

• Lenz’s Law: The direction of the induced emf always opposes the change in magnetic flux that causes it.

Example: If the magnetic flux through a loop increases, the induced current will flow in a direction that creates a magnetic field opposing the increase.

Motional EMF

Mechanisms of Induction

Motional emf arises when a conductor moves through a magnetic field, changing the area or orientation of the circuit relative to the field.

• Changing Area: Moving a coil/loop into a steady magnetic field.

• Rotating Loop: Rotating a loop in a steady magnetic field.

• Changing Area Inside Field: Changing the area of a coil/loop inside a steady magnetic field.

Formula: $\Phi_m = B A \cos(\theta(t))$ $\theta = \omega t$, where $\omega$ is angular velocity (rad/s), $t$ is time.

Electric Generator

Principle of Operation

Electric generators convert mechanical energy into electrical energy using electromagnetic induction.

• Magnetic Flux: $\Phi_m = B A \cos(\theta(t))$

• Induced EMF: $\varepsilon = \frac{d\Phi_m}{dt} = B A \omega \sin(\omega t)$

• For N turns: $\varepsilon = N B A \omega \sin(\omega t)$

• Maximum amplitude: $\varepsilon_0 = N B A \omega$

Example: The induced voltage in a generator is sinusoidal (alternating current, ac). The period $T$ is given by $T = \frac{2\pi}{\omega}$.

Electric Generators vs. Motors

Energy Conversion

• Generator: Converts mechanical energy into electrical energy. Requires an external mechanical force to spin the rotor.

• Motor: Converts electrical energy into mechanical energy. Electric current in coils produces torque, causing rotation.

Comparison Table:

Device	Input	Output
Generator	Mechanical energy	Electrical energy
Motor	Electrical energy	Mechanical energy

Pulsed DC Voltage in Generators

Split Ring Commutator

Some generator setups use a split ring commutator to produce a pulsed direct current (dc) voltage. The commutator reverses the connection every half turn, resulting in a voltage that pulses but does not change direction.

• Current Direction: Follows the rotation and commutator switching.

• Voltage Output: Pulsed dc, as shown in the voltage vs. time graph.

Motional EMF: Conducting Rod Pulled in a Circuit

Analysis and Equations

When a conducting rod moves through a magnetic field, an emf is induced across its ends.

• Area Enclosed: $A = l x$ (where $l$ is length, $x$ is position)

• Magnetic Flux: $\Phi_m = B l x$

• Rod Velocity: $v = \frac{dx}{dt}$

• Induced EMF: $\varepsilon = -\frac{d\Phi_m}{dt} = -B l \frac{dx}{dt} = -B l v$

• Ohm’s Law: $I = \frac{\varepsilon}{R}$

Direction of Current:

• Pushing the rod to the right increases the area and magnetic flux.

• Induced current opposes the increase (Lenz’s law), generating a magnetic field out of the page.

• Current flows counterclockwise around the loop.

Forces:

• Applied force $F_{\text{applied}}$ pushes the rod.

• Magnetic force $F_m = q \vec{v} \times \vec{B}$ acts on charges, opposing the motion.

• Magnetic force restores original flux, consistent with conservation of energy.

Open Circuit Case: Even without a closed loop, a current can be induced in the rod, with positive charges moving to one end and negative charges to the other, analogous to a battery.

Induced Electric Fields

Nature and Properties

Changing magnetic flux induces an electric field ($\vec{E}$) that does work on conduction electrons.

• Faraday’s Law (Integral Form): $\varepsilon = \oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_m}{dt}$

• Induced electric fields are non-conservative ($\oint \vec{E} \cdot d\vec{l} \neq 0$), forming closed loops.

• Direction opposes change in magnetic flux (Lenz’s law).

Contrast: Electrostatic fields are conservative ($\oint \vec{E} \cdot d\vec{l} = 0$).

Magnetic Damping due to Eddy Currents

Formation and Effects

Eddy currents are loops of induced current in conductors due to motional emf. These currents produce significant drag, called magnetic damping, which opposes the motion.

• Induced current opposes the change in magnetic flux.

• Magnetic force from eddy currents slows down the motion of the conductor.

• Occurs when the conductor enters or leaves the magnetic field.

Example: A metal bob swinging through a magnetic field experiences drag due to eddy currents, slowing its motion.

Eddy Currents in Slotted Metal Plates

Cancellation Effects

In slotted metal plates, eddy currents form small loops that can cancel each other, reducing the overall effect of magnetic damping.

Applications of Eddy Currents

Technological Uses

• Electromagnetic Braking: Used in trains and roller coasters for smooth, contactless braking.

• Induction Heating: Heats metals efficiently for industrial processes.

• Metal Sorting and Identification: Separates metals in recycling plants.

• Metal Detectors: Detects metallic objects underground or in luggage.

Excursus: Gradients and Differential Operators

Mathematical Tools (If Time Allows)

Gradients and differential operators are essential for describing changes in fields and fluxes in electromagnetism. They allow precise mathematical formulation of physical laws such as Faraday’s Law and Maxwell’s equations.