LEC 6: Electromagnetic Induction & Inductance: Study Notes for Physics 2101A

Electromagnetic Induction & Inductance

Overview of Electromagnetic Induction

Electromagnetic induction is the process by which a changing magnetic field induces an electromotive force (emf) in a conductor. This principle is fundamental to many electrical devices and is described by Faraday's Law of Induction.

• Faraday's Law: The induced emf in a circuit is proportional to the rate of change of magnetic flux through the circuit.

• Lenz's Law: The direction of the induced emf opposes the change in magnetic flux that produced it.

• Applications: Generators, transformers, induction heating, and electromagnetic braking.

Motional emf: Conducting Rod Without Circuit

Even without a closed loop, a moving conductor in a magnetic field can experience an induced emf. Imagining a closed rectangle, the analysis is similar to a battery with positive and negative poles.

• Charge Separation: Positive charges are pushed to one end (positive pole), negative charges to the other (negative pole).

• Current Flow: In the imaginary circuit, current flows from the positive to the negative pole.

• Formula: For a rod of length $l$ moving at velocity $v$ perpendicular to a magnetic field $B$, the emf is:

$ext{emf} = B l v$

Moving Conducting Materials in Earth's Magnetic Field

Any conducting material moving at an angle to Earth's magnetic field experiences a motional emf, though the effect is small due to Earth's weak field.

• Earth's Magnetic Field: $B_{Earth} hickapprox 5 \times 10^{-5} \text{T}$

• Example Calculation: For a 1.0 m rod moving at 3.0 m/s:

$ext{emf} = B_{Earth} l v = (5 \times 10^{-5}) \times 3.0 \times 1.0 = 150 \mu V$

• Comparison: A typical AA battery has 1.5 V.

Induced Electric Fields

Changing magnetic flux induces an emf, which does work on conduction electrons. The work is done by the induced electric field $\vec{E}$, and Faraday's law can be written in terms of this field:

$\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$

• Non-Conservative Fields: Induced electric fields are non-conservative, forming closed loops (unlike electrostatic fields).

• Direction: The induced field opposes the change in magnetic flux (Lenz's law).

Magnetic Damping Due to Eddy Currents

Motional emf in a conductor can cause circulating currents called eddy currents. These produce magnetic damping, a drag force that opposes motion.

• Eddy Currents: Circulating currents induced in conductors by changing magnetic fields.

• Magnetic Damping: The drag force slows down the motion of the conductor.

• Opposition: Induced current opposes the change in magnetic flux.

Eddy Currents Induced in a Slotted Metal Bob

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

• Neighboring Loops: Eddy currents in adjacent slots flow in opposite directions, leading to cancellation.

Applications of Eddy Currents

Eddy currents have several practical applications in technology and industry.

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

• Induction Heating: Heats metals efficiently for industrial processes.

• Metal Sorting and Identification: Used in recycling and manufacturing.

• Metal Detectors: Detect hidden metallic objects by sensing eddy currents.

Inductance

Inductance is a property of a device that quantifies how effectively it induces an emf in another device or in itself due to changing current.

• Unit: Henry (H), where $1 \text{H} = 1 \text{V} \cdot \text{s} / \text{A}$

• Example: Wireless charging devices use inductance to transfer energy.

Mutual Inductance

Mutual inductance occurs when a changing current in one circuit induces an emf in a nearby circuit.

• Magnetic Flux: The magnetic flux through each circuit varies due to the changing current in the other circuit.

• Mutually Induced emf: Each circuit experiences an emf due to the other's changing current.

Mutual Inductance: Magnetic Flux Through Coils

For two tightly wound coils with $N_1$ and $N_2$ turns:

• Magnetic flux through coil 2 due to current $I_1$ in coil 1: $N_2 \Phi_{21}$

• Magnetic flux through coil 1 due to current $I_2$ in coil 2: $N_1 \Phi_{12}$

Mutual Inductance: Definition and Units

The mutual inductance $M_{21}$ of coil 2 with respect to coil 1 is:

$M_{21} = \frac{N_2 \Phi_{21}}{I_1}$

Similarly, for coil 1 with respect to coil 2:

$M_{12} = \frac{N_1 \Phi_{12}}{I_2}$

• Equality: $M_{12} = M_{21} = M$

• Unit: Henry (H)

Mutual Inductance and emf

The induced emf in coil 1 due to changing current in coil 2:

$\epsilon_1 = -M \frac{dI_2}{dt}$

And in coil 2 due to coil 1:

$\epsilon_2 = -M \frac{dI_1}{dt}$

• Other contributions may exist due to self-inductance.

Self-Inductance

Self-inductance occurs when a changing current in a coil induces an emf in itself. This is described by the following equations:

• Total emf in coil 2: $\epsilon_2 = M_{21} \frac{dI_1}{dt} + M_{22} \frac{dI_2}{dt}$

• Total emf in coil 1: $\epsilon_1 = M_{12} \frac{dI_2}{dt} + M_{11} \frac{dI_1}{dt}$

• Self-inductance coefficients: $M_{22} = -L_2$, $M_{11} = -L_1$

Self-Inductance of a Single Coil

For a single coil, the self-induced emf is:

$\epsilon = -L \frac{dI}{dt}$

• Opposition: The induced emf opposes the change in current.

• External Voltage: $V = L \frac{dI}{dt}$

• Proportionality: The proportionality constant $L$ is the self-inductance.

Similarity to Newton's Law

There is an analogy between self-inductance in circuits and Newton's law of motion. The equations governing both systems have similar forms.

Particle	Circuit (Coil)
Force ($F$)	Potential difference ($V$)
Velocity ($v$)	Current ($I$)
Mass ($m$)	Inductance ($L$)
$p = m \frac{dv}{dt}$ (momentum)	$LI$ (magnetic energy)
$\frac{1}{2} m v^2$ (kinetic energy)	$\frac{1}{2} L I^2$ (magnetic energy)

Example: The behavior of a coil in an electric circuit can be compared to the motion of a mass under force, with inductance playing the role of mass.

Additional info: These notes cover topics from Chapter 29 (Electromagnetic Induction), Chapter 30 (Inductance), and related applications, as outlined in the Physics college course chapters.