Electromagnetic Induction (Chapter 30) – Study Notes

Electromagnetic Induction

Introduction

Electromagnetic induction is the process by which an electromotive force (emf) is generated in a circuit due to a changing magnetic field. This phenomenon was discovered independently by Michael Faraday and Joseph Henry in 1831. The induced emf can drive a current in a closed conducting loop, and its magnitude depends on the rate of change of the magnetic flux through the loop.

Fundamental Concepts

Area Vector

The area vector \( \vec{A} = A \hat{n} \) is defined as a vector perpendicular to a surface, with magnitude equal to the area of the surface. It is used to calculate magnetic flux through a surface.

• Units: square meters (m2)

• Direction: perpendicular to the surface

Magnetic Flux

Magnetic flux (\( \Phi_B \)) quantifies the amount of magnetic field passing through a given area. It is a key quantity in electromagnetic induction.

• Definition: \( \Phi_B = \vec{A} \cdot \vec{B} \)

• If the field is uniform and the surface is flat: \( \Phi_B = AB \cos \theta \)

• General case: \( \Phi_B = \int \vec{B} \cdot d\vec{A} \)

• SI unit: Weber (Wb), where 1 Wb = 1 T·m2

Faraday's Law of Induction

Statement and Mathematical Formulation

Faraday's law states that the induced emf in a circuit is proportional to the rate of change of magnetic flux through the circuit.

• Single loop: $\varepsilon = - \frac{d\Phi_B}{dt}$

• Multiple loops (N turns): $\varepsilon = -N \frac{d\Phi_B}{dt}$

• Magnetic flux: $\Phi_B = \int \vec{B} \cdot d\vec{A}$

The negative sign indicates the direction of the induced emf opposes the change in flux (Lenz's Law).

Ways to Change Magnetic Flux

• Changing the magnitude of the magnetic field (\( B \))

• Changing the area (\( A \)) enclosed by the loop

• Changing the angle (\( \theta \)) between the magnetic field and the normal to the loop

• Any combination of the above

Example: Rotating a loop in a uniform magnetic field changes \( \theta \), thus changing the flux and inducing an emf.

Motional emf

Definition and Mechanism

Motional emf is the emf induced in a conductor moving through a constant magnetic field. The moving charges experience a magnetic force, leading to charge separation and an electric field inside the conductor.

• Force on charge: $\vec{F} = q \vec{v} \times \vec{B}$

• Equilibrium condition: $qE = qvB \implies E = vB$

• Potential difference: $\Delta V = E \ell = B v \ell$ (where \( \ell \) is the length of the conductor)

If the direction of motion is reversed, the polarity of the induced potential difference is also reversed.

Sliding Conducting Bar Example

When a conducting bar moves through a uniform magnetic field, the flux through the loop changes as the bar moves, inducing an emf and current.

• Flux: $\Phi_B = B \ell x$

• Induced emf: $\varepsilon = - \frac{d\Phi_B}{dt} = -B \ell v$

• Current (if resistance is \( R \)): $I = \frac{\varepsilon}{R} = \frac{B \ell v}{R}$

Mechanical energy is converted into electrical energy as someone pushes the bar to maintain constant velocity against the magnetic force.

• Power delivered to resistor: $P = F_{app} v = (I \ell B) v = \frac{\varepsilon^2}{R}$

Lenz's Law

Statement and Physical Interpretation

Lenz's Law states that the direction of the induced current is such that it creates a magnetic field opposing the change in magnetic flux through the loop.

• Mathematical form: $\varepsilon = - \frac{d\Phi_B}{dt}$

• The induced current tends to keep the original magnetic flux through the circuit from changing.

Example: If a conducting bar slides to the right, increasing the flux into the page, the induced current produces a magnetic field out of the page (counterclockwise current).

Applications and Examples

Induced emf in Coils and Loops

• For a coil with \( N \) turns, each of area \( A \), in a changing magnetic field:

$\varepsilon = -N \frac{d\Phi_B}{dt}$ $\Phi_B = NBA$ $\frac{d\Phi_B}{dt} = N A \frac{dB}{dt}$

Example: If a coil of 216 turns (each side 18 cm) is exposed to a field changing from 0 to 0.50 T in 0.68 s, the induced emf is calculated using the above formulas.

Flexible Loop Example

• If the area of a loop changes in a constant magnetic field, the induced emf is:

$\varepsilon = -N \frac{d\Phi_B}{dt}$ $\Phi_B = BA$ $\frac{d\Phi_B}{dt} = B \frac{dA}{dt}$

Rotating Loop and Generators

• When a loop rotates in a magnetic field, the flux changes sinusoidally:

$\Phi_B = BA \cos \theta$ $\theta = \omega t$ $\varepsilon = -N \frac{d\Phi_B}{dt}$

This principle is used in AC generators, where mechanical work is converted into electrical energy.

Transformers

Transformers use electromagnetic induction to change the voltage and current in AC circuits. They consist of primary and secondary coils wound around a common core.

Quantity	Primary Coil	Secondary Coil
Number of turns	\( N_p \)	\( N_s \)
Voltage	\( V_p \)	\( V_s \)
Current	\( I_p \)	\( I_s \)

• Transformer equations:

$\frac{V_s}{V_p} = \frac{N_s}{N_p}$ $V_p I_p = V_s I_s$ (assuming 100% efficiency)

Example: If a transformer steps up voltage from 120 V to 1000 V, and the primary coil has 400 turns, the number of turns in the secondary coil is:

$N_s = N_p \frac{V_s}{V_p}$

Induced emf and Electric Fields

Electric Field from Changing Magnetic Field

A changing magnetic field induces an electric field, even in the absence of a conducting loop. This induced electric field is non-conservative, meaning the line integral of \( \vec{E} \cdot d\vec{s} \) over a closed loop is not zero.

• Integral form of Faraday's Law: $\oint \vec{E} \cdot d\vec{s} = - \frac{d\Phi_B}{dt}$

• Induced electric field lines form closed loops, similar to magnetic field lines.

Example: Inside a solenoid with increasing magnetic field, the induced electric field is tangential to circles centered on the axis.

Inductance

Self-Inductance and Inductors

Inductance is a property of a circuit element (usually a coil) that quantifies its ability to oppose changes in current by generating a back emf.

• Inductance (L): $L = \frac{N \Phi_B}{I}$

• Back emf: $\varepsilon = -L \frac{dI}{dt}$

• Inductance depends on the geometry of the coil, not on the current or flux.

Example: For a solenoid of length \( l \), radius \( r \), and \( N \) turns:

$L = \mu_0 \frac{N^2 A}{l}$ where \( A = \pi r^2 \) is the cross-sectional area.

Summary Table: Key Equations

Concept	Equation (LaTeX)	Description
Magnetic Flux	$\Phi_B = \int \vec{B} \cdot d\vec{A}$	Amount of magnetic field through area
Faraday's Law	$\varepsilon = - \frac{d\Phi_B}{dt}$	Induced emf from changing flux
Motional emf	$\varepsilon = B \ell v$	Emf from moving conductor
Lenz's Law	$\varepsilon = - \frac{d\Phi_B}{dt}$	Direction opposes change in flux
Transformer	$\frac{V_s}{V_p} = \frac{N_s}{N_p}$	Voltage ratio equals turns ratio
Inductance	$L = \mu_0 \frac{N^2 A}{l}$	Inductance of solenoid

Additional info:

• Special relativity provides a unified explanation for the equivalence of mechanical and electrical induction mechanisms.

• Induced electric fields are fundamentally different from electrostatic fields, as they are non-conservative and form closed loops.