Electromagnetic Induction (Chapter 21) – College Physics Study Notes

Electromagnetic Induction

Overview and Goals

Electromagnetic induction is a fundamental concept in physics describing how a changing magnetic field can induce an electric current in a conductor. This chapter covers Faraday’s law, Lenz’s law, motional electromotive force (emf), eddy currents, mutual and self-inductance, transformers, magnetic field energy, and R-L and L-C circuits.

• Faraday’s Law: Explains how a time-varying magnetic field induces an emf.

• Lenz’s Law: Determines the direction of the induced current.

• Motional emf: emf generated by the motion of a conductor in a magnetic field.

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

• Mutual and Self-Inductance: Quantifies how circuits induce emf in themselves or other circuits.

• Transformers: Devices that use induction to change voltage levels.

• Magnetic Field Energy: Energy stored in magnetic fields.

• R-L and L-C Circuits: Circuits involving resistors, inductors, and capacitors.

Magnetic Flux and Induction

Magnetic Flux

Magnetic flux (ΦB) through a surface quantifies the number of magnetic field lines passing through that surface. It is a key concept for understanding induction.

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

• For uniform fields: $\Phi_B = B \cdot A \cdot \cos\theta$

• Angle Dependence: Flux is maximum when the field is perpendicular to the surface ($\theta = 0$), and zero when parallel ($\theta = 90^\circ$).

• Example: A coil in a changing magnetic field will experience a change in flux, leading to induced emf.

Does the Field Induce a Current?

A current is induced in a conductor only if the magnetic flux through the conductor changes over time.

• Stationary Magnet: No current is induced if the magnet and coil are stationary.

• Moving Magnet or Coil: Current is induced when either the magnet or coil moves, changing the flux.

• Example: Moving a magnet into or out of a coil induces a current detected by a galvanometer.

Faraday’s Law of Induction

Faraday’s Law

Faraday’s law quantifies the induced emf in a circuit due to a changing magnetic flux.

• Mathematical Form: $\mathcal{E} = -\frac{d\Phi_B}{dt}$

• Negative Sign: Indicates the direction of induced emf opposes the change in flux (Lenz’s law).

• Example: If the flux through a coil changes by $0.02\,\text{T} \cdot \text{m}^2$ in $0.1\,\text{s}$, the induced emf is $\mathcal{E} = -0.2\,\text{V}$.

Lenz’s Law

Lenz’s law determines the direction of the induced current: it always opposes the change in magnetic flux that produced it.

• Right-Hand Rule: Used to find the direction of induced current.

• Example: If a north pole of a magnet approaches a coil, the induced current creates a north pole facing the magnet to oppose its approach.

Motional Electromotive Force (emf)

Motional emf

Motional emf is generated when a conductor moves through a magnetic field, cutting across field lines.

• Formula: $\mathcal{E} = B \ell v$

• Variables: $B$ = magnetic field strength, $\ell$ = length of conductor, $v$ = velocity perpendicular to $B$.

• Example: A rod of length $0.5\,\text{m}$ moving at $2\,\text{m/s}$ in a $0.3\,\text{T}$ field: $\mathcal{E} = 0.3 \times 0.5 \times 2 = 0.3\,\text{V}$.

Eddy Currents

Definition and Applications

Eddy currents are loops of induced current in conductors exposed to changing magnetic fields. They can cause energy loss and are used in applications like metal detectors and electric meters.

• Formation: Occur in bulk conductors when the magnetic field changes.

• Effects: Can produce heating and magnetic drag.

• Applications: Metal detectors use eddy currents to detect concealed objects.

Inductance

Mutual Inductance

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

• Formula: $\mathcal{E}_2 = -M \frac{dI_1}{dt}$

• Variables: $M$ = mutual inductance, $I_1$ = current in coil 1.

• Example: The Tesla coil uses mutual inductance to transfer energy between coils.

Self-Inductance

Self-inductance is the property of a circuit whereby a changing current induces an emf in itself.

• Formula: $\mathcal{E}_L = -L \frac{dI}{dt}$

• Variables: $L$ = self-inductance, $I$ = current in the coil.

• Example: Inductors in circuits resist changes in current due to self-inductance.

Transformers

Operation and Equations

Transformers use mutual induction to change voltage levels between circuits. They consist of two coils wound on a common core.

• Primary Coil: Receives input voltage.

• Secondary Coil: Delivers output voltage.

• Transformer Equation: $\frac{V_s}{V_p} = \frac{N_s}{N_p}$

• Variables: $V_s$ = secondary voltage, $V_p$ = primary voltage, $N_s$ = secondary turns, $N_p$ = primary turns.

• Example: Power transmission uses transformers to step up or step down voltages.

Energy in Magnetic Fields

Energy Storage

Inductors store energy in their magnetic fields when current flows through them.

• Formula: $U = \frac{1}{2} L I^2$

• Variables: $U$ = energy stored, $L$ = inductance, $I$ = current.

• Example: Energy stored in an inductor can be released when the current decreases.

R-L and L-C Circuits

R-L Circuits

An R-L circuit contains a resistor and an inductor. The current and voltage change over time when the circuit is switched on or off.

• Current Growth: $I(t) = \frac{\mathcal{E}}{R} \left(1 - e^{-\frac{R}{L}t}\right)$

• Time Constant: $\tau = \frac{L}{R}$

• Example: The current in an R-L circuit does not instantly reach its maximum value due to inductance.

L-C Circuits

An L-C circuit contains an inductor and a capacitor. Energy oscillates between the electric field of the capacitor and the magnetic field of the inductor.

• Oscillation Frequency: $f = \frac{1}{2\pi\sqrt{LC}}$

• Application: Used in radio tuners to select frequencies.

• Example: The charge on the capacitor and the current in the inductor oscillate sinusoidally.

Summary Table: Key Equations and Concepts

Concept	Equation	Description
Magnetic Flux	$\Phi_B = B A \cos\theta$	Flux through area $A$ at angle $\theta$ to $B$
Faraday’s Law	$\mathcal{E} = -\frac{d\Phi_B}{dt}$	Induced emf from changing flux
Motional emf	$\mathcal{E} = B \ell v$	emf from moving conductor in $B$
Mutual Inductance	$\mathcal{E}_2 = -M \frac{dI_1}{dt}$	Induced emf in second coil
Self-Inductance	$\mathcal{E}_L = -L \frac{dI}{dt}$	Induced emf in same coil
Transformer	$\frac{V_s}{V_p} = \frac{N_s}{N_p}$	Voltage ratio equals turns ratio
Energy in Inductor	$U = \frac{1}{2} L I^2$	Energy stored in magnetic field
R-L Circuit Current	$I(t) = \frac{\mathcal{E}}{R} \left(1 - e^{-\frac{R}{L}t}\right)$	Current growth in R-L circuit
L-C Circuit Frequency	$f = \frac{1}{2\pi\sqrt{LC}}$	Oscillation frequency

Key Takeaways

• Changing magnetic fields induce electric currents (emf) in conductors.

• Lenz’s law ensures the induced current opposes the change in flux.

• Inductance quantifies the ability of circuits to induce emf in themselves or others.

• Transformers and L-C circuits are practical applications of electromagnetic induction.

Additional info: Some equations and examples have been expanded for clarity and completeness beyond the original slides.