Study Guide: Magnetic Fields, Forces, and Electromagnetic Induction (Chapters 27–29)

Exam Preparation Overview

This study guide covers key concepts from Chapters 27–29, focusing on magnetic fields, magnetic forces, sources of magnetic fields, and electromagnetic induction. It also reviews foundational knowledge from earlier chapters relevant to these topics.

Foundational Concepts from Previous Courses

• Newton's Laws: Fundamental principles describing the relationship between force, mass, and acceleration.

• Kinematic Motion in 1D and 2D: Techniques for solving problems involving displacement, velocity, and acceleration in one and two dimensions.

• Work and Energy:

  • Definition of Work: $W = \vec{F} \cdot \vec{d}$

  • Work-Energy Theorem: $W_{\text{net}} = \Delta K$

  • Potential Energy from Work: $\Delta U = -W_{\text{conservative}}$

• Mathematical Tools: Differentiation and integration of polynomials and trigonometric functions (sine and cosine).

• Quadratic Equation: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Frequency and Angular Frequency: $\omega = 2\pi f$

Review of Key Concepts from Chapters 21–26

• Ohm's Law: $V = IR$

• Power Loss in a Resistor: $P = I^2R = \frac{V^2}{R}$

• Relation Between Electric Field and Potential: $E = \frac{V}{d}$

• Electrical Potential Energy: $U = qV$

Magnetic Field and Forces

Characteristics of Magnets and Magnetic Field Lines

• Magnetic Field Lines: Emerge from the north pole and enter the south pole of a magnet; never cross and form closed loops.

• Field Line Density: Indicates the strength of the magnetic field; closer lines mean a stronger field.

Lorentz Force (Magnetic Force on a Moving Charge)

• Formula: $\vec{F} = q\vec{v} \times \vec{B}$

• Direction: Determined by the right-hand rule.

• Example: A proton moving perpendicular to a uniform magnetic field experiences a force perpendicular to both its velocity and the field.

Motion of Charged Particles in a Magnetic Field

• Circular Motion: When $\vec{v} \perp \vec{B}$, the particle moves in a circle.

• Radius of Path: $r = \frac{mv}{|q|B}$

• Period of Revolution: $T = \frac{2\pi m}{|q|B}$

Force on a Current-Carrying Wire

• Formula: $\vec{F} = I\vec{L} \times \vec{B}$

• Direction: Right-hand rule applies (thumb in direction of current, fingers in direction of $\vec{B}$).

Force and Torque on a Current Loop

• Torque: $\vec{\tau} = \vec{\mu} \times \vec{B}$

• Magnetic Dipole Moment: $\vec{\mu} = I\vec{A}$, where $A$ is the area vector of the loop.

Typical Exam Problems: Magnetic Fields and Forces

• Drawing magnetic field lines for various configurations.

• Analyzing the motion of charged particles in electric and magnetic fields using kinematic equations.

• Calculating the force on a current-carrying wire in a magnetic field.

• Determining the force and torque on a magnetic dipole, including in non-uniform fields.

Sources of Magnetic Field

Biot-Savart Law

• Formula: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$

• Application: Used to calculate the magnetic field produced by a small segment of current-carrying wire.

Magnetic Field from a Straight Wire

• Formula: $B = \frac{\mu_0 I}{2\pi r}$ (at distance $r$ from a long, straight wire)

Magnetic Field from a Current Loop

• On Axis of Loop: $B = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}$ (at distance $x$ along axis)

Force Between Two Parallel Wires

• Formula: $F/L = \frac{\mu_0 I_1 I_2}{2\pi d}$

• Direction: Wires with currents in the same direction attract; opposite directions repel.

Ampère’s Law

• Integral Form: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}$

• Applications: Useful for calculating $B$ in symmetric situations (e.g., solenoids, toroids).

Magnetic Field for Solenoids and Toroids

• Solenoid: $B = \mu_0 n I$ (inside, where $n$ is turns per unit length)

• Toroid: $B = \frac{\mu_0 N I}{2\pi r}$ (inside, at radius $r$)

Types of Magnetism and Atomic View

• Diamagnetism: Weak, negative response to external field; all materials exhibit this to some extent.

• Paramagnetism: Weak, positive response; unpaired electrons align with field.

• Ferromagnetism: Strong, permanent magnetization due to domain alignment (e.g., iron, cobalt, nickel).

• Atomic Picture: Magnetism arises from electron spin and orbital motion.

Typical Exam Problems: Sources of Magnetic Field

• Applying Ampère’s law to solenoids, toroids, and sheets of current.

• Solving for $B$ when current density varies.

• Explaining the atomic basis for different types of magnetism.

Electromagnetic Induction

Faraday’s Law

• Law of Induction: $\mathcal{E} = -\frac{d\Phi_B}{dt}$, where $\mathcal{E}$ is the induced EMF and $\Phi_B$ is the magnetic flux.

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

Lenz’s Law

• Statement: The direction of induced current opposes the change in magnetic flux that produced it.

• Application: Used to determine the direction of induced EMF and current.

Motional EMF

• Formula: $\mathcal{E} = B\ell v$ (for a rod of length $\ell$ moving at velocity $v$ perpendicular to $B$)

• Example: A metal rod sliding on rails in a uniform magnetic field generates an EMF.

Induced Electric Fields

• Changing magnetic fields induce non-conservative electric fields, as described by Faraday’s law.

Eddy Currents

• Circulating currents induced in conductors exposed to changing magnetic fields; cause energy loss as heat.

AC Generators

• Devices that convert mechanical energy to electrical energy using electromagnetic induction.

• Output EMF: $\mathcal{E}(t) = N B A \omega \sin(\omega t)$ for a coil rotating in a uniform $B$ field.

Typical Exam Problems: Electromagnetic Induction

• Calculating EMF or current using Faraday’s law.

• Determining the direction of induced current using Lenz’s law.

• Analyzing the kinematics of sliding or rotating rods in motional EMF problems.

• Calculating the induced electric field from a changing magnetic flux.

Problem Resources

• Sample problems from the textbook and class notes.

• Homework and in-class problems (solutions available on D2L).

• Tutorial problems and sample exams (including those from other instructors).

• Physics Libre Texts library for additional practice and explanations.