Magnetism, Electromagnetic Induction, and Electromagnetic Waves: Exam 2 Study Guide

Chapter 20: Magnetism

Magnetic Field Lines

Magnetic field lines are a visual representation of the direction and strength of a magnetic field. They emerge from the north pole and enter the south pole of a magnet, forming closed loops.

• Direction: Field lines indicate the direction a north magnetic pole would move.

• Density: The closer the lines, the stronger the magnetic field.

• Closed Loops: Unlike electric field lines, magnetic field lines always form closed loops.

• Example: Iron filings around a bar magnet reveal the pattern of magnetic field lines.

Relationship Between Electric Current and Magnetic Fields (Right Hand Rule 1)

Electric currents produce magnetic fields. The direction of the magnetic field around a current-carrying wire is determined by the right hand rule.

• Right Hand Rule 1: If you grasp a wire with your right hand, with your thumb pointing in the direction of the current, your fingers curl in the direction of the magnetic field.

• Formula: The magnetic field at a distance $r$ from a long straight wire carrying current $I$ is given by: $B = \frac{\mu_0 I}{2\pi r}$

• Example: The field around a household electrical wire.

Force Exerted on a Current Element in a Magnetic Field (Right Hand Rule 2)

A current-carrying wire placed in a magnetic field experiences a force. The direction of this force is given by the right hand rule.

• Right Hand Rule 2: Point your fingers in the direction of current, and your palm in the direction of the magnetic field; your thumb points in the direction of the force.

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

• Example: A wire in a laboratory magnet experiences a sideways force.

Force on Electric Charge in Magnetic Field

A moving electric charge in a magnetic field experiences a force perpendicular to both its velocity and the field.

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

• Direction: Determined by the right hand rule for positive charges; opposite for negative charges.

• Example: Electrons in a cathode ray tube are deflected by magnetic fields.

Trajectory and Physics of Electron Moving Through Perpendicular Magnetic Field (Circular Orbit)

When an electron moves perpendicular to a uniform magnetic field, it follows a circular path due to the constant perpendicular force.

• Radius of Orbit: $r = \frac{mv}{qB}$

• Frequency (Cyclotron): $f = \frac{qB}{2\pi m}$

• Example: Electrons in a cyclotron or mass spectrometer.

Magnetic Field from Long Straight Wire and Magnetic Force Between Two Current Carrying Wires

Two parallel wires carrying currents exert forces on each other due to their magnetic fields.

• Magnetic Field: $B = \frac{\mu_0 I}{2\pi r}$

• Force per unit length: $\frac{F}{l} = \frac{\mu_0 I_1 I_2}{2\pi d}$

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

• Example: Busbar wires in electrical substations.

Magnetic Field and Function of Solenoid

A solenoid is a coil of wire that produces a nearly uniform magnetic field inside when current flows.

• Magnetic Field Inside: $B = \mu_0 n I$ where $n$ is the number of turns per unit length.

• Function: Used in electromagnets, relays, and MRI machines.

• Example: Solenoids in car starters.

Chapter 21: Electromagnetic Induction

Faraday's and Lenz's Law

Faraday's Law describes how a changing magnetic field induces an electromotive force (EMF). Lenz's Law states that the induced EMF creates a current whose magnetic field opposes the change.

• Faraday's Law: $\mathcal{E} = -\frac{d\Phi_B}{dt}$

• Lenz's Law: The negative sign indicates opposition to the change in magnetic flux.

• Example: Moving a magnet into a coil induces a current that opposes the motion.

Induced EMF of Moving Conductor

When a conductor moves through a magnetic field, an EMF is induced across its ends.

• Formula: $\mathcal{E} = B l v$ where $l$ is the length of the conductor and $v$ its velocity perpendicular to $B$.

• Example: A rod sliding on rails in a magnetic field.

Mutual and Self Inductance (M and L)

Inductance quantifies the ability of a circuit to induce EMF in itself (self-inductance) or another circuit (mutual inductance).

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

• Self Inductance: $\mathcal{E} = -L \frac{dI}{dt}$

• Example: Transformers use mutual inductance to transfer energy between coils.

Self Inductance of Solenoid

The self-inductance of a solenoid depends on its geometry and number of turns.

• Formula: $L = \mu_0 n^2 A l$ where $A$ is cross-sectional area and $l$ is length.

• Example: Inductors in electronic circuits.

Energy Stored in Magnetic Field

Magnetic fields store energy, especially in inductors.

• Formula: $U = \frac{1}{2} L I^2$

• Example: Energy stored in a solenoid when current flows.

LR Circuit

An LR circuit contains an inductor and resistor. The current changes over time when voltage is applied.

• Time Constant: $\tau = \frac{L}{R}$

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

• Example: Switching on an inductor in a DC circuit.

Chapter 22: Electromagnetic Waves

Displacement Current

The displacement current is a term added by Maxwell to account for changing electric fields in situations where no actual current flows, allowing for the continuity of Ampère's Law.

• Formula: $I_d = \varepsilon_0 \frac{d\Phi_E}{dt}$

• Example: Charging capacitor: changing electric field between plates acts like a current.

Production of EM Waves and Relationship Between Speed of Light, Electric Permittivity, and Magnetic Permeability

Electromagnetic waves are produced by oscillating electric and magnetic fields. The speed of light depends on the electric permittivity and magnetic permeability of free space.

• Formula: $c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$

• Example: Radio waves, light, and X-rays are all EM waves.

Relationship Between Speed of Light, Wavelength, and Frequency

The speed of light relates to the wavelength and frequency of electromagnetic waves.

• Formula: $c = \lambda f$

• Example: Visible light with wavelength $500$nm has frequency$f = \frac{c}{\lambda}$.

EM Spectrum

The electromagnetic spectrum encompasses all types of EM waves, classified by wavelength or frequency.

• Order (increasing frequency): Radio, Microwave, Infrared, Visible, Ultraviolet, X-ray, Gamma ray.

• Applications: Communication (radio), medical imaging (X-ray), sterilization (UV).

Energy Stored and Transported by EM Waves

EM waves carry energy and momentum. The energy density is shared between electric and magnetic fields.

• Energy Density: $u = \varepsilon_0 E^2 + \frac{1}{\mu_0} B^2$

• Intensity: $I = \frac{c \varepsilon_0 E^2}{2}$

• Example: Sunlight delivers energy to Earth's surface.

Radiation Pressure and Momentum from EM Waves

EM waves exert pressure and transfer momentum when absorbed or reflected.

• Radiation Pressure: $P = \frac{I}{c}$ for complete absorption; $P = \frac{2I}{c}$ for reflection.

• Momentum: $p = \frac{E}{c}$

• Example: Solar sails use radiation pressure for propulsion.

Additional info:

• Conceptual examples and figures referenced (e.g., Example 21-3, Figure 21-11) are standard textbook illustrations of Faraday's Law and induced EMF, typically showing a magnet moving in a coil and a rod moving in a magnetic field.

• Problems listed (e.g., 31, 85, 14, 46, 84, 50, 51, 58) are likely practice questions from a textbook, reinforcing the concepts above.