Electromagnetism and AC Circuits

Magnetic Forces and Fields

Magnetic forces and fields are fundamental concepts in electromagnetism, describing the interactions between moving charges, currents, and magnetic materials.

• Magnetic Force (Moving Charge): The force on a charge moving in a magnetic field is given by:

• Particle Motion in a Magnetic Field: The radius of the circular path is:

• Magnetic Force (Conductor): For a current-carrying wire:

• Torque on a Current-Carrying Loop:

• Magnetic Field of a Long, Straight, Current-Carrying Conductor:

• Magnetic Field of a Solenoid:

• Magnetic Field at Center of Current-Carrying Circular Loop:

• Magnetic Force Between Current-Carrying Conductors:

• Magnetic Flux:

Electromagnetic Induction

Electromagnetic induction describes how a changing magnetic field induces an electromotive force (EMF) in a conductor.

• Faraday's Law: The induced EMF is proportional to the rate of change of magnetic flux:

• Lenz's Law: The direction of induced EMF opposes the change in flux.

• Motional EMF:

• Alternating Current (AC) Circuits: Voltage and current vary sinusoidally:

AC Circuits: Frequency, Period, and Transformers

AC circuits involve time-varying voltages and currents, characterized by frequency and period. Transformers are used to change voltage levels in AC circuits.

• Frequency:

• Period:

• Transformers: The ratio of voltages is proportional to the ratio of turns:

Inductance and Magnetic Field Energy

Inductors store energy in magnetic fields and oppose changes in current.

• Inductance:

• Magnetic Field Energy:

Electric Field Energy and LC Circuits

LC circuits consist of inductors and capacitors, exhibiting oscillatory behavior.

• Electric Field Energy:

• LC Circuit Frequency:

• LC Circuit Period:

Impedance and LRC Circuits (AC)

Impedance is the total opposition to current in an AC circuit, combining resistance, inductive reactance, and capacitive reactance.

• Impedance:

• Phase Angle:

Electromagnetic Waves

Electromagnetic waves are oscillations of electric and magnetic fields that propagate through space.

• Wave Equation:

• Speed of Light:

• Energy Density:

• Intensity:

• Photon Energy:

Thin Film Interference

Thin film interference occurs when light waves reflect off the surfaces of a thin film, producing constructive or destructive interference.

• Constructive Interference (2D): ,

• Destructive Interference (2D): ,

Constants and Geometry

Physical constants and geometric formulas are essential for calculations in electromagnetism.

• Speed of Light: m/s

• Elementary Charge: C

• Vacuum Permittivity: C/N·m$^2$

• Vacuum Permeability: T·m/A

• Area of a Circle:

Practice Questions: Applications of Electromagnetism

Magnetism and Magnetic Forces

• Splitting Magnets: When a permanent magnet is cut, each piece retains both north and south poles due to the alignment of magnetic domains.

• Electron Beam in Electric Field: The force on an electron in an electric field is , directed opposite to the field for electrons.

• Current-Carrying Wire in Magnetic Field: The force on a wire in a magnetic field is , with direction given by the right-hand rule.

Induction and Faraday's Law

• Induced EMF in Moving Conductors: A conductor moving in a magnetic field experiences an induced EMF, with direction determined by Lenz's Law.

• Changing Magnetic Flux: The direction of induced current depends on whether the magnetic flux is increasing or decreasing.

AC Circuits and Transformers

• AC Source Period: The period of an AC source is .

• Inductance of Solenoid:

• LC Circuit Frequency:

• Transformer Voltage Ratio:

Electromagnetic Waves and Optics

• Maxwell's Equations: The second equation, , states that the net magnetic flux through any closed surface is zero, implying no magnetic monopoles exist.

• Photon Energy and Wavelength: Photons with shorter wavelengths have higher energy ().

• Refraction and Total Internal Reflection: The critical angle for total internal reflection is .

Sample Table: Electromagnetic Wave Properties

The following table summarizes the relationships between electric field, magnetic field, and energy density in electromagnetic waves.

Additional info:

• Some questions reference thin film interference and optics, which are typically covered in the latter part of an introductory physics course.

• Constants and geometric formulas are provided for quick reference during calculations.