🧲AP Physics 2 (2025) Unit 12 – Magnetism and Electromagnetism

Key Concepts and Definitions

• Magnetism is a force of attraction or repulsion that acts at a distance between particles with magnetic properties (magnetic dipoles)
• Magnetic fields are regions around magnets or current-carrying wires where magnetic forces can be detected and measured
  • Represented by magnetic field lines that show the direction and strength of the field at each point
• Magnetic flux ($\Phi_B$) is the total magnetic field passing through a given area, measured in webers (Wb)
  • Calculated using the equation $\Phi_B = \vec{B} \cdot \vec{A}$, where $\vec{B}$ is the magnetic field and $\vec{A}$ is the area vector
• Electromagnetic induction is the production of an electromotive force (emf) and current in a conductor by a changing magnetic field
• Faraday's law states that the induced emf in a closed loop is equal to the negative rate of change of the magnetic flux through the loop
  • Mathematically expressed as $\varepsilon = -\frac{d\Phi_B}{dt}$, where $\varepsilon$ is the induced emf and $\frac{d\Phi_B}{dt}$ is the rate of change of magnetic flux
• Lenz's law determines the direction of the induced current, stating that it flows to oppose the change in magnetic flux that produced it
• Permeability ($\mu$) is a measure of a material's ability to support the formation of a magnetic field within itself, affecting the strength of the magnetic field in the material

Magnetic Fields and Forces

• Moving charges (electric currents) create magnetic fields around them, with the field strength proportional to the current
• The direction of the magnetic field around a current-carrying wire can be determined using the right-hand rule
  • Point your thumb in the direction of the current, and your fingers will curl in the direction of the magnetic field
• Magnetic fields exert forces on moving charges and current-carrying wires, with the force perpendicular to both the magnetic field and the velocity of the charge or direction of the current
• The magnetic force on a moving charge is given by $\vec{F} = q\vec{v} \times \vec{B}$, where $q$ is the charge, $\vec{v}$ is the velocity, and $\vec{B}$ is the magnetic field
• The magnetic force on a current-carrying wire is $\vec{F} = I\vec{L} \times \vec{B}$, where $I$ is the current and $\vec{L}$ is the length of the wire
• Magnetic fields can be created by permanent magnets, which have north and south poles that attract or repel each other
  • Opposite poles (north and south) attract, while like poles (north-north or south-south) repel
• Earth's magnetic field acts like a giant bar magnet, with field lines extending from the magnetic south pole to the magnetic north pole
  • The magnetic poles do not coincide with the geographic poles, and the magnetic field is tilted about 11 degrees from Earth's rotational axis

Electromagnetic Induction

• A changing magnetic flux through a loop induces an emf and current in the loop, known as electromagnetic induction
• The induced emf depends on the rate of change of the magnetic flux, as described by Faraday's law ($\varepsilon = -\frac{d\Phi_B}{dt}$)
  • A faster change in magnetic flux results in a larger induced emf
• The direction of the induced current is determined by Lenz's law, which states that the induced current flows in a direction to oppose the change in magnetic flux that produced it
• Transformers use electromagnetic induction to change the voltage and current of AC power
  • Consist of two coils (primary and secondary) wound around a common iron core
  • A changing current in the primary coil induces an emf in the secondary coil, with the voltage ratio determined by the ratio of the number of turns in each coil
• Generators convert mechanical energy into electrical energy using electromagnetic induction
  • A coil of wire rotates in a magnetic field, inducing an emf and current in the coil
  • The induced emf alternates as the coil rotates, producing AC power
• Eddy currents are induced currents in bulk conductors caused by changing magnetic fields, often leading to energy losses due to heating

Magnetic Materials and Properties

• Magnetic materials can be classified as diamagnetic, paramagnetic, or ferromagnetic based on their response to external magnetic fields
  • Diamagnetic materials (copper, silver) weakly repel magnetic fields and have a negative magnetic susceptibility
  • Paramagnetic materials (aluminum, platinum) are weakly attracted to magnetic fields and have a positive magnetic susceptibility
  • Ferromagnetic materials (iron, nickel, cobalt) are strongly attracted to magnetic fields and can retain their magnetic properties even after the external field is removed
• Magnetic domains are regions within a ferromagnetic material where the magnetic dipoles are aligned in the same direction
  • In an unmagnetized material, the domains are randomly oriented, resulting in no net magnetic field
  • When exposed to an external magnetic field, the domains align, causing the material to become magnetized
• Magnetic hysteresis is the dependence of a ferromagnetic material's magnetization on its previous magnetic history
  • Represented by a hysteresis loop, which shows the relationship between the applied magnetic field and the resulting magnetization
• Curie temperature is the temperature above which a ferromagnetic material loses its ferromagnetic properties and becomes paramagnetic
  • Occurs because thermal energy disrupts the alignment of the magnetic dipoles
• Magnetic shielding involves using materials with high magnetic permeability (mu-metal) to redirect magnetic fields away from sensitive devices or areas

Applications of Magnetism

• Magnetic compasses use Earth's magnetic field to determine direction, with the needle aligning itself with the field lines
• Electric motors convert electrical energy into mechanical energy using the interaction between magnetic fields and current-carrying wires
  • Consist of a coil of wire (armature) that rotates between the poles of a permanent magnet or electromagnet (stator)
  • The magnetic force on the current-carrying armature causes it to rotate, producing mechanical motion
• Loudspeakers use the interaction between a current-carrying coil and a permanent magnet to convert electrical signals into sound waves
  • The coil is attached to a diaphragm, which vibrates in response to the changing current, producing sound
• Magnetic levitation (maglev) trains use strong magnetic fields to lift and propel the train above a guideway, reducing friction and allowing high-speed travel
• Magnetic Resonance Imaging (MRI) uses strong magnetic fields and radio waves to create detailed images of the body's internal structures
  • Hydrogen atoms in the body align with the magnetic field and absorb and emit radio waves, providing information about the tissue density and composition
• Magnetohydrodynamics (MHD) studies the behavior of electrically conducting fluids (plasmas) in the presence of magnetic fields
  • Applications include plasma confinement in fusion reactors and propulsion systems for spacecraft

Electromagnetic Waves

• Electromagnetic waves are self-propagating waves composed of oscillating electric and magnetic fields that travel through space at the speed of light
• The electric and magnetic fields in an electromagnetic wave are perpendicular to each other and to the direction of wave propagation
• Electromagnetic waves can be characterized by their wavelength, frequency, and energy
  • Wavelength ($\lambda$) is the distance between two consecutive crests or troughs of the wave
  • Frequency ($f$) is the number of wave cycles that pass a fixed point per unit time, measured in hertz (Hz)
  • The relationship between wavelength, frequency, and the speed of light ($c$) is given by $c = \lambda f$
• The electromagnetic spectrum is the range of all possible frequencies and wavelengths of electromagnetic waves, including radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays
  • Different regions of the spectrum have different properties and applications, such as communication, heating, and medical imaging
• Electromagnetic waves carry energy and momentum, which can be absorbed, reflected, or transmitted by matter
  • The energy carried by an electromagnetic wave is proportional to its frequency, as described by the equation $E = hf$, where $h$ is Planck's constant

Mathematical Models and Equations

• The Biot-Savart law describes the magnetic field ($\vec{B}$) generated by a current-carrying wire, given by $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$
  • $\mu_0$ is the permeability of free space, $I$ is the current, $d\vec{l}$ is a small segment of the wire, and $\hat{r}$ is the unit vector pointing from the wire segment to the point where the field is being calculated
• Ampère's circuital law relates the magnetic field around a closed loop to the electric current passing through the loop, expressed as $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$
  • $I_{enc}$ is the total current enclosed by the loop
• Faraday's law of induction states that the induced emf ($\varepsilon$) in a closed loop is equal to the negative rate of change of the magnetic flux ($\Phi_B$) through the loop, given by $\varepsilon = -\frac{d\Phi_B}{dt}$
• The magnetic flux ($\Phi_B$) through a surface is the product of the magnetic field ($\vec{B}$) and the area ($\vec{A}$) of the surface, given by $\Phi_B = \vec{B} \cdot \vec{A}$
• Lenz's law determines the direction of the induced current, stating that it flows to oppose the change in magnetic flux that produced it
• The magnetic force on a moving charge ($\vec{F}$) is given by $\vec{F} = q\vec{v} \times \vec{B}$, where $q$ is the charge and $\vec{v}$ is the velocity
• The magnetic force on a current-carrying wire is $\vec{F} = I\vec{L} \times \vec{B}$, where $I$ is the current and $\vec{L}$ is the length of the wire

Lab Experiments and Demonstrations

• Mapping magnetic fields using iron filings or small compasses to visualize the field lines around bar magnets and current-carrying wires
• Demonstrating the magnetic force on a current-carrying wire by observing the deflection of a wire suspended between the poles of a magnet when a current is passed through it
• Investigating electromagnetic induction by moving a magnet through a coil of wire connected to a galvanometer and observing the induced current
• Building a simple electric motor using a coil of wire, a battery, and a permanent magnet to demonstrate the conversion of electrical energy into mechanical energy
• Constructing a transformer using two coils of wire wound around a common iron core and measuring the input and output voltages to explore the principle of electromagnetic induction
• Observing the effect of magnetic shielding by placing a magnetic compass inside a mu-metal container and noting the reduction in the compass's response to external magnetic fields
• Demonstrating Lenz's law by dropping a strong magnet through a copper tube and observing the reduced acceleration due to the induced currents in the tube
• Exploring the properties of electromagnetic waves by setting up a simple radio transmitter and receiver using a coil of wire, a capacitor, and a diode to send and detect radio waves