🧲AP Physics 2 Unit 5 – Magnetism and Electromagnetic Induction

Fundamental Concepts of Magnetism

• Magnetism arises from the motion of electric charges and the intrinsic magnetic moments of subatomic particles
• Magnetic fields are represented by magnetic field lines, which form continuous loops and never cross each other
• Magnetic field lines start at the north pole of a magnet and end at the south pole
• Magnetic poles always come in pairs (north and south) and cannot be separated
• Magnetic fields can be created by permanent magnets, which are made of ferromagnetic materials (iron, nickel, cobalt) that have been magnetized
  • Ferromagnetic materials contain magnetic domains, which are regions where the magnetic fields of atoms are aligned
• Magnetic fields can also be created by electric currents flowing through a conductor
  • The magnetic field produced by a current-carrying wire can be determined using the right-hand rule
• Magnetic fields are measured in tesla (T) or gauss (G), where 1 T = 10,000 G

Magnetic Fields and Forces

• Moving electric charges (currents) experience a force when placed in a magnetic field, known as the magnetic force
• The magnetic force on a moving charge is perpendicular to both the magnetic field and the velocity of the charge, and its magnitude is given by $F = qvB\sin\theta$
  • $q$ is the charge of the particle
  • $v$ is the velocity of the particle
  • $B$ is the magnetic field strength
  • $\theta$ is the angle between the velocity and the magnetic field
• The direction of the magnetic force on a positive charge can be determined using the right-hand rule
• Current-carrying wires also experience a magnetic force when placed in a magnetic field, with the force given by $F = ILB\sin\theta$
  • $I$ is the current in the wire
  • $L$ is the length of the wire in the magnetic field
• Magnetic fields can exert torques on current loops and magnetic dipoles, causing them to rotate to align with the field
• The magnetic dipole moment ($\mu$) is a measure of the strength and orientation of a magnetic dipole
• The potential energy of a magnetic dipole in a magnetic field is given by $U = -\vec{\mu} \cdot \vec{B}$

Electromagnetic Induction Basics

• Electromagnetic induction is the production of an electromotive force (emf) in a conductor due to a changing magnetic flux
• Magnetic flux ($\Phi_B$) is the amount of magnetic field passing through a surface, given by $\Phi_B = \vec{B} \cdot \vec{A} = BA\cos\theta$
  • $B$ is the magnetic field strength
  • $A$ is the area of the surface
  • $\theta$ is the angle between the magnetic field and the normal to the surface
• A change in magnetic flux through a loop of wire induces an emf and a current in the loop
• The induced emf is proportional to the rate of change of the magnetic flux, as described by Faraday's law
• The direction of the induced current is such that it opposes the change in magnetic flux, as described by Lenz's law
• Electromagnetic induction is the basis for the operation of generators, transformers, and other electrical devices

Faraday's Law and Lenz's Law

• Faraday's law states that the magnitude of the induced emf in a circuit is equal to the rate of change of the magnetic flux through the circuit
  • Mathematically, $\mathcal{E} = -\frac{d\Phi_B}{dt}$, where $\mathcal{E}$ is the induced emf and $\frac{d\Phi_B}{dt}$ is the rate of change of magnetic flux
• The negative sign in Faraday's law indicates that the induced emf opposes the change in magnetic flux, a consequence of Lenz's law
• Lenz's law states that the direction of the induced current is such that it creates a magnetic field that opposes the change in the original magnetic flux
  • This is a consequence of the conservation of energy and the principle of least action
• The induced emf can be increased by increasing the rate of change of the magnetic flux, which can be achieved by
  • Increasing the strength of the magnetic field
  • Increasing the area of the loop
  • Increasing the number of turns in the coil
  • Increasing the speed of the relative motion between the loop and the magnetic field
• Faraday's law and Lenz's law are fundamental to the operation of generators, transformers, and other devices that rely on electromagnetic induction

Applications of Electromagnetic Induction

• Generators convert mechanical energy into electrical energy by using electromagnetic induction
  • A coil of wire is rotated in a magnetic field, inducing an emf and a current in the coil
  • The induced emf alternates as the coil rotates, producing alternating current (AC)
• Electric motors convert electrical energy into mechanical energy by using the magnetic force on current-carrying wires
  • A current-carrying coil is placed in a magnetic field, causing it to rotate due to the magnetic force
• Transformers use electromagnetic induction to change the voltage and current levels of AC
  • Two coils (primary and secondary) are wound around a common iron core
  • An AC current in the primary coil creates a changing magnetic flux in the core, inducing an emf in the secondary coil
  • The voltage ratio between the primary and secondary coils is equal to the ratio of the number of turns in each coil
• Eddy currents are induced currents in bulk conductors caused by changing magnetic fields
  • Eddy currents can be used for braking, heating, and non-destructive testing
  • Eddy currents are minimized in transformer cores and other devices by using laminated cores or ferrite materials
• Electromagnetic induction is also used in induction cooktops, contactless charging, and metal detectors

AC Circuits and Transformers

• Alternating current (AC) is an electric current that periodically reverses direction, in contrast to direct current (DC) which flows in only one direction
• AC is generated by most power plants and is used in power transmission and distribution systems
• AC circuits contain resistors, capacitors, and inductors, which affect the voltage, current, and power in the circuit
  • Resistors oppose the flow of current and dissipate energy as heat
  • Capacitors store energy in electric fields and oppose changes in voltage
  • Inductors store energy in magnetic fields and oppose changes in current
• The voltage and current in an AC circuit are characterized by their amplitude, frequency, and phase
  • The amplitude is the maximum value of the voltage or current
  • The frequency is the number of cycles per second, measured in hertz (Hz)
  • The phase is the relative timing between the voltage and current waveforms
• Transformers are used to step up or step down the voltage in AC circuits
  • The primary coil is connected to the input voltage, and the secondary coil provides the output voltage
  • The voltage ratio is equal to the ratio of the number of turns in the primary and secondary coils
  • Transformers work only with AC because they rely on the constantly changing magnetic flux to induce an emf
• Power transmission systems use high voltages to minimize power losses in the transmission lines
  • Transformers are used to step up the voltage at the power plant and step it down for distribution to homes and businesses

Electromagnetic Waves

• Electromagnetic waves are oscillating electric and magnetic fields that propagate through space at the speed of light
• Electromagnetic waves are produced by accelerating electric charges and can travel through vacuum or matter
• The electric and magnetic fields in an electromagnetic wave are perpendicular to each other and to the direction of propagation
• Electromagnetic waves are characterized by their wavelength, frequency, and amplitude
  • The wavelength ($\lambda$) is the distance between two consecutive crests or troughs of the wave
  • The frequency ($f$) is the number of cycles per second, measured in hertz (Hz)
  • The amplitude is the maximum value of the electric or magnetic field
• The speed of an electromagnetic wave in vacuum is equal to the speed of light, $c = 3 \times 10^8 m/s$
• The relationship between wavelength, frequency, and speed is given by $c = \lambda f$
• The electromagnetic spectrum includes radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays, in order of increasing frequency and decreasing wavelength
• 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 Planck-Einstein relation $E = hf$, where $h$ is Planck's constant

Key Equations and Problem-Solving Strategies

• Magnetic force on a moving charge: $\vec{F} = q\vec{v} \times \vec{B}$
• Magnetic force on a current-carrying wire: $\vec{F} = I\vec{L} \times \vec{B}$
• Magnetic dipole moment: $\vec{\mu} = NIA\hat{n}$
• Torque on a magnetic dipole: $\vec{\tau} = \vec{\mu} \times \vec{B}$
• Magnetic flux: $\Phi_B = \int \vec{B} \cdot d\vec{A}$
• Faraday's law: $\mathcal{E} = -\frac{d\Phi_B}{dt}$
• Transformer voltage ratio: $\frac{V_p}{V_s} = \frac{N_p}{N_s}$
• Electromagnetic wave speed: $c = \lambda f$
• Energy of a photon: $E = hf$
• When solving problems, start by identifying the given information and the quantity to be calculated
• Draw a diagram of the situation, including all relevant vectors and angles
• Determine which equations are applicable based on the given information and the desired quantity
• Substitute the given values into the equations and solve for the unknown variable
• Check the units and the reasonableness of the answer, and consider any special cases or limiting conditions