ap physics c: e&m unit 12 study guides

Key Concepts and Definitions

• Magnetic field $\vec{B}$ represents the region around a magnet or current-carrying wire where magnetic forces can be detected
• Magnetic flux $\Phi_B$ measures the amount of magnetic field passing through a surface (units: webers, Wb)
• Magnetic permeability $\mu$ describes a material's ability to support the formation of a magnetic field within itself
  • Vacuum permeability: $\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$
  • Relative permeability: $\mu_r = \mu / \mu_0$
• Magnetic dipole moment $\vec{\mu}$ quantifies the torque experienced by a current loop or bar magnet when placed in an external magnetic field
• Faraday's law states that a changing magnetic flux through a loop induces an electromotive force (emf) in the loop
• Lenz's law predicts the direction of induced current in a loop based on the change in magnetic flux

Magnetic Field Basics

• Magnetic fields are represented by field lines, which form closed loops and never cross each other
• The direction of a magnetic field line at any point is tangent to the field line and points from the north pole to the south pole
• Magnetic field strength decreases with distance from the source (magnet or current-carrying wire)
• Magnetic fields can be uniform (constant strength and direction) or non-uniform (varying strength and/or direction)
• The Earth's magnetic field is approximately a dipole field, with north and south magnetic poles near the geographic poles
• Magnetic fields are measured in teslas (T) or gauss (G), where 1 T = 10,000 G

Sources of Magnetic Fields

• Permanent magnets produce magnetic fields due to the alignment of magnetic domains within the material (ferromagnetic materials like iron, nickel, and cobalt)
• Current-carrying wires generate magnetic fields in accordance with the Biot-Savart law
  • The direction of the magnetic field is determined by the right-hand rule
• Solenoids (tightly wound coils of wire) produce strong, uniform magnetic fields inside the coil when current flows through the wire
  • The magnetic field strength inside a solenoid is given by $B = \mu_0 n I$, where $n$ is the number of turns per unit length and $I$ is the current
• Electromagnets are temporary magnets created by passing current through a coil of wire, often wrapped around a ferromagnetic core (soft iron)
• Superconductors can generate strong magnetic fields due to their ability to carry high currents with zero resistance

Magnetic Forces on Moving Charges

• A charged particle moving in a magnetic field experiences a force perpendicular to both the particle's velocity and the magnetic field (Lorentz force)
  • The magnitude of the force is given by $F = qvB\sin\theta$, where $q$ is the charge, $v$ is the velocity, $B$ is the magnetic field strength, and $\theta$ is the angle between $\vec{v}$ and $\vec{B}$
• The direction of the magnetic force on a moving charge is determined by the right-hand rule
• Charged particles moving parallel to a magnetic field experience no force
• Charged particles moving perpendicular to a magnetic field experience maximum force and follow a circular path
  • The radius of the circular path is given by $r = mv/qB$, where $m$ is the mass of the particle
• Magnetic fields can be used to filter or separate charged particles based on their charge-to-mass ratio (mass spectrometry)

Magnetic Forces on Current-Carrying Wires

• A current-carrying wire in a magnetic field experiences a force due to the interaction between the magnetic field and the moving charges (electrons) in the wire
• The force on a current-carrying wire is given by $\vec{F} = I\vec{L} \times \vec{B}$, where $I$ is the current, $\vec{L}$ is the length vector of the wire, and $\vec{B}$ is the magnetic field
• The direction of the force is determined by the right-hand rule (curl the fingers of the right hand from $\vec{L}$ to $\vec{B}$, and the thumb points in the direction of $\vec{F}$)
• Parallel current-carrying wires experience attractive forces if the currents flow in the same direction and repulsive forces if the currents flow in opposite directions
• The force between two parallel current-carrying wires is given by $F = \mu_0 I_1 I_2 L / 2\pi d$, where $I_1$ and $I_2$ are the currents, $L$ is the length of the wires, and $d$ is the distance between them
• Magnetic forces on current-carrying wires are the basis for electric motors and loudspeakers

Electromagnetic Induction

• Faraday's law of induction states that a changing magnetic flux through a loop induces an electromotive force (emf) in the loop
  • The induced emf is given by $\mathcal{E} = -d\Phi_B/dt$, where $\Phi_B$ is the magnetic flux
• Lenz's law predicts the direction of the induced current, which opposes the change in magnetic flux that produced it
• Motional emf is induced when a conductor moves through a magnetic field
  • The magnitude of the motional emf is given by $\mathcal{E} = BLv$, where $B$ is the magnetic field strength, $L$ is the length of the conductor, and $v$ is the velocity of the conductor perpendicular to the field
• Transformers use electromagnetic induction to change the voltage and current levels in AC circuits
  • 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 that create magnetic fields opposing the change in the original magnetic field (used in braking systems)

Applications and Real-World Examples

• Magnetic Resonance Imaging (MRI) uses strong magnetic fields and radio waves to create detailed images of the body's internal structures
• Particle accelerators (cyclotrons, synchrotrons) use magnetic fields to guide and accelerate charged particles for research in physics and medicine
• Maglev trains use powerful electromagnets to levitate and propel the train, reducing friction and allowing for high-speed travel
• Electric generators convert mechanical energy into electrical energy through electromagnetic induction (wind turbines, hydroelectric power plants)
• Electromagnetic braking systems in vehicles use eddy currents to slow down the vehicle without physical contact between the brake pads and rotors
• Hall effect sensors measure magnetic fields and are used in various applications (brushless DC motors, proximity sensors, current sensors)

Problem-Solving Strategies

• Identify the type of problem (magnetic field, force, induction) and the relevant given information
• Draw a clear diagram showing the magnetic field, current, or moving charge, and label all known quantities
• Determine the appropriate equation or principle to apply based on the problem type and given information
• Use the right-hand rules to determine the directions of magnetic fields, forces, and induced currents
• Solve the equation for the desired quantity, paying attention to units and significant figures
• Check the reasonableness of the answer by considering the magnitude and direction of the result, and compare it to known values or physical intuition
• Practice solving a variety of problems to develop a strong understanding of the concepts and problem-solving techniques
• When faced with a complex problem, break it down into smaller, manageable steps and solve each step systematically