💡AP Physics C: E&M Unit 5 – Electromagnetism

Key Concepts and Foundations

• Electromagnetism fundamental force of nature responsible for interactions between charged particles and magnetic fields
• Electric charges come in two types: positive and negative
  • Like charges repel each other while opposite charges attract
  • Charge is conserved meaning it cannot be created or destroyed
• Electric fields exert forces on charged particles proportional to the charge and field strength
  • Represented by field lines pointing in the direction of the force on a positive test charge
• Magnetic fields produced by moving charges or magnetic dipoles (bar magnets)
  • Represented by field lines forming closed loops
• Electromagnetic waves self-propagating transverse waves with oscillating electric and magnetic fields
  • Include radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays
• Lorentz force law describes the force on a charged particle moving in an electromagnetic field: $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$

Electrostatics and Coulomb's Law

• Electrostatics deals with the forces and fields associated with stationary electric charges
• Coulomb's law states that the force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them: $F = k \frac{|q_1q_2|}{r^2}$
  • $k$ is Coulomb's constant equal to $8.99 \times 10^9 \frac{N \cdot m^2}{C^2}$
  • Force is attractive for opposite charges and repulsive for like charges
• Electric field strength at a point is defined as the force per unit charge: $\vec{E} = \frac{\vec{F}}{q}$
  • Units are newtons per coulomb (N/C) or volts per meter (V/m)
• Principle of superposition states that the total electric field at a point is the vector sum of the fields due to individual charges
• Electric dipole consists of two equal and opposite charges separated by a small distance
  • Experiences a torque in an external electric field trying to align it with the field direction

Electric Fields and Gauss's Law

• Electric flux is a measure of the number of electric field lines passing through a surface
  • Calculated as the integral of the electric field over the surface: $\Phi_E = \int \vec{E} \cdot d\vec{A}$
• Gauss's law relates the electric flux through a closed surface to the total charge enclosed: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$
  • $\epsilon_0$ is the permittivity of free space equal to $8.85 \times 10^{-12} \frac{C^2}{N \cdot m^2}$
• Gauss's law is useful for determining the electric field of highly symmetric charge distributions (spheres, cylinders, planes)
• Electric field inside a conductor is always zero in electrostatic equilibrium
  • Charges reside on the surface of a conductor
• Shielding effect occurs when a conducting shell surrounds a cavity
  • External fields have no effect on the interior of the cavity

Electric Potential and Capacitance

• Electric potential energy is the work done to move a charge in an electric field: $U = qV$
  • Measured in joules (J) or electron volts (eV)
• Electric potential is the potential energy per unit charge: $V = \frac{U}{q}$
  • Measured in volts (V) which is joules per coulomb (J/C)
• Potential difference between two points is the work per unit charge required to move a positive test charge between them: $\Delta V = -\int \vec{E} \cdot d\vec{l}$
• Equipotential surfaces are surfaces on which all points are at the same potential
  • Electric field lines are always perpendicular to equipotential surfaces
• Capacitance is a measure of a conductor's ability to store electric charge: $C = \frac{Q}{V}$
  • Measured in farads (F) which is coulombs per volt (C/V)
• Parallel plate capacitor consists of two conducting plates separated by an insulating material (dielectric)
  • Capacitance is proportional to the plate area and inversely proportional to their separation: $C = \frac{\epsilon_0 A}{d}$
• Energy stored in a capacitor is given by: $U = \frac{1}{2}CV^2 = \frac{1}{2}\frac{Q^2}{C}$

Current, Resistance, and Circuits

• Electric current is the rate of flow of electric charge: $I = \frac{dQ}{dt}$
  • Measured in amperes (A) which is coulombs per second (C/s)
• Current density is the current per unit cross-sectional area: $\vec{J} = \frac{I}{A}$
  • Related to the electric field and conductivity by Ohm's law: $\vec{J} = \sigma \vec{E}$
• Resistance is a measure of a material's opposition to current flow: $R = \frac{V}{I}$
  • Measured in ohms ($\Omega$) which is volts per ampere (V/A)
• Resistivity is an intrinsic property of a material that determines its resistance: $\rho = \frac{RA}{l}$
  • Measured in ohm-meters ($\Omega \cdot m$)
• Series circuits have components connected end-to-end
  • Equivalent resistance is the sum of individual resistances: $R_{eq} = R_1 + R_2 + ...$
• Parallel circuits have components connected side-by-side
  • Reciprocal of equivalent resistance is the sum of reciprocals of individual resistances: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$
• Kirchhoff's laws describe the behavior of current and potential in circuits
  • Junction rule: sum of currents entering a junction equals sum of currents leaving
  • Loop rule: sum of potential differences around any closed loop is zero

Magnetic Fields and Forces

• Magnetic fields produced by moving charges or magnetic dipoles
  • Represented by field lines forming closed loops
• Magnetic field at a point due to a current-carrying wire given by the Biot-Savart law: $d\vec{B} = \frac{\mu_0}{4\pi}\frac{I d\vec{l} \times \hat{r}}{r^2}$
  • $\mu_0$ is the permeability of free space equal to $4\pi \times 10^{-7} \frac{T \cdot m}{A}$
• Magnetic force on a moving charge is perpendicular to both the field and velocity: $\vec{F} = q\vec{v} \times \vec{B}$
  • Direction given by the right-hand rule
• Magnetic force on a current-carrying wire is given by: $\vec{F} = I\vec{l} \times \vec{B}$
  • Used in electric motors to convert electrical energy to mechanical energy
• Magnetic dipole moment is a measure of the strength and orientation of a magnetic dipole: $\vec{\mu} = I\vec{A}$
  • Experiences a torque in an external magnetic field trying to align it with the field direction: $\vec{\tau} = \vec{\mu} \times \vec{B}$
• Ampère's law relates the magnetic field around a closed loop to the current passing through it: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$
  • Useful for determining the magnetic field of highly symmetric current distributions (long straight wires, solenoids, toroids)

Electromagnetic Induction

• Faraday's law states that a changing magnetic flux induces an electromotive force (emf) in a conductor: $\mathcal{E} = -\frac{d\Phi_B}{dt}$
  • Negative sign indicates that the induced emf opposes the change in flux (Lenz's law)
• Magnetic flux is the integral of the magnetic field over a surface: $\Phi_B = \int \vec{B} \cdot d\vec{A}$
  • Units are webers (Wb) which is tesla-square meters ($T \cdot m^2$)
• Induced emf can be increased by increasing the number of turns in a coil (transformers)
  • Ratio of secondary to primary voltage is equal to the ratio of turns: $\frac{V_s}{V_p} = \frac{N_s}{N_p}$
• Motional emf is induced in a conductor moving through a magnetic field: $\mathcal{E} = Blv$
  • Used in generators to convert mechanical energy to electrical energy
• Eddy currents are induced currents in bulk conductors that create a opposing magnetic field
  • Can be used for braking or heating
• Inductance is a measure of a conductor's opposition to changing current: $L = \frac{\Phi_B}{I}$
  • Units are henries (H) which is webers per ampere (Wb/A)
• Energy stored in an inductor is given by: $U = \frac{1}{2}LI^2$

Maxwell's Equations and EM Waves

• Maxwell's equations are a set of four fundamental equations that describe all classical electromagnetic phenomena
  • Gauss's law for electric fields: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$
  • Gauss's law for magnetic fields: $\oint \vec{B} \cdot d\vec{A} = 0$ (no magnetic monopoles)
  • Faraday's law: $\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$
  • Ampère-Maxwell law: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$
• Electromagnetic waves are self-propagating oscillations of electric and magnetic fields
  • Predicted by Maxwell's equations
  • Travel at the speed of light in vacuum: $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8 m/s$
• EM waves are transverse waves with the electric and magnetic fields perpendicular to each other and the direction of propagation
  • Can be polarized meaning the electric field oscillates in a particular plane
• Energy carried by EM waves is given by the Poynting vector: $\vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B}$
  • Intensity is the average power per unit area: $I = \frac{1}{2} c \epsilon_0 E_0^2$
• Spectrum of EM waves includes radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays
  • Different regions have different wavelengths, frequencies, and photon energies related by: $c = \lambda f$ and $E = hf$

Applications and Problem-Solving

• Electrostatics problems often involve calculating electric fields, forces, and potentials for various charge distributions
  • Use Coulomb's law, electric field superposition, Gauss's law, and electric potential energy
• Capacitance problems involve determining the capacitance, stored charge, and energy of different capacitor configurations
  • Use the definition of capacitance, parallel plate capacitor equation, and series/parallel combinations
• Current and resistance problems analyze the behavior of current, voltage, and power in DC circuits
  • Use Ohm's law, resistor combinations, Kirchhoff's laws, and power dissipation ($P = IV$)
• Magnetism problems calculate magnetic fields, forces, and torques in various scenarios
  • Use the Biot-Savart law, magnetic force equations, magnetic dipole moments, and Ampère's law
• Induction problems examine the induced emf, current, and power in conductors experiencing a changing magnetic flux
  • Use Faraday's law, Lenz's law, motional emf, and transformer equations
• Electromagnetic wave problems involve determining the characteristics and energy transport of EM waves
  • Use the wave equation, Poynting vector, intensity, and photon energy
• In all problems, it's crucial to:
  • Identify the relevant concepts and equations
  • Draw clear diagrams and define variables
  • Check units and use dimensional analysis
  • Solve symbolically before plugging in numbers
  • Verify the reasonableness of the answer