https://teddyrookbook.com/physics/electricity-and-magnetism/Recent content in Electricity and Magnetism on Teddy Rook Bookhttps://teddyrookbook.com/img/share.jpghttps://teddyrookbook.com/img/share.jpgHugoenhttps://teddyrookbook.com/physics/electricity-and-magnetism/charge-densities/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/charge-densities/λ, σ and ρ describe charge spread along a line, over a surface or through a volume, so dq = λ dx. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/field-of-rod/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/field-of-rod/Add up dE = k dq/r² as components to find the field of a charged rod. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/field-of-ring-disk/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/field-of-ring-disk/On the axis of a ring E = kQz/(z² + R²)^(3/2), and a disk is built from rings. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/electric-flux/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/electric-flux/Φ_E = ∫E · dA counts the field passing through a surface, using the surface's normal. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/gauss-law/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/gauss-law/The total flux through any closed surface equals q_enclosed/ε₀. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/gauss-spheres/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/gauss-spheres/Find the fields of point charges, charged shells and uniformly charged balls, inside and out. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/gauss-lines-planes/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/gauss-lines-planes/A long line gives E = λ/(2πε₀r) and a large sheet gives E = σ/(2ε₀), the same at every distance. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/conductors/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/conductors/Inside a conductor E = 0, excess charge sits on the surface, and just outside E = σ/ε₀. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/potential-from-field/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/potential-from-field/ΔV = −∫E · dl, so potential falls along the direction of the field. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/potential-of-distributions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/potential-of-distributions/V = k∫dq/r adds up as a scalar for rods, rings and disks. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/field-from-potential/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/field-from-potential/E = −dV/dr, and in three dimensions E = −(∂V/∂x, ∂V/∂y, ∂V/∂z). Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/energy-of-charges/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/energy-of-charges/Add kqᵢqⱼ/rᵢⱼ over every pair to find the energy needed to assemble the charges. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/energy-conservation-charges/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/energy-conservation-charges/A charge released near fixed charges trades potential energy for kinetic energy: qΔV + ΔK = 0. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/charge-sharing/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/charge-sharing/Connected conductors come to the same potential, so two spheres share charge in proportion to their radii. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/capacitance-geometry/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/capacitance-geometry/Find C with Gauss's law and ΔV for parallel-plate, cylindrical and spherical capacitors. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/field-energy/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/field-energy/The energy of a charged capacitor lives in its field, with density u = ½ε₀E². Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/dielectrics/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/dielectrics/A dielectric weakens the field by κ and raises C to κC; with the battery connected Q grows, but disconnected V drops. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/drift-velocity/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/drift-velocity/I = nqv_dA and J = I/A; electrons drift along at a fraction of a millimeter per second, even though a signal travels nearly at light speed. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/resistance-integral/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/resistance-integral/Add dR = ρ dx/A(x) along a conductor whose cross section changes. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/multiloop-circuits/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/multiloop-circuits/Write junction and loop equations and solve them as a system. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/rc-charging/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/rc-charging/The loop rule gives a differential equation whose solution is q = Cε(1 − e^(−t/RC)). Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/rc-discharging/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/rc-discharging/A capacitor discharges as q = Q₀e^(−t/RC), and all of its stored energy ends up as heat in the resistor. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/lorentz-force/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/lorentz-force/F = q(E + v × B), computed with cross products. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/crossed-fields/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/crossed-fields/Only charges with v = E/B pass straight through crossed fields, and the radius of the path then sorts them by mass-to-charge ratio. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/hall-effect/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/hall-effect/A current crossing a magnetic field pushes its carriers sideways, building a voltage that reveals their sign. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/force-on-wire-integral/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/force-on-wire-integral/dF = I dl × B, integrated along bent wires. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/loop-torque/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/loop-torque/A loop has magnetic moment μ = NIA, and τ = μ × B turns motors and meters. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/biot-savart/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/biot-savart/Each piece of current makes dB = (μ₀/4π) I dl × r̂/r². Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/loop-field/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/loop-field/At the center of a loop B = μ₀I/(2R), and on the axis the field falls off with distance. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/amperes-law/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/amperes-law/∮B · dl = μ₀I_enclosed around any closed path. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/ampere-applications/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/ampere-applications/Ampère's law gives B = μ₀Ir/(2πR²) inside a wire, μ₀nI in a solenoid and μ₀NI/(2πr) in a toroid. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/changing-flux/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/changing-flux/Φ = ∫B · dA can change because B, the area or the angle changes with time. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/faraday-calculus/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/faraday-calculus/ε = −dΦ/dt for loops that rotate, grow or sit in changing fields. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/generators/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/generators/A coil of N turns spinning at ω in a field B makes ε = NBAω sin ωt. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/induced-electric-field/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/induced-electric-field/A changing magnetic field makes a curling electric field: ∮E · dl = −dΦ/dt, with or without a wire. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/magnetic-braking/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/magnetic-braking/The force on an induced current opposes the motion, so a rod on rails coasts to a stop or falls at a terminal speed. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/inductance/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/inductance/ε = −L dI/dt, and a long solenoid has L = μ₀N²A/ℓ. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/mutual-inductance/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/mutual-inductance/A changing current in one coil induces an emf in another, so a transformer gives V_s/V_p = N_s/N_p. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/rl-circuits/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/rl-circuits/The current rises as I = (ε/R)(1 − e^(−Rt/L)), with time constant L/R. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/magnetic-energy/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/magnetic-energy/An inductor stores U = ½LI², with field energy density u = B²/(2μ₀). Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/lc-circuits/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/lc-circuits/Energy sloshes between the capacitor and the inductor at ω = 1/√(LC), like a mass on a spring. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/rlc-circuits/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/rlc-circuits/Resistance damps LC oscillations, just as friction damps a mass on a spring. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/ac-rms/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/ac-rms/V = V₀ sin ωt, and the rms value V₀/√2 gives the same average power as a steady voltage. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/reactance/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/reactance/X_C = 1/(ωC) and X_L = ωL; current leads the voltage in a capacitor and lags it in an inductor. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/impedance/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/impedance/In a series RLC circuit Z = √(R² + (X_L − X_C)²) and tan φ = (X_L − X_C)/R. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/ac-resonance-power/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/ac-resonance-power/At ω₀ = 1/√(LC) the current peaks, and the average power is I_rms·V_rms·cos φ. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/displacement-current/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/displacement-current/A changing electric field acts like a current in Ampère's law, which fills the gap of a charging capacitor. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/maxwells-equations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/maxwells-equations/Four laws in integral form hold all of electricity and magnetism. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/em-waves-maxwell/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/em-waves-maxwell/In empty space the fields travel as a wave at c = 1/√(μ₀ε₀), with E = cB. Electricity and Magnetism · Physics Ladderhttps://teddyrookbook.com/physics/electricity-and-magnetism/em-wave-energy/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/electricity-and-magnetism/em-wave-energy/Intensity I = ½cε₀E₀² flows along the Poynting vector, and light presses on surfaces with pressure I/c. Electricity and Magnetism · Physics Ladder