United Grade 12 AP Physics C: Electricity and Magnetism
Chapters: 6
1. Electric Charges, Fields, and Gauss's Law
Electric Charge and Electric Force · Conservation of Electric Charge and the Process of Charging · Electric Fields · Electric Fields of Charge Distributions · Electric Flux · Gauss's Law
- Electric Charges and Fields – Charge comes in two kinds, is conserved and comes in whole-number packets of e = 1.6 × 10⁻¹⁹ C. Two point charges push or pull with F = kq₁q₂/r² (k = 9 × 10⁹ N m² C⁻²). Forces and fields from many charges add as vectors (superposition). The field E = F/q₀ of a point charge is kq/r²; field lines show it. A dipole (±q a distance 2a apart) has moment p = q·2a; in a uniform field it feels zero net force but a torque τ = pE sinθ. Electric flux Φ = E·A, and Gauss's law says the flux out of any closed surface is q_enclosed/ε₀, which quickly gives E for a long wire (λ/2πε₀r), a plane sheet (σ/2ε₀) and a thin spherical shell (kq/r² outside, 0 inside).
2. Electric Potential
Electric Potential Energy · Electric Potential · Conservation of Electric Energy
- Electrostatic Potential and Capacitance – Electrostatic potential V at a point is the work done per unit positive charge to bring it from infinity: V = kq/r for a point charge, and potentials of many charges add as plain numbers. Potential difference V_B − V_A = W_AB/q. Equipotential surfaces join equal-V points; E is perpendicular to them and E = −dV/dr. Potential energy of two charges is U = kq₁q₂/r, and of a dipole in a field U = −pE cosθ. Conductors have free charges (E inside = 0); dielectrics have bound charges that polarise and cut the field to E₀/K. A capacitor stores charge Q = CV. A parallel plate capacitor has C = ε₀A/d, and KC with a dielectric. In series 1/C = 1/C₁ + 1/C₂ + …; in parallel C = C₁ + C₂ + …. Energy stored U = ½CV² = Q²/2C = ½QV.
3. Conductors and Capacitors
Electrostatics with Conductors · Redistribution of Charge Between Conductors · Capacitors · Dielectrics
- Electrostatic Potential and Capacitance – Electrostatic potential V at a point is the work done per unit positive charge to bring it from infinity: V = kq/r for a point charge, and potentials of many charges add as plain numbers. Potential difference V_B − V_A = W_AB/q. Equipotential surfaces join equal-V points; E is perpendicular to them and E = −dV/dr. Potential energy of two charges is U = kq₁q₂/r, and of a dipole in a field U = −pE cosθ. Conductors have free charges (E inside = 0); dielectrics have bound charges that polarise and cut the field to E₀/K. A capacitor stores charge Q = CV. A parallel plate capacitor has C = ε₀A/d, and KC with a dielectric. In series 1/C = 1/C₁ + 1/C₂ + …; in parallel C = C₁ + C₂ + …. Energy stored U = ½CV² = Q²/2C = ½QV.
- Electrostatics: Induction, Polarisation and Sharing of Charge – Electrostatics is the study of charges at rest. Objects get charged by friction (rubbing), by contact (conduction) or by induction (bringing a charge near, without touching). In a conductor, free electrons move: a nearby charge pushes them to one side (induction), and earthing then leaves the conductor with the opposite charge. In an insulator (dielectric), electrons cannot travel, but each molecule stretches into a tiny dipole and lines up with the field (polarisation); that is why a charged comb attracts neutral paper. When charged conductors are joined by a wire, charge flows until their potentials are equal. For spheres far apart, V = kq/r, so the charge divides in the ratio of the radii: q₁/q₂ = r₁/r₂, and the smaller sphere has the larger surface charge density. Total charge is always conserved.
4. Electric Circuits
Electric Current · Simple Circuits · Resistance, Resistivity, and Ohm's Law · Electric Power · Compound Direct Current Circuits · Kirchhoff's Loop Rule · Kirchhoff's Junction Rule · Resistor-Capacitor (RC) Circuits
- Current in a Metal: Drift Velocity and Mobility – In a metal, free electrons move very fast in random directions, so on average they go nowhere. An electric field adds a small, steady shift opposite to the field: the drift velocity vd = eEτ/m. The current is I = n e A vd, and the current density is j = n e vd. Mobility μ = vd/E tells how easily a charge drifts. Drift speed is only about a millimetre per second, yet a bulb lights at once because the field is set up in the whole wire almost instantly.
- EMF, Internal Resistance, Power and Combination of Cells – A cell does work on charges; the work per coulomb is its emf ε. Inside the cell there is a small internal resistance r. With current I, the terminal voltage is V = ε − Ir and I = ε/(R + r). Electrical power P = VI = I²R = V²/R, and energy W = Pt. Power delivered to R is largest when R = r. Cells in series: εeq = ε1 + ε2, req = r1 + r2. Cells in parallel: εeq = (ε1r2 + ε2r1)/(r1 + r2), 1/req = 1/r1 + 1/r2.
- Ohm's Law, Resistivity and Effect of Temperature – Using drift velocity, V = IR with R = ml/(ne²τA) = ρl/A. Resistivity ρ = m/(ne²τ) depends only on the material and temperature; conductivity σ = 1/ρ. Ohm's law in vector form is j = σE. Metals, wires and resistors give a straight V–I line (ohmic); bulbs, diodes and devices like GaAs give curved or one-way graphs (non-ohmic). For metals ρ rises with temperature: ρT = ρ0[1 + α(T − T0)]; alloys like nichrome change very little; semiconductors get lower ρ when hot.
- Energy in Systems and Electric Power – A system is the set of objects you choose to study. Its total energy stays the same unless energy crosses the system boundary as work, heat or radiation: ΔE_system = W + Q. Inside, energy moves between stores such as kinetic, gravitational, elastic, thermal, chemical and electric. Power is the rate of energy transfer, P = ΔE/Δt, in watts. In a circuit the electric power is P = IV; for a resistor this also equals I²R and V²/R. Real devices waste part of the input as heat, so efficiency η = useful output ÷ input is always less than 100%. Designing a device means choosing the input store, the output store and cutting the waste.
- Kirchhoff's Rules and the Wheatstone Bridge – Junction rule: at any junction, the sum of currents entering equals the sum leaving (ΣI = 0), because charge is conserved. Loop rule: around any closed loop, the algebraic sum of potential changes is zero (ΣΔV = 0), because energy is conserved. Sign rules: a resistor crossed along the current gives −IR; a cell crossed from − to + gives +ε. A Wheatstone bridge of four resistors P, Q, R, S is balanced (no galvanometer current) when P/Q = R/S; this lets us find an unknown resistance, as in the metre bridge.
- Electrostatic Potential and Capacitance – Electrostatic potential V at a point is the work done per unit positive charge to bring it from infinity: V = kq/r for a point charge, and potentials of many charges add as plain numbers. Potential difference V_B − V_A = W_AB/q. Equipotential surfaces join equal-V points; E is perpendicular to them and E = −dV/dr. Potential energy of two charges is U = kq₁q₂/r, and of a dipole in a field U = −pE cosθ. Conductors have free charges (E inside = 0); dielectrics have bound charges that polarise and cut the field to E₀/K. A capacitor stores charge Q = CV. A parallel plate capacitor has C = ε₀A/d, and KC with a dielectric. In series 1/C = 1/C₁ + 1/C₂ + …; in parallel C = C₁ + C₂ + …. Energy stored U = ½CV² = Q²/2C = ½QV.
5. Magnetic Fields and Electromagnetism
Magnetic Fields · Magnetism and Moving Charges · Magnetic Fields of Current-Carrying Wires and the Biot-Savart Law · Ampère's Law
- Magnetic Field due to a Current: Biot-Savart Law and Ampere's Law – A current makes a magnetic field around it (Oersted, 1820). The Biot-Savart law gives the field of a tiny piece of wire: dB = (μ₀/4π)·I dl sinθ / r². Adding all pieces of a circular loop gives B = μ₀NI / 2R at the centre and μ₀NIR² / 2(R² + x²)^{3/2} on the axis. Ampere's circuital law ∮B·dl = μ₀I gives B = μ₀I / 2πr for a long straight wire. A long solenoid has a strong, uniform field B = μ₀nI inside and almost none outside.
- Lorentz Force, Torque on a Loop and the Moving Coil Galvanometer – A charge q moving with velocity v feels F = q(E + v × B), the Lorentz force. The magnetic part qvB sinθ is at right angles to v, so it bends the path into a circle (r = mv/qB) without changing speed. A wire feels F = IL × B. Parallel wires attract if the currents are in the same direction: F/L = μ₀I₁I₂/2πd. A loop in a field feels a torque τ = m × B with magnetic moment m = NIA, so a loop acts as a magnetic dipole. A moving coil galvanometer uses this torque: its deflection φ = (NAB/k)I. A small shunt turns it into an ammeter; a large series resistance turns it into a voltmeter.
6. Electromagnetic Induction
Magnetic Flux · Electromagnetic Induction · Induced Currents and Magnetic Forces · Inductance · Circuits with Resistors and Inductors (LR Circuits) · Circuits with Capacitors and Inductors (LC Circuits)
- Electromagnetic Induction – When the magnetic flux through a coil changes, an emf is produced in it: ε = −N dΦ/dt (Faraday). The minus sign is Lenz's law: the induced current always opposes the change that made it. Moving a rod in a field gives motional emf ε = Blv. A changing current makes an emf in its own coil (self-induction, L) and in a nearby coil (mutual induction, M).
- Force on a Current-Carrying Conductor, Motor and Induction – A wire carrying current inside a magnetic field feels a push. The push is biggest when the wire is at 90° to the field and zero when it is parallel. Fleming's left-hand rule gives its direction. A motor uses this push to spin a coil; a generator does the reverse and makes current by moving a coil or magnet.
- LR Circuits: How an Inductor Slows a Current – An inductor is a coil that opposes any change in the current through it by making a back emf ε_L = −L dI/dt (L in henry). In a circuit with a battery ε, a resistor R and an inductor L in series, the current cannot jump. When the switch closes it grows as I = (ε/R)(1 − e^(−t/τ)), where the time constant τ = L/R. At t = τ the current is about 63% of its final value ε/R, and after about 5τ it is practically steady; then the inductor acts like a plain wire. If the battery is removed and the loop is closed through R, the current decays as I = I₀ e^(−t/τ), falling to 37% after one τ. The energy that keeps it going was stored in the inductor's magnetic field, U = ½LI². Kirchhoff's loop rule with the inductor gives ε − IR − L dI/dt = 0.
- Alternating Current – An alternating current (AC) changes size and direction again and again: I = I₀ sin ωt. Its rms value is I₀/√2, the steady DC that gives the same heating. A resistor keeps V and I in step; an inductor makes I lag by 90° (Xʟ = ωL); a capacitor makes I lead by 90° (Xᴄ = 1/ωC). In a series LCR circuit Z = √(R² + (Xʟ − Xᴄ)²), resonance happens when Xʟ = Xᴄ, and average power is P = Vrms Irms cos φ.