Electricity Study Guide

📖 Core Concepts Electric charge – property of particles (‑e⁻, +p⁺). Smallest unit = elementary charge e = $1.602 × 10^{-19}\,\text{C}$. Like charges repel, opposite attract (Coulomb’s law). Electric current (I) – flow of charge, measured in amperes (A). Conventional direction = direction a positive charge would move. Electric field (𝐄) – vector field that exerts force on a test charge; $E = F/q$. Field lines start on + charges, end on – charges; zero inside a hollow conductor (Faraday cage). Electric potential & voltage (V) – work per unit charge to move a test charge; $1\;\text{V}=1\;\text{J/C}$. Difference between two points drives current. Ohm’s Law – relationship for resistive elements: $V = IR$. Power (P) – rate of energy transfer: $P = VI = I^{2}R = V^{2}/R$ (watts, W). Electromagnetic induction – a changing magnetic flux $\PhiB$ induces an emf: $\mathcal{E} = -\dfrac{d\PhiB}{dt}$ (Faraday’s law). AC vs. DC – DC flows one direction; AC reverses periodically (usually sinusoidal). --- 📌 Must Remember Coulomb’s law: $F = k\displaystyle\frac{|q{1}q{2}|}{r^{2}}$. Elementary charge: $e = 1.602\,176\,634\times10^{-19}\,\text{C}$. Ohm’s Law: $V = IR$. Capacitance: $Q = CV$ → $XC = \dfrac{1}{2\pi f C}$ (reactive impedance). Inductor voltage: $V = L\dfrac{dI}{dt}$ → $XL = 2\pi f L$. Power formulas: $P = VI = I^{2}R = V^{2}/R$. Faraday’s law (induced emf): $\mathcal{E} = -\dfrac{d\PhiB}{dt}$. Maxwell’s insight: Changing electric fields ↔ magnetic fields → electromagnetic waves travel at $c = 3.00\times10^{8}\,\text{m/s}$. Safety threshold (perception): ≈ 0.1–1 mA (mains frequency). --- 🔄 Key Processes Generating AC electricity (generator): Rotate a coil in a magnetic field → magnetic flux changes → emf $\mathcal{E}= -N\frac{d\PhiB}{dt}$ → sinusoidal voltage. Transforming voltage (ideal transformer): Primary/secondary windings: $Vs/Vp = Ns/Np$, $Is/Ip = Np/Ns$. Charging a capacitor: Apply voltage → charge accumulates: $Q(t)=C V(1-e^{-t/RC})$ (RC charging curve). Discharging an inductor: Interrupt current → induced emf opposes change: $V = L\,dI/dt$, current decays exponentially: $I(t)=I0 e^{-tR/L}$. Power delivery in transmission: Step‑up voltage → lower current → $P=VI$ constant, reduces $I^2R$ losses. --- 🔍 Key Comparisons Resistor vs. Capacitor Resistor: dissipates energy as heat, $V=IR$, current instantaneously follows voltage. Capacitor: stores energy in electric field, $I = C\,dV/dt$, blocks DC steady‑state. Capacitor vs. Inductor Capacitor: opposes changes in voltage, passes high‑frequency current. Inductor: opposes changes in current, passes low‑frequency (DC) current. DC vs. AC DC: constant polarity, simple power calculations. AC: sinusoidal, characterized by RMS values, enables easy transformation of voltage. Series vs. Parallel Resistors Series: $R{\text{eq}} = \sum Ri$, same current, voltages add. Parallel: $1/R{\text{eq}} = \sum 1/Ri$, same voltage, currents add. --- ⚠️ Common Misunderstandings “Current flows from positive to negative.” – Conventional current does, but electron flow is opposite. “Capacitors store charge indefinitely.” – Real capacitors leak; charge decays over time. “Higher voltage always means more danger.” – Danger depends on resulting current through the body; low voltage can be lethal if the current path is low resistance. “Inductors block AC.” – Inductors impede rapid changes; they pass low‑frequency (including DC) relatively freely. --- 🧠 Mental Models / Intuition Water analogy: Voltage = water pressure, current = flow rate, resistance = pipe diameter. Magnetic flux change = “pumping” electrons: Moving a magnet or coil “stirs” the magnetic field, which “pushes” charges (induced emf). RC time constant ($\tau = RC$): Time for a capacitor to charge to 63 % of its final voltage; think of it as “how long the capacitor holds the charge before it feels the voltage.” --- 🚩 Exceptions & Edge Cases Superconductors: Zero resistance → Ohm’s law $V=IR$ gives $V=0$ even with large $I$. Non‑linear components (diodes, transistors): $V$–$I$ relationship is exponential, not obeying Ohm’s law. High‑frequency AC: Skin effect reduces effective conductor cross‑section, increasing AC resistance. Dielectric breakdown: Capacitor voltage exceeds material rating → sudden loss of insulating property. --- 📍 When to Use Which Calculate voltage drop: Use Ohm’s law ($V=IR$) for resistive circuits; for reactive circuits use impedance $Z = \sqrt{R^{2}+(XL-XC)^{2}}$. Select energy storage: Need rapid burst → capacitor or superconducting magnetic storage. - Need long‑term, high energy density → batteries or chemical (hydrogen). Choose transformer ratio: If you need to transmit 10 kV over long distance, step‑up to 100 kV (ratio 10:1) to cut current tenfold. Decide between motor types: Fixed speed, high torque → induction motor. Variable speed, precise control → brushless DC or stepper motor. --- 👀 Patterns to Recognize Series RLC resonance: When $XL = XC$, impedance is minimum → large current surge at resonant frequency $f0 = \frac{1}{2\pi\sqrt{LC}}$. Voltage division in series: $Vk = V{\text{total}} \times \frac{Rk}{\sum Ri}$. Current division in parallel: $Ik = I{\text{total}} \times \frac{1/Rk}{\sum 1/Ri}$. Power loss in transmission: Scales with $I^{2}R$ → always look for ways to reduce current (step‑up voltage). --- 🗂️ Exam Traps Confusing RMS vs. peak values: RMS voltage of a sinusoid is $V{\text{RMS}} = V{\text{peak}}/\sqrt{2}$. Sign of induced emf: Faraday’s law includes a negative sign (Lenz’s law); the induced emf always opposes the change in flux. Capacitor in DC steady state: It acts as an open circuit (no current), not a short. Inductor in AC steady state: It behaves like a resistor with reactance $XL = 2\pi f L$, not a perfect short. Misapplying Ohm’s law to diodes/transistors: Their $V$–$I$ curves are non‑linear; use the diode equation $I = IS(e^{V/(nVT)}-1)$. ---