Chapter 21 — Electricity: Charges, Fields, Potential, and Circuits Summary & Study Notes

⚡ Electric Charge and Conservation

• Charge (q) is a fundamental property of matter. There are two types: positive and negative. Like charges repel; opposite charges attract.
• Elementary charge is $e = 1.602\times10^{-19}\ \text{C}$. Macroscopic charges are integer multiples of $e$.
• Conservation of charge: total charge in an isolated system is constant. Charges may be transferred but not created/destroyed.
• Materials: conductors allow free movement of charges (e.g., metals), insulators do not, and semiconductors lie between.

🧲 Coulomb's Law and Electric Force

• Coulomb's law gives the electrostatic force between two point charges: $F = k \frac{q_1 q_2}{r^2}$, where $k = 8.99\times10^{9}\ \text{N}\cdot\text{m}^2/\text{C}^2$.
• The force is along the line joining charges and is attractive for opposite signs, repulsive for like signs.
• In vector form: $\vec{F}${12} = k \frac{q_1 q_2}{r^2}\hat{r}{12}.

🧭 Electric Field (E)

• The electric field is defined as the force per unit positive test charge: $\vec{E} = \frac{\vec{F}}{q_{test}}$.
• Field of a point charge: $\vec{E} = k \frac{q}{r^2}\hat{r}$.
• Superposition principle: fields from multiple charges add vectorially.
• Field lines: point away from positive charges, toward negative. Density of lines indicates magnitude.

📦 Gauss's Law

• Gauss's law: $\Phi_E = \oint \vec{E}\cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$, where $\epsilon_0 = 8.85\times10^{-12}\ \text{C}^2/\text{N}\cdot\text{m}^2$.
• Useful for finding $\vec{E}$ when there is high symmetry: spherical, cylindrical, planar.
• Conductors in electrostatic equilibrium: electric field inside is zero; any excess charge resides on the surface.

🔋 Electric Potential and Potential Energy

• Electric potential energy $U$ between charges relates to work required to assemble them.
• Electric potential (voltage) $V$ at a point is potential energy per unit charge: $V = \frac{U}{q}$.
• For a point charge: $V = k \frac{q}{r}$ (chosen zero at infinity).
• Potential difference between two points: $\Delta V = V_B - V_A = -\int_A^B \vec{E}\cdot d\vec{l}$.
• Conservative nature: line integral of $\vec{E}$ is path independent.

🧮 Capacitance and Capacitors

• Capacitance $C$ is $C = \frac{Q}{V}$; measures ability to store charge per unit potential.
• Parallel-plate capacitor: $C = \epsilon_0 \frac{A}{d}$ (ideal, ignoring edge effects).
• With a dielectric of constant $\kappa$: $C = \kappa \epsilon_0 \frac{A}{d}$.
• Energy stored in a capacitor: $U = \frac{1}{2}CV^2 = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}QV$.
• Capacitors in series: $\frac{1}{C_{eq}} = \sum \frac{1}{C_i}$. In parallel: $C_{eq} = \sum C_i$.

🔌 Electric Current and Microscopic Picture

• Current (I) is rate of charge flow: $I = \frac{dQ}{dt}$. Direction of conventional current is flow of positive charge.
• Current density $\vec{J}$: current per unit area, $I = \int \vec{J}\cdot d\vec{A}$.
• Drift velocity $v_d$ relates to current: $v_d = \frac{I}{n q A}$, where $n$ is charge carrier density, $q$ carrier charge, $A$ cross-sectional area.

🔩 Resistance and Ohm's Law

• Ohm's law (macroscopic): $V = IR$, where $R$ is resistance.
• Resistivity relation: $R = \rho \frac{L}{A}$, where $\rho$ is resistivity and $\sigma = 1/\rho$ is conductivity.
• Temperature dependence: for many metals, $\rho(T) = \rho_0[1 + \alpha (T - T_0)]$.

🔁 Circuits: Series, Parallel, Kirchhoff

• Series resistors: $R_{eq} = \sum R_i$; same current, voltages add.
• Parallel resistors: $\frac{1}{R_{eq}} = \sum \frac{1}{R_i}$; same voltage, currents add.
• Kirchhoff's laws: junction rule (sum of currents = 0) and loop rule (sum of potential changes around loop = 0).
• For capacitors: in series charge on each is same; in parallel voltage across each is same.

⏱ RC Circuits and Time Dependence

• Charging a capacitor through resistor: $Q(t) = Q_{max}[1 - e^{-t/(RC)}]$, $I(t) = \frac{Q_{max}}{RC} e^{-t/(RC)}$.
• Discharging: $Q(t) = Q_0 e^{-t/(RC)}$, $I(t) = -\frac{Q_0}{RC} e^{-t/(RC)}$.
• Time constant $\tau = RC$: after $t = \tau$, capacitor charges to ~63% of final value.

⚡ Power and Energy in Circuits

• Instantaneous power delivered by source: $P = IV$.
• Using Ohm's law: $P = I^2 R = \frac{V^2}{R}$.
• Energy dissipated in time $t$: $W = \int P dt$.

🔍 Useful Problem-Solving Tips

• Use superposition for fields and potentials from multiple charges (potentials add scalarly).
• Choose Gaussian surfaces that match symmetry for Gauss's law.
• Keep track of signs for work and potential (positive work against field increases potential).
• For circuits, label currents and apply Kirchhoff consistently; check units.

📘 Quick Formula Summary

• Coulomb: $F = k \frac{q_1 q_2}{r^2}$
• Field (point charge): $E = k \frac{q}{r^2}$
• Potential (point): $V = k \frac{q}{r}$
• Gauss: $\oint \vec{E}\cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$
• Capacitance: $C = \frac{Q}{V}$, $C_{parallel} = \sum C_i$, $\frac{1}{C_{series}} = \sum \frac{1}{C_i}$
• Energy in capacitor: $U = \frac{1}{2}CV^2$
• Current: $I = \frac{dQ}{dt}$, $v_d = \frac{I}{nqA}$
• Ohm: $V = IR$, $R = \rho \frac{L}{A}$
• RC time constant: $\tau = RC$

Keep a sheet of these formulas and practice vector addition of fields and potential integrals to master problems.