Current Electricity
⚑ Panic box
Ohmic conductor — $V=IR$ only when resistance stays constant under the conditions.
Drift — $\vec v_d=-\dfrac{e\tau}{m}\vec E$; electron drift is opposite $\vec E$.
Cell under load — $V=\varepsilon-Ir$ while supplying current; open circuit: $V=\varepsilon$.
Kirchhoff — junction: current in = current out; closed loop: signed potential changes sum to zero.
Wheatstone null — $I_g=0 \Rightarrow \dfrac{R_1}{R_2}=\dfrac{R_3}{R_4}$.
1. Current, drift & conduction
- Current — $I=\dfrac{\Delta Q}{\Delta t}$ for steady flow; instantaneously $I=\dfrac{dQ}{dt}$.
- Conventional current — direction positive charge moves; in metals, electron drift is opposite.
- Current density — $\vec j=nq\vec v_d$; for electrons, $q=-e$; uniform normal flow: $I=|j|A$.
- Drift speed — $\vec v_d=-\dfrac{e\tau}{m}\vec E$ for electrons; $v_d=\dfrac{I}{neA}$ in magnitude.
- Relaxation time — $\tau$, average time between successive carrier collisions.
- Mobility — $\mu=\dfrac{|v_d|}{E}=\dfrac{e\tau}{m}$ for electrons; unit $\mathrm{m^2,V^{-1},s^{-1}}$.
Random thermal motion → average velocity zero → no net current; applied field adds a small average drift to the random motion. Collisions repeatedly interrupt acceleration, producing steady average drift.
2. Resistance, resistivity & Ohm’s law
- Ohm’s law — $V=IR$; $V$ is proportional to $I$ only for an ohmic conductor under fixed physical conditions.
- Resistance — $R=V/I$, unit $\Omega=\mathrm{V/A}$; depends on material, dimensions and temperature.
- Geometry — $R=\rho\dfrac{\ell}{A}$; longer conductor → larger $R$; larger cross-section → smaller $R$.
- Resistivity — $\rho=RA/\ell$, material property; unit $\Omega,\mathrm m$.
- Conductivity — $\sigma=1/\rho$; $\vec j=\sigma\vec E$ or $\vec E=\rho\vec j$.
- Drude-model link — $\sigma=\dfrac{ne^2\tau}{m}$; $\rho=\dfrac{m}{ne^2\tau}$, for the chapter’s constant-$n$, constant-$\tau$ model.
- Material classes — metals: low $\rho$ · semiconductors: intermediate; $\rho$ generally falls as temperature rises · insulators: very high $\rho$.
Ohm’s law fails if the $V$–$I$ relation is nonlinear, changes with the sign of $V$, or gives more than one $V$ for a given $I$ (as in GaAs). $V=IR$ can still define a ratio for a device; it does not prove that the device is ohmic.
3. Temperature effects
- Linear-range relation — $\rho_T=\rho_0[1+\alpha(T-T_0)]$; approximately valid over a limited temperature range.
- Temperature coefficient — $\alpha$, fractional change in resistivity per unit temperature change; unit $\mathrm{K^{-1}}$ or $^\circ\mathrm C^{-1}$.
| Material | As temperature rises | Recall point |
|---|---|---|
| Metals | $\rho$ usually increases | $\alpha>0$; more collisions, nearly fixed carrier density |
| Semiconductors / insulators | $\rho$ decreases | increasing carrier density dominates |
| Nichrome, manganin, constantan | $\rho$ changes little | useful for stable resistance |
For a uniform wire, $R$ changes in the same proportion as $\rho$ if its dimensions are treated as unchanged: $R_T=R_0[1+\alpha(T-T_0)]$.
4. Electrical energy, power & transmission
- Electrical energy transferred — $W=IV\Delta t$; in a resistor, dissipated as heat.
- Power — $P=IV=I^2R=\dfrac{V^2}{R}$; choose the form matching known quantities.
- Transmission loss — $P_c=I^2R_c=\dfrac{P^2R_c}{V^2}$ for delivered power $P=VI$; high transmission voltage lowers cable loss.
5. Cells & equivalent sources
- EMF — $\varepsilon$, work supplied per unit charge; open-circuit terminal voltage, not a force.
- Internal resistance — $r$, resistance inside the cell; while discharging, $V=\varepsilon-Ir$ and $I=\dfrac{\varepsilon}{R+r}$ for external resistance $R$.
- Maximum theoretical current — $I_{\max}=\varepsilon/r$ at $R=0$; practical cells may have a lower safe limit.
| Cell combination | Equivalent emf | Equivalent internal resistance |
|---|---|---|
| Series, aiding | $\varepsilon_{eq}=\sum_i\varepsilon_i$ | $r_{eq}=\sum_i r_i$ |
| Series, opposing | algebraic sum; opposing emf is negative | $r_{eq}=\sum_i r_i$ |
| Parallel | $\varepsilon_{eq}=\dfrac{\sum_i(\varepsilon_i/r_i)}{\sum_i(1/r_i)}$ | $\dfrac{1}{r_{eq}}=\sum_i\dfrac{1}{r_i}$ |
Parallel-cell formula uses consistent polarity signs; for two cells, $\varepsilon_{eq}=\dfrac{\varepsilon_1r_2+\varepsilon_2r_1}{r_1+r_2}$.
6. Circuit rules & bridge
- Junction rule — $\sum I_{in}=\sum I_{out}$; charge does not build up at a steady-current junction.
- Loop rule — $\sum \Delta V=0$ around any closed loop; potential returns to its starting value.
- Current sign — choose branch directions first; a negative result means actual flow is opposite the chosen arrow.
- Cell sign in a loop — crossing $-\to+$ gives $+\varepsilon$; crossing $+\to-$ gives $-\varepsilon$.
- Resistor sign in traversal — along current: $-IR$; against current: $+IR$.
- Mark a trial direction for each branch current.
- Use junction equations to relate currents.
- Choose independent closed loops; add signed voltage changes.
- Solve simultaneous equations; reverse any negative-current arrow.
Wheatstone bridge — four arms $R_1,R_2,R_3,R_4$; galvanometer joins the other pair of vertices.
- Balanced bridge: $I_g=0$ → $\dfrac{R_1}{R_2}=\dfrac{R_3}{R_4}$ → $R_1R_4=R_2R_3$.
- Unknown arm, when $R_4$ is unknown: $R_4=\dfrac{R_2R_3}{R_1}$.
- Meter bridge — practical resistance measurement using the balanced-bridge principle.
7. Traps
Current is a vector→ current is scalar; its arrow marks direction, not vector addition.All electrons drift together→ random motion remains; drift is only their small average motion.$V=IR$ guarantees Ohm’s law→ it defines resistance; Ohm’s law requires a linear $V$–$I$ relation under fixed conditions.EMF is a force→ it is energy supplied per unit charge, measured in volts.Terminal voltage equals $\varepsilon$ under load→ while discharging, internal resistance gives a drop $Ir$.Use a guessed current direction as certain→ a negative solved value means the actual direction is reversed.