Each piece resistance: $r = R/8$
4 pieces in parallel: $R_1 = \dfrac{r}{4} = \dfrac{R/8}{4} = \dfrac{R}{32}$
Two such parallel groups in series:
$$R_{eff} = R_1 + R_1 = \frac{R}{32} + \frac{R}{32} = \frac{R}{16}$$Series: $R_{eff} = R_1+R_2+\cdots+R_n$. Current same through all; voltages add. Parallel: $1/R_{eff} = 1/R_1+1/R_2+\cdots$. Voltage same across all; currents add. For $n$ equal resistors $r$: series gives $nr$, parallel gives $r/n$. When a wire of resistance $R$ is cut into $n$ equal pieces: each piece $= R/n$ (resistance proportional to length).
$V = IR$. Resistivity $\rho$: $R = \rho L/A$. $\rho$ depends on material and temperature. $\rho(T) = \rho_0[1+\alpha(T-T_0)]$. Conductors: $\alpha > 0$ (resistance increases with T). Semiconductors: $\alpha < 0$ (resistance decreases with T). Superconductors: $\rho = 0$ below critical temperature $T_c$.
KCL (Junction rule): sum of currents entering = sum leaving ($\sum I = 0$). Conservation of charge. KVL (Loop rule): sum of EMF = sum of IR drops in any closed loop ($\sum \mathcal{E} = \sum IR$). Conservation of energy. Used to solve complex networks. Wheatstone bridge balanced: $P/Q = R/S$, no current through galvanometer.
Four resistors $P$, $Q$, $R$, $S$ and a galvanometer $G$. Balanced condition: $P/Q = R/S$ (no current through $G$). Applications: Metre bridge (finding unknown resistance), Potentiometer (measuring EMF, internal resistance). Post Office Box: precision resistance measurement. Sensitivity: maximum when all four resistances are equal.
Wire of uniform resistance with steady current. Potential drops uniformly along length. Used to: compare EMFs: $\mathcal{E}_1/\mathcal{E}_2 = l_1/l_2$. Measure internal resistance: $r = R(l_1/l_2 - 1)$. Advantages over voltmeter: draws no current at balance → measures true EMF (not terminal voltage). Accuracy depends on uniformity of wire and stability of driver cell.
Terminal voltage $V = \mathcal{E} - Ir$. $r$ = internal resistance. At no load ($I=0$): $V = \mathcal{E}$ (open circuit). At full load: $V < \mathcal{E}$ (voltage drops across internal resistance). Maximum power transfer: when external resistance $R = r$ (internal). Power $P = I^2 R = \mathcal{E}^2 R/(R+r)^2$.
Power $P = VI = I^2R = V^2/R$. Energy $E = Pt$ (joules) $= Pt/3600000$ kWh. 1 unit of electricity = 1 kWh. Electrical heating: Joule heating $H = I^2Rt$. Fuse: thin wire melts when $I^2R$ heating exceeds melting point. Rating: maximum safe current. MCB (Miniature Circuit Breaker): electromagnetic trip mechanism, reusable.
4-band code: first two bands = first two digits, third = multiplier, fourth = tolerance. Mnemonic: B B ROY Great Britain Very Good Wife. Black=0, Brown=1, Red=2, Orange=3, Yellow=4, Green=5, Blue=6, Violet=7, Grey=8, White=9. Gold tolerance = ±5%, Silver = ±10%. Example: Red Red Red Gold = $22 \times 10^2 \pm 5\%$ = 2200 Ω ±5%.