Current Electricity and Circuits

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Current Electricity — Key Facts

Electric current is the flow of charge. By convention, current direction is the direction of flow of positive charge (opposite to electron flow).

$$I = \frac{Q}{t}$$

SI unit: Ampere (A), where 1 A = 1 C/s

Ohm’s Law: $$V = IR$$

where V = potential difference (volts), I = current (amps), R = resistance (ohms)

Resistance: $$R = \frac{\rho L}{A}$$

where ρ = resistivity (Ω·m), L = length (m), A = cross-sectional area (m²)

Resistivity of common materials (at 20°C):

• Copper: 1.68 × 10⁻⁸ Ω·m
• Iron: 9.71 × 10⁻⁸ Ω·m
• nichrome: 1.10 × 10⁻⁶ Ω·m

Series and Parallel:

Configuration	Same through	Same across
Series	Current (I)	Voltage (V)
Parallel	Voltage (V)	Current (I)

Series: $R_{eq} = R_1 + R_2 + …$ Parallel: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + …$

⚡ ECAT Exam Tip: For power dissipation: $P = VI = I^2R = \frac{V^2}{R}$. In series, the element with highest resistance dissipates most power. In parallel, the element with lowest resistance dissipates most power.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding…

EMF and Internal Resistance:

EMF (electromotive force) is the energy provided per coulomb of charge: $$\varepsilon = \frac{W}{q}$$

Real battery has internal resistance r. Terminal voltage: $$V = \varepsilon - Ir$$

Power transferred to load: $P = VI = I(\varepsilon - Ir) = I\varepsilon - I^2r$

Maximum power transfer occurs when $R = r$ (load equals internal resistance).

Kirchhoff’s Laws:

Junction Law (KCL): At any junction, sum of currents entering = sum leaving $$\sum I_{in} = \sum I_{out}$$

Loop Law (KVL): Around any closed loop, sum of potential differences = 0 $$\sum V = 0$$

Sign Convention:

• Current entering a junction: positive
• EMF (positive terminal to negative): +ε
• EMF (negative terminal to positive): -ε
• Resistor in direction of current: -IR
• Resistor against direction of current: +IR

Wheatstone Bridge:

When $R_1/R_2 = R_3/R_4$, the bridge is balanced and no current flows through the middle resistor.

This principle is used in metre bridge and Carey Foster bridge experiments.

Potentiometer:

A potentiometer measures emf without drawing current (ideal voltmeter).

Principle: If uniform potential gradient exists along wire, potential difference is proportional to length.

$$V \propto l$$

For comparing emf of two cells: $\frac{\varepsilon_1}{\varepsilon_2} = \frac{l_1}{l_2}$

⚡ ECAT Exam Tip: In metre bridge problems, the unknown resistance $R = \frac{l_1}{l_2} \times S$ where S is the known resistance in the left gap.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Solving Complex Circuits:

Loop Analysis:

1. Assign currents to each branch
2. Write KVL equations for each independent loop
3. Solve using Cramer’s rule or matrix method

Node Analysis:

1. Assign node voltages
2. Write KCL equations at each node
3. Solve for node voltages

Delta-Star Transformation:

Sometimes circuits cannot be simplified directly. Use delta-star (π-T) transformation:

$$R_Y = \frac{R_\Delta R_C}{R_A + R_B + R_C}$$

$$R_A = \frac{R_{AB} \cdot R_{AC}}{R_{AB} + R_{BC} + R_{CA}}$$

(and cyclic permutations)

This is useful for finding equivalent resistance of complex networks.

RC Circuits:

Charging: $q = Q(1 - e^{-t/RC})$, $V = V_0(1 - e^{-t/RC})$ Discharging: $q = Qe^{-t/RC}$, $V = V_0e^{-t/RC}$

Time constant: $\tau = RC$ (time to reach 63% of final value)

Thermoelectric Effects:

When temperature differences cause electric currents or vice versa:

1. Seebeck Effect: Temperature difference → electric current (thermoelectric generator)
2. Peltier Effect: Electric current → temperature difference (refrigeration)
3. Thomson Effect: Temperature gradient along conductor with current → absorption/emission of heat

Thermoelectric power (Seebeck coefficient): $S = dV/dT$

Cell Combinations:

Series: n identical cells

• Net emf: $n\varepsilon$
• Net internal resistance: $nr$
• Maximum current: $I_{max} = \frac{n\varepsilon}{R + nr}$

Parallel: n identical cells

• Net emf: $\varepsilon$
• Net internal resistance: $r/n$
• Useful for providing larger current capacity

Mixed grouping: For maximum current through external resistance R: $$I = \frac{n\varepsilon}{R + nr/n} \quad \text{when } m \text{ cells in series, } n \text{ in parallel}$$

Optimal arrangement: $mR = nr$ (external resistance equals total internal resistance per branch)

⚡ ECAT 2024 Analysis: Questions on potentiometer, metre bridge, and RC charging/discharging appeared in recent papers. For circuits with multiple loops, always check number of independent equations: for n loops, you need n independent KVL equations.

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