Current Electricity

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Current Electricity — Key Facts

Electric current is the rate of flow of charge: $$I = \frac{Q}{t}$$

Unit: Ampere (A). Conventional current direction = direction of positive charge flow (opposite to electron flow).

Drift Velocity: $$v_d = \frac{I}{nAq}$$

where n = number density of free electrons, A = cross-sectional area.

Typical drift velocity in copper wire: ~0.1 mm/s (very slow!)

Ohm’s Law: $$V = IR$$

where R = resistance = ρL/A

Resistivity (ρ) and Conductivity (σ): $$\rho = \frac{1}{\sigma}$$ $$R = \frac{\rho L}{A}$$

Temperature dependence: $\rho_T = \rho_0[1 + \alpha(T - T_0)]$

where α = temperature coefficient of resistance.

⚡ JEE Exam Tip: For superconductors, α is negative and at very low temperatures, ρ becomes exactly zero. For metals, α > 0; for semiconductors, α < 0 (resistance decreases with temperature increase).

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding…

Circuit Analysis:

Series: Same current through all elements $$R_{eq} = R_1 + R_2 + …$$

Parallel: Same voltage across all branches $$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + …$$

EMF and Internal Resistance:

Real battery has emf ε and internal resistance r. $$V_{terminal} = \varepsilon - Ir$$

Power delivered to load R: $P = I^2R = \frac{\varepsilon^2 R}{(R+r)^2}$

Maximum power transfer when R = r (load matching condition).

Kirchhoff’s Laws:

Junction Law: $\sum I_{in} = \sum I_{out}$ (conservation of charge) Loop Law: $\sum V = 0$ (conservation of energy)

⚡ JEE Exam Tip: For complex circuits, use symmetry where possible. If circuit is symmetric about an axis, nodes at same potential can be combined. Also use the method of assumed currents — assign directions, if answer comes negative, direction is reversed.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Wheatstone Bridge:

Balanced when: $\frac{R_1}{R_2} = \frac{R_3}{R_4}$

Current through middle resistor = 0 when balanced.

For unbalanced bridge, use delta-star transformation or write mesh equations.

Metre Bridge (Slide Wire Bridge):

Unknown resistance: $R = \frac{l_1}{l_2} \times S$

where S = known resistance in left gap, l₁ and l₂ = lengths of wire on either side of balance point.

Accuracy depends on uniform wire resistance. Key advantage: uses a wire with uniform cross-section, so resistance is proportional to length.

Potentiometer:

Advantages over voltmeter:

• Does not draw current (infinite resistance) — measures true emf
• Can compare emf of two cells directly
• Can measure internal resistance of a cell

Sensitivity increases with:

• Higher potential gradient along wire
• Longer wire
• Lower current through potentiometer wire

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

For measuring internal resistance: balance point found first with switch open (R = ∞), then with switch closed (R finite).

Delta-Star Transformation:

For complex networks that cannot be simplified directly: $$R_Y = \frac{R_\Delta R_C}{R_A + R_B + R_C}$$

And the reverse: $$R_{AB} = \frac{R_A R_B + R_B R_C + R_C R_A}{R_C}$$

This transformation can simplify networks of resistors that are neither series nor parallel.

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:

1. Seebeck effect: temperature difference → electric current (thermoelectric generator)
2. Peltier effect: current → temperature difference at junctions (reversible 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): $\varepsilon_{net} = n\varepsilon$, $r_{net} = nr$ Parallel (n identical cells): $\varepsilon_{net} = \varepsilon$, $r_{net} = r/n$

Optimal arrangement for maximum current through external R: $mR = nr$ (m cells in series, n in parallel)

Mixed grouping: if we have N cells total, arranged as m in series and n in parallel (m × n = N), then $I = \frac{n\varepsilon}{R + nr/n}$. Maximum current when $mR = nr$.

⚡ JEE Advanced 2022 Analysis: Questions on metre bridge, potentiometer, and RC circuits appeared in recent papers. For potentiometer, the balance point shifts if the current through the potentiometer wire changes — always account for this when comparing emf or measuring internal resistance.

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