Cells, Kirchhoff's Laws, Bridges and Potentiometer notes: ISC Class 12 Physics

1. Emf, internal resistance and terminal p.d.

Emf E: work done by the cell (non-electrostatic, chemical action) per unit charge taken once round the circuit; unit volt (J C⁻¹). Equal to the terminal p.d. on open circuit.
Internal resistance r: resistance of the electrolyte and electrodes; grows with electrode separation, falls with larger immersed area.
Energy per coulomb: $E = IR + Ir = I(R + r)$ , so the terminal p.d. $V = IR = E - Ir$ while discharging; $V = E + Ir$ while being charged.
V–I graph: straight line, intercept E on the V-axis, slope −r, short-circuit current E/r (e.g. E = 4.5 V, r = 0.30 Ω, R = 2.7 Ω: I = 1.5 A, V = 4.05 V).

V = E − Ir (E = 2.0 V, r = 0.40 Ω)

Intercept on the V-axis = E; slope = −r.

Formulas

Cell on load
$E = I(R + r),\qquad V = E - Ir$

Key terms

Emf: Energy supplied by a source per unit charge passing once round the circuit.
Internal resistance: Resistance of the material inside a cell, through which its own current must pass.
Terminal potential difference: Potential difference across the terminals of a cell; E − Ir while it supplies current I.

2. Grouping of cells

n identical cells in series: emf nE, resistance nr; $I = \dfrac{nE}{R + nr}$ (best when R ≫ r). With some cells reversed, their emfs subtract but their resistances still add.
m identical cells in parallel: emf E, resistance r/m; $I = \dfrac{E}{R + r/m}$ (best when R ≪ r).
Mixed (m rows of n): $I = \dfrac{nE}{R + nr/m}$ , largest when $R = nr/m$ .
Two unequal cells in parallel: $E_{eq} = \dfrac{E_1r_2 + E_2r_1}{r_1 + r_2}$ , $r_{eq} = \dfrac{r_1r_2}{r_1 + r_2}$ . The weaker cell may be charged by the stronger one.

I = nE/(R + nr/m), n = 12/m

Best arrangement: R = nr/m (here 1.8 Ω = 6 × 0.6/2).

Currents as solved in worked example 3.

Formulas

Mixed grouping
$I = \dfrac{nE}{R + nr/m}$
Two cells in parallel
$E_{eq} = \dfrac{E_1r_2 + E_2r_1}{r_1 + r_2},\quad r_{eq} = \dfrac{r_1r_2}{r_1 + r_2}$

Key terms

Mixed grouping: Identical cells arranged as m parallel rows, each of n cells in series.

3. Kirchhoff’s laws

Junction law (conservation of charge): $\sum I = 0$ at any junction.
Loop law (conservation of energy): $\sum E = \sum IR$ round any closed loop.
Signs: −IR going along the current through a resistor, +IR against it; +E crossing a cell from − to +, −E from + to −.
Assume directions; a negative answer means the current flows the other way. Use as many independent loops as there are unknown currents less the junction equations.

Currents as solved in worked example 4.

Formulas

Kirchhoff’s laws
$\sum I = 0,\qquad \sum E = \sum IR$

Key terms

Junction: A point in a circuit where three or more conductors meet.
Loop: Any closed conducting path in a network.

4. Wheatstone bridge and metre bridge

Wheatstone bridge (P, Q in one pair of arms, R, S in the other, galvanometer between their junctions): balanced when $\dfrac{P}{Q} = \dfrac{R}{S}$ (from the loop law with $I_g = 0$ ). Balance is independent of the cell’s emf and the galvanometer’s resistance; the bridge is most sensitive when all four arms are of the same order.
Metre bridge: uniform 100 cm wire forms two arms; unknown X in the left gap, known R in the right, balance at l: $X = R\,\dfrac{l}{100 - l}$ . Aim for l near 50 cm (end resistances matter least); interchange X and R and average.

Worked example 5: balance at 55.0 cm.

Formulas

Bridge balance
$\dfrac{P}{Q} = \dfrac{R}{S}$
Metre bridge
$X = R\,\dfrac{l}{100 - l}$

Key terms

Null point: Position of the jockey at which the galvanometer shows no deflection.

5. Potentiometer: principle, gradient, sensitivity

A uniform wire carrying a steady current: potential drop ∝ length, $V = kl$ . A cell (+ to A) balanced at length l: $E = kl$ , with no current drawn — an ideal voltmeter.
Potential gradient $k = V_{AB}/L = I\rho/A$ (V m⁻¹). Wire of constantan or manganin (high ρ, low temperature coefficient), uniform cross-section.
Sensitivity rises as k falls: longer wire, more resistance in the driver circuit. Balance needs driver emf (p.d. across the wire) > E and both + terminals at A.

Worked example 6: k = 0.40 V m⁻¹, balance at 2.7 m.

V = kl (k = 0.40 V m⁻¹)

Uniform wire, steady current: V ∝ l.

Formulas

Potentiometer
$E = kl,\qquad k = \dfrac{V_{AB}}{L}$

Key terms

Potential gradient: Fall of potential per unit length along the potentiometer wire.
Sensitivity of a potentiometer: Change in balancing length per unit change in emf; greater when the potential gradient is smaller.

6. Potentiometer applications

Comparing emfs (two-way key, same k): $\dfrac{E_1}{E_2} = \dfrac{l_1}{l_2}$ ; neither k nor the internal resistances are needed. Sum-and-difference method: $\dfrac{E_1 + E_2}{E_1 - E_2} = \dfrac{l_a}{l_b}$ .
Internal resistance: open circuit $E = kl_1$ ; with R across the cell, $V = kl_2$ ; $r = R\,\dfrac{l_1 - l_2}{l_2}$ .

Formulas

Comparing emfs
$\dfrac{E_1}{E_2} = \dfrac{l_1}{l_2}$
Internal resistance
$r = R\,\dfrac{l_1 - l_2}{l_2}$

Exam tips

Derive E = I(R + r) from energy per coulomb; V = E + Ir when charging.
Mixed grouping: largest current when R = nr/m.
Kirchhoff: label currents, one sign convention, negative answer = reversed direction.
Potentiometer: smaller k, more sensitive; no balance if the wire’s p.d. is less than E.

1. Emf, internal resistance and terminal p.d.

Formulas

Key terms

2. Grouping of cells

Formulas

Key terms

3. Kirchhoff’s laws

Formulas

Key terms

4. Wheatstone bridge and metre bridge

Formulas

Key terms

5. Potentiometer: principle, gradient, sensitivity

Formulas

Key terms

6. Potentiometer applications

Formulas

Exam tips

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