Two cells of emfs ε₁ and ε₂ and internal resistances r₁ and r₂ are connected in parallel. Derive expressions for the equivalent emf and equivalent internal resistance of the combination.

1. 1 · State the principle. For a cell giving current, the terminal voltage is V = ε − Ir. Cells in parallel share the same V.
2. 2 · Set up. Like terminals joined at B₁ (+) and B₂ (−); currents I₁ and I₂ leave B₁; I = I₁ + I₂ flows in the external circuit.
3. 3 · Draw. [Two cells side by side between B₁ and B₂, each with its r; currents I₁, I₂ and I marked; load R]
4. 4 · Write V and add the currents. V = ε₁ − I₁r₁ and V = ε₂ − I₂r₂, so I = ε₁/r₁ + ε₂/r₂ − V(1/r₁ + 1/r₂).
5. 5 · Rearrange. V = (ε₁r₂ + ε₂r₁)/(r₁ + r₂) − I r₁r₂/(r₁ + r₂).
6. 6 · Real life. Inverter batteries joined in parallel share the load and give a longer backup.
7. 7 · Compare and close. With V = ε_eq − Ir_eq: ε_eq = (ε₁r₂ + ε₂r₁)/(r₁ + r₂) and r_eq = r₁r₂/(r₁ + r₂).

Scoring tip: The comparison with V = ε_eq − Ir_eq earns the final mark, so write it out in full.