Derive the expression for the equivalent resistance of three resistors joined in parallel. Why is the equivalent resistance less than the smallest of them?

1. 1 · Key idea. In parallel, each resistor has the same potential difference V across it, and the main current divides among them.
2. 2 · Draw the circuit. [R₁, R₂ and R₃ as three branches between points A and B; a battery V across A and B; currents I₁, I₂ and I₃ in the branches and I in the main wire]
3. 3 · Branch currents. I₁ = V/R₁, I₂ = V/R₂ and I₃ = V/R₃.
4. 4 · Add the currents. I = I₁ + I₂ + I₃ = V(1/R₁ + 1/R₂ + 1/R₃).
5. 5 · Compare. For the equivalent resistor I = V/R, so 1/R = 1/R₁ + 1/R₂ + 1/R₃.
6. 6 · Why it is smaller. 1/R is larger than each 1/R₁, 1/R₂ or 1/R₃, so R is less than the smallest: each branch gives the current an extra path.
7. 7 · Real life. Home appliances are joined in parallel, so each gets the full mains voltage (230 V) and has its own switch.
8. 8 · Close in one line. So resistors in parallel add as reciprocals, and the result is always below the smallest one.

Scoring tip: Write 'the same potential difference across each' as your first line. It is the key idea and earns the first mark.