Resistors in parallel

Purpose: to obtain a smaller resistance out of several larger ones.

How to measure the equivalent resistance experimentally: connect the three resistors side by side between the same two points, and place the group in a circuit with a battery, a rheostat and an ammeter in the main wire. Adjust the rheostat until a suitable main current I is read. Measure the voltage V across the group with a voltmeter and the current in each branch (I₁ in R₁, I₂ in R₂, I₃ in R₃) with the ammeter. You will find that I = I₁ + I₂ + I₃ and that V / I is smaller than the smallest resistance.

Facts for a parallel connection:

• The same potential difference V is across every branch (all the branches start at one common point and end at another common point).
• The main current splits: I = I₁ + I₂ + I₃.

Derivation: I₁ = V/R₁, I₂ = V/R₂, I₃ = V/R₃, and I = V/R. Substituting into I = I₁ + I₂ + I₃ and dividing by V:

1/R = 1/R₁ + 1/R₂ + 1/R₃

• For two resistors: R = R₁R₂ / (R₁ + R₂) (product over sum). This is quicker than adding fractions; use it whenever you have only two branches.
• For N equal resistors: R = R₀ / N.
• The equivalent resistance is smaller than the smallest resistor. The smallest resistor carries the largest current (currents divide inversely with resistance: I₁ : I₂ = R₂ : R₁). The water analogy fits: the widest pipe (smallest resistance) carries most of the flow, and adding another pipe always makes it easier for the water to get through, so the total resistance falls.
• Advantage: each branch works independently, which is why household appliances are connected in parallel: each gets the full mains voltage, and switching one off does not affect the others.
• For two resistors, the current in each branch can be found from the main current without finding the voltage: I₁ = I × R₂ / (R₁ + R₂) and I₂ = I × R₁ / (R₁ + R₂).