Parallel circuits

What a parallel circuit is

• Components in parallel are connected across each other, providing separate paths for the charge.
• Each path is called a branch.
• If one branch breaks, the others keep working — the charge simply takes the remaining paths.

The three rules

Voltage is the same across every branch.

$$V_T = V_1 = V_2 = \dots$$

• Every branch is connected directly across the same two points, so each experiences the full supply voltage.

Currents add up to the total.

$$I_T = I_1 + I_2 + \dots$$

• The current splits at the junction and recombines afterward.
• This is conservation of charge: nothing is lost at the junction.
• The branch with the smallest resistance takes the largest current, because they all have the same voltage across them and $I = V/R$.

Resistances combine as reciprocals.

$$\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \dots$$

• The total resistance is always smaller than the smallest individual resistance.
• This is the result students find most surprising, and it is worth understanding rather than memorising: adding another branch gives the charge an extra route, making it easier for current to flow overall.
• For exactly two resistors there is a useful shortcut:

$$R_T = \frac{R_1R_2}{R_1 + R_2}$$

• For $n$ identical resistors of resistance $R$: $R_T = R/n$.

Build parallel and series combinations below and compare the currents in each branch:

Consequences worth knowing

• Adding another branch lowers $R_T$, so the total current drawn from the supply increases.
• Existing branches are unaffected — they still have the full supply voltage across them and the same current as before.
  • This is exactly why household wiring is parallel: switching on a jug does not dim the lights, and each appliance gets the full $230$ V.
• Each appliance can be switched independently, and one failure does not stop the rest.