18.3 - Parallel Circuits

Parallel circuits and independence

In a parallel circuit, each component sits on its own loop, or branch, connected separately to the supply.

Components work independently

• Because the branches are separate, components work independently - if one is disconnected, the others keep working.
• This is why parallel circuits are used in cars and house lighting, where lamps and appliances must be switched on and off separately.

Mixed circuits

When components are on the same branch in a mixed circuit, the series-circuit rules apply to that branch.

Potential difference and current rules

Potential difference is the same across every branch

• Each branch gets the full supply potential difference, so the p.d. is the same across all branches: $V_1=V_2=V_3=\dots$
• Because each charge travels down only one branch, it transfers all its energy to the components on that branch.
• Identical bulbs connected in parallel are therefore all equally bright.

The branch currents add up to the total current

• The total current equals the sum of the currents in the separate branches: $I_{\text{total}}=I_1+I_2+\dots$
• At any junction, the total current going in equals the total current coming out, because charge is conserved.

Current in a single branch

• The current in a branch is $I=\frac{V}{R}$, where $V$ is the supply p.d. and $R$ is that branch's resistance.
• The branch with the lowest resistance carries the largest current.
• Changing the resistance of one branch only affects the current in its own branch; the other branches are unchanged.

Why adding a resistor in parallel lowers resistance

• The total resistance of a parallel circuit is always less than the resistance of any individual branch.
• Adding another resistor in parallel lowers the total resistance further.

The reasoning

1. Every branch has the full source p.d. across it.
2. Adding another branch gives the current an extra path, so the current in the existing branches stays the same while new current is drawn.
3. This increases the total current drawn from the supply.
4. Since $R=\frac{V}{I}$ and $V$ is constant, an increase in total current means total resistance decreases.

Formula for total resistance in parallel

Where:

• $R_T$ = Total resistance of the parallel branches, in ohms (Ω)

• $R_1$ = Resistance of the first branch, in ohms (Ω)

• $R_2$ = Resistance of the second branch, in ohms (Ω)

• Add the reciprocals of the branch resistances, then take the reciprocal of the result to find $R_T$.

• The total resistance is always smaller than the smallest individual branch resistance.

Combining with the series rule

For circuits with both series and parallel parts:

1. Calculate the resistance of each branch using the series rule $R=R_1+R_2+\dots$
2. Combine the parallel branches using the reciprocal formula.

Worked example - Total resistance in a hybrid circuit

A circuit is connected across an 18 V supply. Branch 1 has a 4 Ω resistor and an 8 Ω resistor in series. Branch 2 is a single 36 Ω resistor. Find the resistance of branch 1, the total resistance of the circuit, and the total current drawn from the supply.