Resistors in Parallel

Deriving the equation for resistors in parallel

In a parallel circuit, the reciprocal of the combined resistance of two or more resistors is the sum of the reciprocal of the individual resistances
In a parallel circuit:
- The current is split at the junction (and therefore between resistors)
- The potential difference is the same through all resistors
The equation for combined resistors in parallel is derived using Kirchhoff’s laws:

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Derivation of resistors in parallel (2), downloadable AS & A Level Physics revision notes

The reciprocal of the combined resistance is the sum of the reciprocals of the individual resistances

Resistors in parallel diagram, downloadable AS & A Level Physics revision notes

Resistors connected in parallel have the same voltage

$\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} . . .$

This means the combined resistance decreases and is less than the resistance of any of the individual components
For example, If two resistors of equal resistance are connected in parallel, then the combined resistance will halve

The reciprocal of a value is $\frac{1}{v a l u e}$
For example, the reciprocal of a whole number such as 2 equals
- The reciprocal of $\frac{1}{2}$ is 2
If the number is already a fraction, the numerator and denominator are ‘flipped’ round

Reciprocals, downloadable AS & A Level Physics revision notes

The reciprocal of a number is 1 ÷ number

In the case for the resistance R, this becomes $\frac{1}{R}$ . To get the value of R from $\frac{1}{R}$ , you must calculate $\frac{1}{y o u r a n s w e r}$
You can also use the reciprocal button on your calculator (labelled either x^-1 or $\frac{1}{x}$ , depending on your calculator

The circuit below shows 3 resistors connected in parallel.

Which value gives the combined resistance of all the resistors in this circuit?

A. $\frac{5 R}{2}$ B. $\frac{2}{5 R}$ C. $\frac{5}{2 R}$ D. $\frac{2 R}{5}$

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Answer: D

Step 1: Resistors in parallel equation

$\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + . . . . = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}$

Step 2: Substitute in the values given from the equation

$\frac{1}{R_{T}} = \frac{1}{R} + \frac{1}{2 R} + \frac{1}{R} = (1 + \frac{1}{2} + 1) \frac{1}{R} = \frac{5}{2 R}$

Step 3: Calculate the total resistance

$\frac{1}{R_{T}} = \frac{5}{2 R}$

$R_{T} = \frac{2 R}{5}$

The most common mistake is to leave the answer as 1/R_T. Remember to calculate $\frac{1}{a n s w e r}$ to get the value of R_T.