Resistors in Parallel (Cambridge International A-Level Physics)

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:

Resistors in Parallel

• When two or components are connected in parallel: The reciprocal of the combined resistance is the sum of the reciprocals of the individual resistances

Resistors in parallel

Resistors connected in parallel have the same voltage

• The equation for the combined resistance, R of resistors in parallel is:

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

• 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

Maths tip

• The reciprocal of a value is $\frac{1}{value}$

• For example, the reciprocal of a whole number such as 2 equals $\frac{1}{2}$

  • The reciprocal of $\frac{1}{2}$ is 2

• If the number is already a fraction, the numerator and denominator are ‘flipped’ round

Reciprocals

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 $\frac{1}{your answer}$
• You can also use the reciprocal button on your calculator (labelled either x^-1 or $\frac{1}{x}$, depending on your calculator

Worked Example The circuit below shows 3 resistors connected in parallel.

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

A.$\frac{5R}{2}$

B. $\frac{2}{5R}$

C.$\frac{5}{2R}$

D. $\frac{2R}{5}$

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

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

Step 3: Calculate the total resistance

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

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

Exam Hint The most common mistake is to leave the answer as 1/R_T. Remember to calculate $\frac{1}{answer}$ to get the value of R_T.