CAIE A-Level Physics 10.2.5 Deriving Parallel Resistance

Practise deriving the parallel-resistance relationship from common potential difference and branch currents using Kirchhoff’s laws.

Exam points

derive the equivalent resistance of resistors in parallel from Kirchhoff’s laws
show how branch currents and common p.d. combine to obtain the parallel formula

[Maximum number: 2]

Three resistors of resistances $R_{1}, R_{2}$ and $R_{3}$ are connected as shown in Fig. 6.1.

Fig. 6.1

The currents in the resistors are $I_{1}, I_{2}$ and $I_{3}$ . The total current in the combination of resistors is I and the potential difference across the combination is V.

Show that the total resistance R of the combination is given by the equation

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

$I=I_{1}+I_{2}+I_{3}$

B1

$\begin{aligned} (V / R)=\left(V / R_{1}\right)+\left(V / R_{2}\right)+\left(V / R_{3}\right) \text { or } (I / V)=\left(I_{1} / V\right)+\left(I_{2} / V\right)+\left(I_{3} / V\right) \text { and (so) } 1 / R=1 / R_{1}+1 / R_{2}+1 / R_{3} \end{aligned}$

A1

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10.2.5—Kirchhoff’s laws, a formula for the combined resistance of two or more question 1