13.7.1 Resistors in Parallel and Series

When resistors are connected in series, both resistors have the same current passing through them. And the PD across each resistor sum up to be the total PD. So

\displaystyle \displaystyle \begin{aligned}{{V}_{{total}}}&={{V}_{1}}+{{V}_{2}}\\I{{R}_{{eff}}}&=I{{R}_{1}}+I{{R}_{2}}\\{{R}_{{eff}}}&={{R}_{1}}+{{R}_{2}}\end{aligned}

When resistors are connected in parallel, both resistors have the same PD across their terminals. The currents through each resistor sum up to be the total current. So

\displaystyle \displaystyle \begin{aligned}{{I}_{{total}}}&={{I}_{1}}+{{I}_{2}}\\\frac{V}{{{{R}_{{eff}}}}}&=\frac{V}{{{{R}_{1}}}}+\frac{V}{{{{R}_{2}}}}\\\frac{1}{{{{R}_{{eff}}}}}&=\frac{1}{{{{R}_{1}}}}+\frac{1}{{{{R}_{2}}}}\end{aligned}

The derivation can be easily extended to more than two resistors.

Connecting N resistors (with resistances R1, R2, … RN) in series results in an effective resistance of

\displaystyle {{R}_{{eff}}}={{R}_{1}}+{{R}_{2}}+...{{R}_{N}}

Connecting N resistors (with resistances R1, R2, … RN) in parallel results in an effective resistance of

\displaystyle \frac{1}{{{{R}_{{eff}}}}}=\frac{1}{{{{R}_{1}}}}+\frac{1}{{{{R}_{2}}}}+...\frac{1}{{{{R}_{N}}}}

Video Explanation

Parallel and Series

Concept Test

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