Electricity and magnetism

Series and parallel circuits

A series circuit carries the same current through its components; parallel branches share the same voltage. These rules determine equivalent resistance.

Resistors may lie along one path (series) or in separate branches connected to the same two nodes (parallel). Identify circuit nodes first: series elements have no branch between them, while parallel elements share both endpoint nodes.

\text{Series: }R_{\rm eq}=R_1+R_2,\quad I_1=I_2=I;\qquad \text{Parallel: }\frac1{R_{\rm eq}}=\frac1{R_1}+\frac1{R_2},\quad U_1=U_2=U

Definition: Current and voltage in combinations

Series: current is the same and the source voltage equals the sum of resistor drops. Parallel: each branch has the same voltage and total current is the sum of branch currents. For two parallel resistors, $R_{\rm eq}=R_1R_2/(R_1+R_2)$ .

Compare series and parallel circuits; track conventional current, opposite electron drift, and branch measurements.

Current conservation at a junction

At a branching node, charge does not steadily accumulate, so total current entering equals total current leaving. In an unbranched series circuit, the same current passes through every component. Electrons are not consumed by a resistor.

Example: Two parallel resistors

Resistors of $6\,\Omega$ and $3\,\Omega$ are in parallel across a $12$ V source. Find the equivalent resistance and total current.

Solution

$R_{\rm eq}=6\times3/(6+3)=2\,\Omega$ . Thus $I=U/R_{\rm eq}=12/2=6$ A. The branches carry $2$ A and $4$ A, adding to $6$ A.

A mixed network can be solved by reducing series or parallel groups step by step until one equivalent resistance remains. For $4\,\Omega$ and $8\,\Omega$ in series, $R_{eq}=12\,\Omega$ ; across a $12$ V source the current is $1$ A, so the voltage drops are $4$ V and $8$ V. Their sum equals the source voltage, as required by the loop rule.

After reducing a network, check each branch again with Ohm’s law. Parallel resistors share voltage, but the lower-resistance branch carries more current; branch currents must sum to the current entering the junction.

Quick check

In a series circuit, which quantity is the same through every resistor?

Two equal resistors $R$ in parallel have equivalent resistance:

References

Young, Freedman (2019). University Physics