next up previous index
Next: Kirchhoff's Laws Up: Circuits Previous: Resistors in Series

Resistors in Parallel

Suppose now in a section of a circuit we encounter a combination of two resistors as in Fig. 17.5.
Figure 17.5: Two resistors in parallel
\epsfxsize=4 in

These resistors are said to be in parallel, and as before, it is possible to consider them as one single equivalent resistor R eq . To find this equivalent resistor, we exploit the fact from charge conservation that I = I1 + I2 . Using again Ohm's law (17.3), as well as the point that the potential difference across R1 and R2 is the same, we find this becomes

$\displaystyle{\frac{V_{ab}}{R_{\:\rm eq}}}$ = $\displaystyle{\frac{V_{ab}}{R_1}}$ + $\displaystyle{\frac{V_{ab}}{R_2}}$ $\displaystyle\rightarrow$ $\displaystyle{\textstyle\frac{1}{R_{\:\rm eq}}}$ = $\displaystyle{\textstyle\frac{1}{R_1}}$ + $\displaystyle{\textstyle\frac{1}{R_2}}$ (9)

This also is readily extended to the case of multiple resistors R1,R2,...,RN in parallel:

$\displaystyle{\textstyle\frac{1}{R_{\:\rm eq}}}$ = $\displaystyle{\textstyle\frac{1}{R_1}}$ + $\displaystyle{\textstyle\frac{1}{R_2}}$ +...+ $\displaystyle{\textstyle\frac{1}{R_N}}$ (10)