5.1.3. Parallel Circuits¶

$\draw (0,0) to[voltage source, v=$U$] (0,3) to[short, i=$I$] (2,3) to[R=$R_1$, i=$I_1$] (2,0) to[short] (0,0); \draw (2,3) to[short] (4,3) to[R=$R_2$, i=$I_2$] (4,0) to[short] (2,0); \draw (5,0) to (5,0);$

Circuit with two resistors in parallel

Consider a circuit with two resistors in parallel. It is a closed circuit and current will flow. The voltage drop over both resistors is the same and equal to the battery voltage $U$. Using Ohm’s law we then can calculate the current flowing through the resistors:

\[\begin{split}I_1 = \frac{U}{R_1}\\ I_2 = \frac{U}{R_2}\end{split}\]

The total current flowing from the battery through the circuit is the sum of $I_1$ and $I_2$:

\[I = I_1 + I_2 = \frac{U}{R_1} + \frac{U}{R_2}\]

i.e. the two resistors in parallel act as one resistor with resistance $R$ such that

\[\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}\]

Our analysis above can be done for an arbitrary number of resistors in a parallel circuit.

In a parallel connection of resistors the voltage across all resistors is the same, the current flowing through each resistor is dependent on its resistance in relation to all other resistances.