Resistors in parallel
Current
When resistors are connected in parallel, the supply current is equal to the sum of the currents through each resistor. In other words the currents in the branches of a parallel circuit add up to the supply current.
\(I_{S}=I_{1}+I_{2}+I_{3}\)
This relationship expresses the law of conservation of charge. All electrons that set out from the supply must return to the supply and each electron can only pass through one parallel branch.
Potential difference
When resistors are connected in parallel, they have the same potential difference across them. In other words, any components in parallel have the same potential difference across them.
So in the circuit above \(V_{S}=V_{1}=V_{2}=V_{3}\)
Resistance
For the circuit above, the formula for finding the total resistance of resistors in parallel is \(\frac{1}{{{R_P}}} = \frac{1}{{{R_1}}} + \frac{1}{{{R_2}}} + \frac{1}{{{R_3}}}\)
Question
The following resistor network is set up.
Calculate the total resistance of the network.
\(R_{1}=12\Omega\)
\(R_{2}=18\Omega\)
\(R_{3}=6\Omega\)
\(\frac{1}{{{R_T}}} = \frac{1}{{{R_1}}} + \frac{1}{{{R_2}}} + \frac{1}{{{R_3}}}\)
\(\frac{1}{{{R_T}}} = \frac{1}{{{12}}} + \frac{1}{{{18}}} + \frac{1}{{{6}}}\)
\(\frac{1}{{{R_T}}} = \frac{3}{{{36}}} + \frac{2}{{{36}}} + \frac{6}{{{36}}}\)
\(\frac{1}{{{R_T}}} = \frac{11}{{{36}}}\)
\(11R_{T}=36\)
\(R_{T}= \frac{{36}}{{11}}\)
\(R_{T}= 3.27\Omega\)
The total resistance is \(3.27 \Omega\)