A parallel circuit contains four resistors. The total resistance of the circuit is 120 Ω. Three of the resistors have values of 820 Ω, 750 Ω, and 470 Ω. What is the value of the fourth resistor?
The following is the given information:
Resistance, \(\mathrm{R}_{1}=820 \Omega\)
Resistance, \(\mathrm{R}_{2}=750 \Omega\)
Resistance, \(\mathrm{R}_{3}=470 \Omega\)
Total resistance of the circuit is,
\(\mathrm{R}_{T}=120 \Omega\)
The total resistance of a parallel circuit is the reciprocal of the sum of the reciprocals of the
individual branches.
\(\frac{1}{\mathrm{R}_{T}}=\frac{1}{\mathrm{R}_{1}}+\frac{1}{\mathrm{R}_{2}}+\frac{1}{\mathrm{R}_{3}}+\frac{1}{\mathrm{R}_{4}}\)
\(\frac{1}{120 \Omega}=\frac{1}{820 \Omega}+\frac{1}{750 \Omega}+\frac{1}{470 \Omega}+\frac{1}{R_{4}}\)
\(0.00833=0.001219+0.00133+0.002127+\frac{1}{\mathrm{R}_{4}}\)
\(0.00833=0.004676+\frac{1}{\mathrm{R}_{4}}\)
\(\frac{1}{\mathrm{R}_{4}}=0.00833-0.004676\)
\(\frac{1}{\mathrm{R}_{4}}=0.003654\)
\(\mathrm{R}_{4}=\frac{1}{0.003654}\)
\(\mathrm{R}_{4}=273.6728 \Omega\)
Thus, fourth resistor value is,
\(\mathrm{R}_{4}=273.6728 \Omega\)