Judge the equivalent resistance when the following are connected in parallel:
(a) 1 Ω and 106 Ω
(b) 1 Ω and 103 Ω, and 106 Ω
In a parallel connection, the equivalent resistance is given by
\(\frac{1}{R}\) = \(\frac{1}{R_1} \) + \(\frac{1}{R_2}\) + \(\frac{1}{R_3}\) + ........
(a) R1 = 1Ω and R2 = 106 Ω
\(\frac{1}{R}\) = \(\frac{1}{R_1} \) + \(\frac{1}{R_2}\)
\(\frac{1}{R}\) = \(\frac{1}{1} \) + \(\frac{1}{10^6}\)
\(\frac{1}{R}\) = \(\frac{10^6 + 1}{10^6}\)
R = \(\frac{10^6}{10^6 + 1}\)
R = \(\frac{10^6}{1000000 + 1}\)
R = \(\frac{10^6}{1000001}\)Ω
R = \(\frac{1000000}{1000001}\)
R ≈ 1Ω
Equivalent Resistance is R = \(\frac{10^6}{1000001}\)Ω
Approximately but less than 1Ω, because in a parallel combination of resistors, the equivalent resistance is less than the least resistance.
(b) R1 = 1Ω and R2 = 103 Ω and R3 = 106 Ω
\(\frac{1}{R}\) = \(\frac{1}{R_1} \) + \(\frac{1}{R_2}\) + \(\frac{1}{R_3}\)
\(\frac{1}{R}\) = \(\frac{1}{1} \) + \(\frac{1}{10^3}\)+ \(\frac{1}{10^6}\)
\(\frac{1}{R}\) = \(\frac{10^6 + 10^3 + 1}{10^6}\)
\(\frac{1}{R}\) = \(\frac{1000000 + 1000 + 1}{10^6 + 1}\)
\(\frac{1}{R}\) = \(\frac{1001001}{10^6}\)
R = \(\frac{10^6}{1001001}\)Ω
R = \(\frac{1000000}{1001001}\)
R ≈ 0.999Ω
Equivalent Resistance is R = \(\frac{10^6}{1001001}\)Ω