The total resistance in a parallel circuit is always less than the smallest branch resistance. Which statement is true?

The total resistance in a parallel circuit is always less than the smallest branch resistance. Which statement is true?

• A. True
• B. False
• C. It depends on values
• D. It equals the sum of branch resistances

Answer: (A)

In parallel circuits, adding more paths for current lowers the overall resistance. The total resistance is found from 1/R_eq = sum over all branches of 1/R_i. Since every branch resistance is a positive number, each 1/R_i is positive, and the sum is larger than any single 1/R_i. That makes R_eq smaller than the smallest branch resistance. For example, with two branches of 4 Ω and 6 Ω, the equivalent resistance is R_eq = (4×6)/(4+6) = 2.4 Ω, which is less than both 4 Ω and 6 Ω. In typical parallel networks with multiple finite branches, you always get a value smaller than the smallest branch. (If some branches are open, the equality can occur only if all but one branch are open, but with two or more finite paths, the equivalent resistance is strictly less.)

In parallel circuits, adding more paths for current lowers the overall resistance. The total resistance is found from 1/R_eq = sum over all branches of 1/R_i. Since every branch resistance is a positive number, each 1/R_i is positive, and the sum is larger than any single 1/R_i. That makes R_eq smaller than the smallest branch resistance. For example, with two branches of 4 Ω and 6 Ω, the equivalent resistance is R_eq = (4×6)/(4+6) = 2.4 Ω, which is less than both 4 Ω and 6 Ω. In typical parallel networks with multiple finite branches, you always get a value smaller than the smallest branch. (If some branches are open, the equality can occur only if all but one branch are open, but with two or more finite paths, the equivalent resistance is strictly less.)