What is the total resistance in parallel circuits?

What is the total resistance in parallel circuits?

• A. 1/R total = 1/R1 + 1/R2 + 1/R3 + ...
• B. R total = R1 + R2 + R3
• C. R total = (R1 * R2) / (R1 + R2)
• D. R total is the maximum of the resistors

Answer: (A)

In parallel, the voltage across each resistor is the same and currents add up. Each branch carries I_n = V/R_n, so the total current is I_total = sum(V/R_n) = V * sum(1/R_n). The total resistance is R_total = V/I_total = 1 / [sum(1/R_n)]. That means the reciprocal of the total resistance equals the sum of the reciprocals of each resistor: 1/R_total = 1/R1 + 1/R2 + 1/R3 + ... This general rule shows why the total resistance in parallel is lower than any individual resistor. For just two resistors, this also yields R_total = (R1*R2)/(R1+R2), which is a specific case of the general formula. The other options describe series behavior or misinterpret the parallel setup.

In parallel, the voltage across each resistor is the same and currents add up. Each branch carries I_n = V/R_n, so the total current is I_total = sum(V/R_n) = V * sum(1/R_n). The total resistance is R_total = V/I_total = 1 / [sum(1/R_n)]. That means the reciprocal of the total resistance equals the sum of the reciprocals of each resistor: 1/R_total = 1/R1 + 1/R2 + 1/R3 + ... This general rule shows why the total resistance in parallel is lower than any individual resistor.

For just two resistors, this also yields R_total = (R1*R2)/(R1+R2), which is a specific case of the general formula. The other options describe series behavior or misinterpret the parallel setup.