Which formula expresses the total resistance in parallel circuits?

Which formula expresses 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 a parallel circuit, the same voltage is across every resistor, so the current through each branch is I_i = V/R_i. The total current is the sum of those branch currents: I_total = V(1/R1 + 1/R2 + 1/R3 + ...). Since the total resistance relates to the total current by V = I_total R_total, it follows that R_total = V / I_total = 1 / (1/R1 + 1/R2 + 1/R3 + ...). This is why the total resistance in parallel is given by the reciprocal sum. This general rule covers any number of resistors in parallel. For two resistors, you can also use R_total = (R1*R2)/(R1+R2), which is a specific case of the same idea, but the general expression uses the sum of reciprocals. The other ideas don’t fit: adding resistances gives the total in series, while the total in parallel is always less than the smallest individual resistance, not the maximum.

In a parallel circuit, the same voltage is across every resistor, so the current through each branch is I_i = V/R_i. The total current is the sum of those branch currents: I_total = V(1/R1 + 1/R2 + 1/R3 + ...). Since the total resistance relates to the total current by V = I_total R_total, it follows that R_total = V / I_total = 1 / (1/R1 + 1/R2 + 1/R3 + ...). This is why the total resistance in parallel is given by the reciprocal sum.

This general rule covers any number of resistors in parallel. For two resistors, you can also use R_total = (R1*R2)/(R1+R2), which is a specific case of the same idea, but the general expression uses the sum of reciprocals. The other ideas don’t fit: adding resistances gives the total in series, while the total in parallel is always less than the smallest individual resistance, not the maximum.