What Is the Resistance of a Parallel Circuit?

Equivalent Resistance

The strategy for determining the current in a parallel circuit entails determining the equivalent, or effective, resistance of the parallel segment of the circuit. The question reduces to: If you replaced the parallel section with a resistor that left the current elsewhere in the circuit the same, what would its resistance be?

The formula for such a resistor is 1/Rt = 1/R1 + 1/R2 + .... Here Rt is the resistance of the equivalent resistor, or the total resistance. R1, R2 and so on are the resistance of the resistors that are parallel to each other. For example, if two resistors are parallel to each other and their resistances are 1 ohm and 2 ohms, their effective resistance would be the reciprocal of 1/1 + ½, which is 3/2.

Suppose the resistance before the current splits up into parallel paths and resistance after these paths recombine sum to Rs, denoting the series section's total resistance. Then the total resistance for the circuit is Rs + Rt.

You can then calculate the current through the series sections of the circuit using the equation V=I(Rs+Rt), where V represents the electromotive force (emf) propelling the current, I, of electrons. So if the circuit is connected to a 9-volt battery, Rs is 2 ohms, say, and the parallel resistors are as stated above, then I = 9/(2+1.5) = 2.57 amps, after rounding.

Suppose a circuit splits into two parallel branches at point A and recombine at point B. The potential drop, V, between A and B must be the same regardless of which of the two branches the electrons go down. If this were not the case, then current would increase or decrease through the two branches until the drop was even through both branches. In other words, electrons in one branch would exit the branch and have enough emf driving them to back up into the other branch. Suppose the resistor in each branch, R1 and R2, have currents i1 and i2 flowing through them. Then V=i1*R1 and V=i2*R2. The currents must sum to the current, i, before and after the circuit splits up into parallel branches. Therefore, i = i1 + i2 = V/R1 + V/R2 = V(1/R1 + 1/R2). So the equivalent resistance Rt, which has the property that V=i*Rt, can be found by combining these two equations. In other words, V/Rt = i = V(1/R1 + 1/R2). So 1/Rt = 1/R1 + 1/R2.