6.2 Circuit Laws Applied to a Parallel Circuit

Now that you have seen some circuit laws in the Glossary, it is a good idea to go through them a little more, so that they can be understood in practice.

Let us begin with a simple circuit of two resistors in parallel. These resistors have been labeled respectively R₁ and R₂:

Since there are two junctions in the circuit, the current is probably not the same everywhere in the circuit. We will label the three branches of the circuit I₁, I₂ and I_T. Their directions are indicated on the following diagram:

Kirchoff’s Current Law states that the sum of the currents going into a junction is equal to the sum of the currents going out of a junction. Therefore, for this circuit, we can say that I_T (the current going into the junction) = I₁ + I₂ (the sum of the current leaving the junction.

We now move on to consider the p.d.’s of the various circuit components. The p.d. across any wire component is usually so small compared to the E.M.F. and other p.d.’s in the circuit that it is negligible, and therefore considered to be 0. However, the potential difference across the two resistors is sure to be greater. According to Ohm’s Law, V (the p.d. across a component) = R x I. Therefore, using the symbols we already have, the potential difference across the two resistors can be stated as R₁I₁ and R₂I₂. Now we ask you to recall Kirchoff’s Voltage Law: the sum of the E.M.F.’s of any given circuits equals the sum of the potential differences of the components in it. A parallel circuit has more than one route for current to take. Each of these routes is like a separate circuit. The circuit we have been discussing can be considered as circuit A (in green), and circuit B (in blue):

Since there is only one component in each circuit the p.d. across it must equal the E.M.F. of the battery. Therefore, Kirchoff’s Law for circuit A is E.M.F. (of the cell) = R₁I₁ , and for the second circuit it is E.M.F. = R₂I₂. Combining these two equations, we get E.M.F. = R₁I₁ = R₂I₂.

A third element of the circuit to be considered is resistance. In a series circuit, the total resistance of a circuit is the sum of the resistance of each of the components. However, this is not the case for a parallel circuit. The total resistance of a parallel circuit is equal to the voltage across the parallel segment divided by the current going through it. Or, in more mathematical terms, R_T = V_T/I_T. It is possible, however, to derive an equation for finding out the total resistance without having to know the two numbers V_T and I_T.

Looking at the resistance in our circuit again, we find R₁ = V₁/I₁ and R₂ = V₂/I₂, where V’s indicate the voltage across the resistors:

Solving for currents, we get I₁ = V₁ / R₁, and I₂ = V₂ / R₂. The currents of each of the branches can be added to get the total current, giving us I_T = V₁ / R₁ + V₂ / R₂. Now, since I_T = V_T/R_T, this can be substituted into the previous equation, leaving it only in terms of resistance and voltage. V_T / R_T = V₁ / R₁ + V₂ / R₂. For our circuit, we know that the voltage across the parallel segment, V_T, is equal to the E.M.F. of the power source, which in turn equals V₁ and V_2. Canceling these three from the equation leaves 1/ R_T = 1/ R₁ + 1/ R₂, or:

For n resistors in parallel, simply solve for the total resistance using 1/R_T = 1/R₁ + . . .+ 1/R_n.

By trying out some numbers in these equations, you may find that the total resistance is always less than each of the individual ones. This is because it is easier for charges to travel when there are more possible paths to travel on. The effect of this is that the overall resistance is decreased.