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Combining Resistors

Resistors in Series

A battery is connected to two light bulbs in series. This means that the current flows through one light bulb first, and then the other. The current feels resistance in each light bulb. We want to find the combined resistance of the two light bulbs.

To make things simple, we represent each light bulb by a rectangular box. This is the standard way to represent a resistance. The rectangular box symbol is called a resistor.

The purpose is to find a value for the combined resistance. Before we can do this, we need a very clear meaning for combined resistance: It is the resistance of a single resistor that gives the same current as the two original resistors.

Suppose that the resistances of the two original resistors are R₁ and R₂. The combined resistance is given by this formula:

R = R₁ + R₂.

To see why, consider the potential difference across each resistor. Let these be V₁ and V₂ respectively. Since the resistors are in series, the current through both of them must be the same. Applying the Ohm's law to both of them gives

V₁ = IR₁ and V₂ = IR₂.

Adding left side to left side, and right side to right side, gives

V₁ + V₂ = IR₁ + IR₂.

(V₁ + V₂) is just the potential difference V across the two resistors. So

V = IR₁ + IR₂.

Dividing both sides by I gives

V / I = R₁ + R₂.

Comparing this with the Ohm's law formula V/I = R, we see that the right side gives the combined resistance:

R = R₁ + R₂.

Resistors in Parallel

Next, consider another case where the two light bulbs are connected in parallel to the battery. This means that current from the battery splits into two parts, each part going through a different light bulb. Again, we represent the resistance of each light bulb by a resistor symbol.

In this case, the currents through the the resistors can be different. Let the currents be I₁ and I₂ respectively. Since both resistors are connected to the common points A and C, the potential difference V across each resistor is the same. Write the Ohm's law in the form I = V/R, and apply to each resistor:

I₁ = V / R₁ and I₂ = V / R₂.

Adding these gives:

I₁ + I₂ = V(1 / R₁ + 1 / R₂).

The left side is the sum of currents in the resistors, which is equal to the original current I before it splits. So

I = V(1 / R₁ + 1 / R₂).

Compare this with the Ohm's law formula for the combined resistance: 

I = V/R. 

Comparing the right sides of the two equations above gives

1/R = 1/R₁ + 1/R₂