Resistors in Series and Parallel

Voltage across Resistors in Series and Parallel

We can learn more about resistors in series and parallel by considering the voltage drops across them when a current flows.

The total voltage drop across resistors connected in series is equal to the sum of the voltage drops across each resistor. For example, consider a circuit with a battery and two resistors of the same resistance \( R \) connected in series. If the voltage supplied by the battery is \( V \) then the voltage drop across each resistor will be \( \frac V2 \).

Fig. 4 - The total voltage drop across resistors in series is equal to the sum of the voltage drops across each resistor. The voltage drops can be measured by using voltmeters.

On the other hand, the same voltage will be supplied to each resistor for a circuit with a battery and two resistors connected in parallel. If the battery's voltage is \( V \), then the voltage drop across each resistor will also be \( V \).

Fig. 5 - All of the resistors in a parallel connection take all of the voltage from the battery.

These two cases generalize easily to any number of resistors. For resistors in series, the supplied voltage is given by

$$V=V_1+V_2+...+V_N,$$

where the subscripts indicate the resistor. For resistors in parallel:

$$V=V_1=V_2=...=V_N.$$

Current Through Resistors in Series and Parallel

The amount of current flowing through resistors is different when they are connected in series and when they are connected in parallel.

For resistors connected in series, the same current flows through all of them, as there are no junctions between the resistors where the current can split.

$$I=I_1=I_2=...=I_N.$$

For resistors connected in parallel, the current splits between them. For \( N \) resistors in parallel, the total current through them is given by

$$I=I_1+I_2+...+I_N.$$

Resistors in Series and Parallel Formulae

To find the formulae for the resistance of resistors in series and parallel, we need to use Ohm's law, which states that, for an Ohmic conductor, the relationship between its voltage, current and resistance is

$$V=IR.$$

Fixed resistors are Ohmic conductors and obey Ohm's law. For resistors in series, the supplied voltage is given in terms of the voltage drops across the resistors by

$$V=V_1+V_2+...+V_N.$$

The total resistance of the resistors will be given by the rearranged version of Ohm's law:

$$R_T=\frac VI.$$

The current is the same through each resistor for a series combination, so

$$R_T=\frac{V_1}{I}+\frac{V_2}{I}+...\frac{V_N}{I}.$$

Each term is simply the resistance of each resistor, so the total resistance for a series combination is equal to the sum of the resistances of the resistors!

$$R_T=R_1+R_2+...R_N.$$ The same process can be repeated to find the total resistance of a parallel combination. For resistors in parallel, the current is shared between them and the total current is equal to

$$I=I_1+I_2+...+I_N.$$

The total current will be equal to the voltage supplied divided by the total resistance:

$$I=\frac {V}{R_T}.$$

The voltage drop across each resistor is equal to \( V \) so the current through each resistor can be expressed in a similar way. For example, for the first resistor:

$$I_1=\frac{V}{R_1},$$

so the following expression can be written (both terms are equal the total current):

$$\frac {V}{R_T}=\frac{V}{R_1}+\frac{V}{R_2}+...+\frac{V}{R_N}.$$

The \( V \) on either side cancels out and leaves the equation for the resistance of a combination of parallel resistors.

$$\frac 1{R_T}=\frac{1}{R_1}+\frac{1}{R_2}+...+\frac{1}{R_N}.$$

Difference Between Resistors in Series and Parallel

There are several key differences between resistors in parallel and series connections:

• The total voltage drop across a series connection is equal to the sum of the individual voltage drops. There is the same voltage drop across each resistor in a parallel connection.
• The current is the same through each resistor in a series connection. The current is shared between the resistors in a parallel connection.
• Adding more resistors in series increases the total resistance because the current has to flow through each resistor. Adding more resistors in parallel decreases the total resistance because there are more paths for the current to pass through.

Rules for Resistors in Series and Parallel

The rules for resistors in series and parallel connections are summarized in the table below and should be remembered.

Equivalent Resistors in Series and Parallel

Consider two resistors in parallel that each have a resistance of \( 2R \), as shown in the circuit below.

Fig. 6 - Two \( 2R \) resistors placed in parallel have the same resistance as one resistor of resistance \( R \).

The total resistance of the combination can be calculated using the parallel resistors formula with \( N=2 \):

$$\frac 1{R_T}=\frac{1}{R_1}+\frac{1}{R_2}.$$

Both of the resistances are \( R \) so this becomes

$$\frac 1{R_T}=\frac{1}{2R}+\frac{1}{2R}=\frac 2R.$$

Rearranging this expression gives the total resistance as \( R_T=R \). This shows that two resistors with a resistance \( 2R \) connected in parallel have the same total resistance as a single resistor of resistance \( R \).

Examples of Resistors in Series and Parallel

The formulae for resistors in series and parallel connections can be useful in practice problems. In the following practice problems, the resistance of the battery will be assumed to be negligible.

Combining Resistors in Series and Parallel

Throughout this article, it may have seemed that we have been assuming that there is one resistor on each branch in parallel combinations. However, even if there is more than one, the series connection formula can be used to find the overall resistance of multiple resistors on a branch so that they can be treated as one resistor when using the parallel resistor formula.