Resistors can be connected in two ways in a circuit. The connections are:

**1**. Connection of resistors in series.

**2.** Connection of resistors in parallel.

### Resistance in Series:

When three resistors, R_{1}, R_{2} and R_{3} are connected in series, i.e. one end to another end to form a linear network, the same amount of current flows through them when they are connected to a source of e.m.f.

The total resistance, R (effective resistance) is given by V = IR_{eff}

From the circuit diagram, the voltage V is connected across the three resistors. This voltage is shared across each of the resistors.

Remember that the voltages drop across each of the resistors are different because the resistances of the resistors are different. Therefore, Let the Voltage of the cell connected across the three resistors be V.

Let the voltage drop across resistor R_{1} be V_{1}.

Let the voltage drop across resistor R_{2} be V_{2}, also,

Let the voltage drop across resistor R_{3} be V_{3}.

Therefore the total voltage can then be found;

V = V_{1} + V_{2} + V_{3}

IR_{eff} = \( \scriptsize \pm R1 \pm R2 \pm R3 \)

Using Ohmâ€™s law:

V = IR_{eff}

V_{1 }= IR_{1}

V_{2} = IR_{2}

V_{3} = IR_{3}

Since the same current flows through the resistors,

Then IR_{eff }= IR_{1 }+ IR_{2} + IR_{3}

Factorising I we then have:

IR_{eff }= I(R_{1} + R_{2} + R_{3})

We then divide both sides by I

= \( \frac {IR_{eff}}{I} = \frac {I \left ( R_1 + R_2 + R_3 \right)}{I} \)

âˆ´ R_{eff } = R_{1} + R_{2} + R_{3}

The total resistance is greater than the highest resistance in the circuit.

### Example 1:

Two resistors of value 3Î© and 5Î© are connected in series. Find the effective resistance.

**Solution:**

R_{total }= R_{1} + R_{2}

** **= 3Î© + 5Î©

= 8Î©

### Example 2:

From the diagram below, calculate the total resistance of the circuit.

**Solution:**

From the diagram, R_{1} = 8 Î©, R_{2} = 5 Î©, R_{3} = 13 Î©.

Formula: R_{total }= R_{1} + R_{2} + R_{3}

Substitution: R_{total }= 8Î© + 5Î© + 13Î©

â‡’ Total resistance of the circuit R_{total }= 26Î©

### Example 3:

7 Ohms, 9 Ohms and R resistors are connected in series. If the total resistance of the circuit is 45, calculate the value of the third resistor R.

**Solution:**

**Data given in the question:**

R_{1} = 7 ohms, R_{2} = 9 ohms, R_{3} = ?

**Formula: **R_{total }= R_{1} + R_{2} + R_{3}

**Substitute the values: **

45 = 7 + 9 + R_{3}.

â‡’ 45 = 16 + R_{3}

**Make R _{3} the subject: **

R_{3} = 45 – 16.

â‡’ R = 29 Î©

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