Table of Contents

**Combination Of Resistances**

**Series Combination**

When two or more resistances are joined end-to-end so that the same current flows through each of them, they are said to be connected in series.

When a series combination of resistances is connected to a battery, the same current (I) flows through each of them.

**Law of combination of resistances in series **

The law of combination of resistances in series states that when a number of resistances are connected in series, their equivalent resistance is equal to the sum of the individual resistances*. *Thus, if R_{1}, R_{2}, R_{3} …, etc. are combined in series, then the equivalent resistance (R) is given by,

R = R_{1} + R_{2} + R_{3} + … ….(i)

**Derivation of mathematical expression of resistances in series combination **

Let, R_{1}, R_{2} and R_{3} be the resistances connected in series, I be the current flowing through the circuit, i.e., passing through each resistance, and V_{1}, V_{2} and V_{3} be the potential difference across R_{1,} R_{2} and R_{3}, respectively. Then, from Ohm’s law,

V_{1} = IR_{1}, V_{2} = IR_{2} and V_{3} = IR_{3} …(ii)

If, V is the potential difference across the combination of resistances then,

V = V_{1} + V_{2} + V_{3} …(iii)

If, R is the equivalent resistance of the circuit, then V = IR …(iv)

Using Eqs. (i) to (iv) we can write,

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

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

or, IR = I (R_{1} + R_{2} + R_{3})

or, R = R_{1} + R_{2} + R_{3}

Therefore, when resistances are combined in series, the equivalent resistance is higher than each individual resistance.

**Some results about series combination :**

- When two or more resistors are connected in series, the total resistance of the combination is equal to the sum of all the individual resistances.
- When two or more resistors are connected in series, the same current flows through each resistor.
- When a number of resistors are connected in series, the voltage across the combination (i.e. voltage of the battery in the circuit), is equal to the sum of the voltage drop (or potential difference) across each individual resistor.

**Parallel Combination **

When two or more resistances are connected between two common points so that the same potential difference is applied across each of them, they are said to be connected is parallel*.*

When such a combination of resistance is connected to a battery, all the resistances have the same potential difference across their ends.

**Derivation of mathematical expression of parallel combination :**

Let, V be the potential difference across the two common points A and B. Then, from Ohm’s law

Current passing through R_{1},I_{1} = V/R_{1} …(i)

Current passing through R_{2},I_{2} = V/R_{2} …(ii)

Current passing through R_{3},I_{3} = V/R_{3} …(iii)

If R is the equivalent resistance, then from Ohm’s law, the total current flowing through the circuit is given by,

I = V/R …(iv)

and I = I_{1} + I_{2} + I_{3} …(v)

Substituting the values of I,I_{1},I_{2} and I_{3} in Eq. (v),

Cancelling common V term, one gets

The equivalent resistance of a parallel combination of resistance is less than each of all the individual resistances.

**Important results about parallel combination :**

(i) Total current through the circuit is equal to the sum of the currents flowing through it.

(ii) In a parallel combination of resistors the voltage (or potential difference) across each resistor is the same and is equal to the applied voltage i.e. v1 = v_{2} = v_{3} = v :

(iii) Current flowing through each resistor is inversely proportional to its resistances, thus higher the resistance of a resistors, lower will be the current flowing through it.