Combination Of Resistances

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.

Series Combination
Series Combination

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 R1, R2, R3 …, etc. are combined in series, then the equivalent resistance (R) is given by,

R = R1 + R2 + R3 + …                  ….(i)

Derivation of mathematical expression of resistances in series combination 

Let, R1, R2 and R3 be the resistances connected in series, I be the current flowing through the circuit, i.e., passing through each resistance, and V1, V2 and V3 be the potential difference across R1, R2 and R3, respectively. Then, from Ohm’s law,

V1 = IR1, V2 = IR2 and V3 = IR3 …(ii)

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

V = V1 + V2 + V3                  …(iii)

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

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

IR = V = V1 + V2 + V3

= IR1 + IR2 + IR3

or,        IR = I (R1 + R2 + R3)

or,        R = R1 + R2 + R3

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

Some results about series combination :

  1. 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.
  2. When two or more resistors are connected in series, the same current flows through each resistor.
  3. 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.

Parllel Combination
Parllel Combination

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 R1,I1 = V/R1    …(i)

Current passing through R2,I2 = V/R2    …(ii)

Current passing through R3,I3 = V/R3     …(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 = I1 + I2 + I3                    …(v)

Substituting the values of I,I1,I2 and I3 in Eq. (v),

                                \frac{{\rm{V}}}{{\rm{R}}} = \frac{{\rm{V}}}{{{{\rm{R}}_{\rm{1}}}}} + \frac{{\rm{V}}}{{{{\rm{R}}_{\rm{2}}}}} + \frac{{\rm{V}}}{{{{\rm{R}}_{\rm{3}}}}}...\left( {vi} \right)

Cancelling common V term, one gets

                                \frac{{\rm{1}}}{{\rm{R}}} = \frac{{\rm{1}}}{{{{\rm{R}}_{\rm{1}}}}} + \frac{{\rm{1}}}{{{{\rm{R}}_{\rm{2}}}}} + \frac{{\rm{1}}}{{{{\rm{R}}_{\rm{3}}}}}

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 = v2 = v3 = 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.