Combination of Resistance
There are two ways of connecting the resistance that is, the series combination and is parallel combination.
Series Combination
The combination, in which the resistance are connected end to end with each other, is called series combination.
Let \[{{R}_{1}},\,\,{{R}_{2}}\] and \[{{R}_{3}}\] be three resistance, connected in series across a battery of potential V volt, as shown in the figure above. Now suppose \[{{V}_{1}}\] be the potential difference across the resistance \[{{R}_{1}};\,\,{{V}_{2}}\] be the potential difference across \[{{R}_{2}};\] and \[{{V}_{3}}\] be the potential difference across \[{{R}_{3}}\]. The total potential across the three resistance is given by
\[\mathbf{V=}{{\mathbf{V}}_{\mathbf{1}}}\mathbf{+}{{\mathbf{V}}_{\mathbf{2}}}\mathbf{+}{{\mathbf{V}}_{\mathbf{3}}}\]
But by Ohms law, \[V=I\times R\]
Since the same current I flows through the three resistance \[{{R}_{1}},\,\,{{R}_{2}}\]and \[{{R}_{3}}\], so by ohms law
\[{{V}_{1}}=I\times {{R}_{1}},\,\,{{V}_{2}}=I\times {{R}_{2}}\]and\[{{V}_{3}}=I\times {{R}_{3}}\]
Therefore, \[I\times R=I\times {{R}_{1}}+I\times {{R}_{2}}+I\times {{R}_{3}}\]
\[\Rightarrow \,\,I\times R=I\times \,({{R}_{1}}+{{R}_{2}}+{{R}_{3}})\,\Rightarrow \,\,R={{R}_{1}}+{{R}_{2}}+{{R}_{3}}\]
Hence, the resultant resistance is equivalent to the sum of all individual resistance connected in series.
Characteristics
- If different resistances are joined with each other in such a way that there is only one path for the flow of electric current, then the combination of such resistances is called Series Combination.
- In series combination current through each resistor is constant.
- In series combination potential difference across each resistor is different depending upon the value of resistance.
- Equivalent resistance of circuit is equal to the sum of individual resistances. The disadvantage of the series combination is that if one component is fused, then the other components of circuit will not function.
Parallel Combination
When all the resistance are connected between the two common ends of the circuits, it is called a parallel connection. Let \[{{R}_{1}},\,\,{{R}_{2}}\] and \[{{R}_{3}}\] be the three resistance connected in parallel, as shown in the circuit diagram given above, across a potential difference of V volt. In this case, the potential difference across the ends of all the three resistance will remain same. However the current flowing through each resistance will not be the same. Suppose \[{{I}_{1}},\,\,{{I}_{2}}\] and \[{{I}_{3}}\] be the current flowing through the three resistance \[{{R}_{1}},\,\,{{R}_{2}}\] and \[{{R}_{3}}\] respectively. Then the total current flowing in the circuit is
\[I={{I}_{1}}+{{I}_{2}}+{{I}_{3}}\]
But by Ohm's law, \[I=\frac{V}{R}\]
Therefore, \[{{I}_{1}}=\frac{V}{{{R}_{1}}},\,\,{{I}_{2}}=\frac{V}{{{R}_{2}}}\] and \[{{I}_{3}}=\frac{V}{{{R}_{3}}}\]
Putting these values in the above equation we get,
\[\frac{V}{R}=\frac{V}{{{R}_{1}}}+\frac{V}{{{R}_{2}}}+\frac{V}{{{R}_{3}}}\]
\[\Rightarrow \,\frac{V}{R}=V\,\left( \frac{1}{{{R}_{1}}}+\frac{1}{{{R}_{2}}}+\frac{1}{{{R}_{3}}} \right)\] \[\Rightarrow \,\,\frac{1}{R}=\frac{1}{{{R}_{1}}}+\frac{1}{{{R}_{2}}}+\frac{1}{{{R}_{3}}}\]
Hence the equivalent resistance is equal to the sum of reciprocal of individual resistance.
Characteristics
- If there are more than one path for the flow of current in a circuit then the combination of resistances is called Parallel Combination.
- In parallel combination current through each resistor is different.
- Potential difference across each resistor is constant.
- Equivalent resistance of circuit is always less than more...