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# RESISTANCES (OR RESISTORS) IN PARALLEL

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### RESISTANCES (OR RESISTORS) IN PARALLEL

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• The reciprocal of the combined resistance of a number of resistance connected in parallel is equal to the sum of the reciprocal of all the individual resistances. For ex. If no. of resistances are connected in parallel then their combined resistance R is given by formula

• When a number of resistances are connected in parallel then their combined resistance is less than the smallest individual resistance.

KEY POINTS

(1) When a number of resistances are connected in parallel then the potential difference across each resistance is the same which is equal to the voltage of the battery applied.

(2) When a number of resistances are connected parallel, then the sum of the current flowing through all the resistances is equal to the total current flowing in the circuit.

DERIVATION

Suppose the total current flowing in the circuit is I, then the current passing through resistance R1 will be I1 and the current passing through the resistance R2 will be I2 it is obvious that

Total current, I=I1+I2 $latex \_\_\_\_\_\_\left( 1 \right)$

Applying ohm’s law to whole circuit

$latex \text{I=}\frac{\text{V}}{\text{R}}\,\,\,\,\,\,\,\,\,\_\_\_\_\_\_\_\left( \text{2} \right)$

Applying separately to resistance $latex {{\text{R}}_{\text{1}}}$ and $latex {{\text{R}}_{\text{2}}}$

$latex \begin{array}{l}\text{I=}\frac{\text{V}}{{{\text{R}}_{\text{1}}}}\,\,\,\,\_\_\_\_\_\_\_\_\_\_\_\left( \text{3} \right)\\{{\text{I}}_{\text{2}}}\text{=}\frac{\text{V}}{{{\text{R}}_{\text{2}}}}\_\_\_\_\_\_\_\_\_\_\_\left( \text{4} \right)\end{array}$

Now putting values of $latex \text{I,}\,{{\text{I}}_{\text{1}}}$ and $latex {{\text{I}}_{\text{2}}}$ from equation (2) (3), (4) in equation,

We get

$latex \begin{array}{l}\frac{\text{V}}{\text{R}}\text{=}\frac{\text{V}}{{{\text{R}}_{\text{1}}}}\text{+}\frac{\text{V}}{{{\text{R}}_{\text{2}}}}\\\text{V}\left[ \frac{\text{1}}{\text{R}} \right]\text{=V}\left[ \frac{\text{1}}{{{\text{R}}_{\text{1}}}}\text{+}\frac{\text{1}}{{{\text{R}}_{\text{2}}}} \right]\end{array}$

Cancelling V from both sides

$latex \frac{\text{1}}{\text{R}}\text{=}\frac{\text{1}}{{{\text{R}}_{\text{1}}}}\text{+}\frac{\text{1}}{{{\text{R}}_{\text{2}}}}$

Thus, if two resistances $latex {{\text{R}}_{\text{1}}}$ and $latex {{\text{R}}_{\text{2}}}$ are connected in parallel, then their resultant resistance R is given by the formula:

$latex \frac{\text{1}}{\text{R}}\text{=}\frac{\text{1}}{{{\text{R}}_{\text{1}}}}\text{+}\frac{\text{1}}{{{\text{R}}_{2}}}$