Question

A wire of resistance R1 is cut into five equal pieces. These five pieces of wire are then connected in parallel. If the resultant resistance of this combination be R2, then the ratio R1R2 is
(a) 125
(b) 15
(c) 5
(d) 25; A wire of resistance R1 is cut into five equal pieces. These five pieces of wire are then connected in parallel. If the resultant resistance of this combination be R2, then the ratio R1R2 is
(a) 125
(b) 15
(c) 5
(d) 25

Answer 1

The ratio R1: R2 is Option (d) i.e. 25.

Explanation:

Given : A wire of resistance R1 is cut into five equal pieces.

Then each piece has a resistance of

Then each piece has a resistance of $\frac{R1}{5}$

And given that the resultant resistance of this combination be R2.

Then the parallel combination is given by

$\frac {1} {R2} = \frac {1} { \frac{R1}{5} } + \frac {1} { \frac{R1}{5} } + \frac {1} { \frac{R1}{5} } + \frac {1} { \frac{R1}{5} } + \frac {1} { \frac{R1}{5} } \:$

$\frac{1}{R2} = \frac{5}{R1} + \frac{5}{R1} + \frac{5}{R1} + \frac{5}{R1} + \frac{5}{R1} \:$

$\frac{1}{R2} = \frac{25}{R1}$

$\frac{R1}{R2} = 25$

The ratio R1: R2 is Option (d) i.e. 25.

Answer 2

Ratio of Resistances is 1:25

Explanation:

• The Resistance of each part will be $\frac {R}{5}$

• Equivalent Resistance will be  

$\frac {1}{R'} = \frac {5}{R} + \frac {5}{R} +\frac {5}{R} + \frac {5}{R} + \frac {5}{R}$ ""(1)

As per Parallel Connection of Resistances -

$\frac {1}{R_P} = \frac {1}{R_1} + \frac {1}{R_2} + \frac {1}{R_3} .... \frac {1}{R_n}$

Using relation (1)-

$R' = \frac {R}{25}$

$\frac {R'}{R} = \frac {1}{25}$