Question

A wire of resistance R is cut into five equal pieces are connected in parallel and equivalent resistances of the combination are R’. Calculate their ratio

Answer 1

Answer:

Let the length of resistance R be L

Let the resistance of each part after cutting be R

s

Since resistance is proportional to the length of the resistor,

R

s

=R/5.....................(i)

For resistors in parallel,

R

′

1

=

R

1

1

+

R

2

1

+

R

3

1

+

R

4

1

+

R

5

1

⟹R

′

=R

s

/5................(ii)

From (i) and (ii),

R

′

=R/25

R/R

′

=25

Answer 2

Answer:

$\frac{R}{R´} = 25$

Explanation:

Let r be the resistance of the small pieces of wire.

IN SERIES

When the five pieces were a single wire(same as the five pieces connected in series), the resistance is R. So

$R = r + r + r + r +r$

$⇒R = 5r ----------(1)$

IN PARALLEL

When the 5 pieces are connected in parallel, effective resiatance is R´. So

$\frac{1}{R´} = \frac{1}{r} + \frac{1}{r} + \frac{1}{r} + \frac{1}{r} + \frac{1}{r}$

$⇒\frac{1}{R´} = \frac{5}{r}$

$⇒R´ = \frac{r}{5}----------(2)$

RATIO

Dividing (1) and (2), we get

$\frac{R}{R´} = \frac{5r}{\frac{r}{5}}$

$⇒\frac{R}{R´} = \frac{5r\times5}{r}$

$⇒\frac{R}{R´} = 25$