A wire of radius r has resistance R. If it is stretched to a radius r/2, its resistance becomes:

A wire of radius

r

has resistance

R

. If it is stretched to a radius

r / 2

, its resistance becomes

16 R .

Initial radius of wire,

r_{1} = r

and final radius of wire,

r_{2} = \frac{r}{2} = 0.5 r .

l_{1} A_{1} = length and area of cross section of first wire

l_{2} A_{2} = length and area of cross section of second wire

Since the volume of the wire after stretch remains constant, therefore

l_{1} A_{1} = l_{2} A_{2}

\frac{l_{1}}{l_{2}} = \frac{A_{1}}{A_{2}}

\frac{l_{1}}{l_{2}} = \frac{r_{2}^{2}}{r_{1}^{2}}

\frac{l_{1}}{l_{2}} = {(\frac{0.5 r}{r})}^{2}

\frac{l_{1}}{l_{2}} = \frac{1}{4}

The resistance

(R) = ρ \frac{l}{A} \propto \frac{l}{A}

.
Therefore

\frac{R_{1}}{R_{2}} = \frac{l_{1}}{l_{2}} \times \frac{A_{2}}{A_{1}}

[

R_{1} = resistance of first wire, R_{2} = resistance of second wire]

\frac{R_{1}}{R_{2}} = \frac{1}{4} \times \frac{1}{4}

\frac{R_{1}}{R_{2}} = \frac{1}{16}

R_{2} = 16 R_{1} = 16 R

Also Check:

A wire of radius $r$ has resistance $R$ . If it is stretched to a radius $r / 2$ , its resistance becomes: