The electric resistance of a certain wire of iron is R. If its length and radius are both doubled, then

The electric resistance of a certain wire of iron is R. If its length and radius are both doubled, then

1. A

the resistance will be doubled and the specific resistance will be halved

2. B

the resistance will be halved and the specific resistance will remain unchanged

3. C

the resistance will be halved and the specific resistance will be doubled

4. D

the resistance and the specific resistance, will both remain unchanged

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Solution:

The formula for resistance of wire is

$R=\frac{\rho l}{A}$

where $\rho$ = specific resistance of the wire

$⇒R\propto \frac{l}{A}$

Substituting these values in Eq. (i), we have

$\frac{{\mathrm{R}}_{1}}{{\mathrm{R}}_{2}}=2\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{and}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\mathrm{R}}_{2}\text{\hspace{0.17em}}=\frac{\mathrm{R}}{2}$

Therefore, resistance will be halved.
Now the specific resistance of the wire does not depend on the geometry of the wire, hence, it remains unchanged.

Aliter:

$=\frac{1}{2}$

$⇒\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\mathrm{R}}_{2}\text{\hspace{0.17em}\hspace{0.17em}}=\text{\hspace{0.17em}\hspace{0.17em}}\frac{{\mathrm{R}}_{1}}{2},$  specific resistance doesn't depend upon length and radius.

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