A) the resistance will be doubled and the specific resistance will be halved
B) the resistance will be halved and the specific resistance will remain unchanged
C) the resistance will be halved and the specific resistance will be doubled
D) the resistance and the specific resistance, will both remain unchanged
Correct Answer: B
Solution :
The formula for resistance of wire is \[R=\frac{\rho l}{A}\] where\[\rho =\]specific resistance of the wire \[\Rightarrow \]\[R\propto \frac{l}{A}\] \[\Rightarrow \]\[R\propto \frac{1}{{{r}^{2}}}\] \[(\because \,A=\pi {{r}^{2}})\] \[\therefore \] \[\frac{{{R}_{1}}}{{{R}_{2}}}=\frac{{{l}_{1}}}{{{l}_{2}}}\times \frac{r_{2}^{2}}{r_{1}^{2}}\] (i) Given, \[{{l}_{1}}=l,{{l}_{2}}=2l,\,{{r}_{1}}=r,{{r}_{2}}=2r,{{R}_{1}}=R\] Substituting these values in Eq. (i), we have \[\frac{{{R}_{1}}}{{{R}_{2}}}=\frac{l}{2l}\times \frac{{{(2r)}^{2}}}{{{r}^{2}}}\] \[\frac{{{R}_{1}}}{{{R}_{2}}}=2\,\,\And \,{{R}_{2}}=\frac{R}{2}\] Therefore, resistance will be halved. Now the specific resistance of the wire does not de- pend on the geometry of the wire hence, it remains unchanged. Aliter: \[R\propto \frac{l}{{{r}^{2}}}\Rightarrow \frac{{{R}_{2}}}{{{R}_{1}}}=\frac{{{l}_{2}}}{{{l}_{1}}}\times \frac{r_{1}^{2}}{r_{2}^{2}}=\left( \frac{2}{1} \right)\times {{\left( \frac{1}{2} \right)}^{2}}\] \[\Rightarrow \]\[{{R}_{2}}=\frac{{{R}_{1}}}{2},\]specific resistance doesn't depend upon length and radius.
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