Question

A uniform wire of length I and radius r has a resistance of 100 $\Omega$ It is recast into a wire of radius .$\frac{r}{2}$ The resistance of new wire will be -

• A. 1600 $\Omega$
• B. 100 $\Omega$
• C. 200 $\Omega$
• D. 400 $\Omega$

Answer 1

The correct option is A 1600 $\Omega$

$\begin{array}{c}Accordingtoqueation.........\\ l,r\to resis\tan cy\left(R\right)=100\\ and,radiusoflwire\left(\frac{r}{2}\right)\to R_2[\begin{array}{c}wehaveresistanceformula:\\ R=\frac{\rho l}{A}\end{array}\\ R=\frac{\rho l}{A}=\frac{\rho l}{\pi r^2}\\ \Rightarrow R=\frac{\rho lA}{A.A}=\frac{\rho v}{A^2}=\frac{\rho v}{{\pi }^2{r}^4}\\ \Rightarrow R\propto \frac{1}{r^4}\\ \Rightarrow R_1=\frac{k}{r^4}\\ \Rightarrow R_2=\frac{k}{{\left(\frac{r}{2}\right)}^4}=\frac{16k}{r^4}\\ \therefore R_2=16R_1=16\times 100=1600\Omega \\ sotheoptionisA.\end{array}$