Question

A uniform wire of length $ l $ and radius $ r $ has resistance $100\, \Omega $ . It is recasted into a thin wire of (i) length 2l (ii) radius $ \frac{r}{2.} $ The resistance of the new wire in each case will be :

• A. 200 $ \Omega $ 400 $ \Omega $
• B. 100 $ \Omega $ 200 $ \Omega $
• C. 400 $ \Omega $ 160 $ \Omega $
• D. 400 $ \Omega $ 1600 $ \Omega $

Answer 1

Answer: (D)

(i) Since, given wire of fixed mass is recanted to twice the length
$ R\propto {{l}^{2}} $ .....(i)
Let R' be the new resistance
$ R'\propto l{{'}^{2}} $ ...(2)
Hence, $ \frac{R'}{R}={{\left( \frac{l'}{l} \right)}^{2}} $ or
$ R'={{\left( \frac{{{l}_{1}}'}{l} \right)}^{2}}R $
Here, $ R=100\Omega \,\,l'=2l $
So, $ R'={{\left( \frac{2l}{l} \right)}^{2}}\times 100=400\Omega $ (ii)
Now in terms of radius $ R\propto \frac{1}{{{r}^{4}}} $
Here, $ r'=\frac{r}{2} $
Hence, $ \frac{R'}{R}={{\left( \frac{r}{r'} \right)}^{4}} $ or
$ R'={{\left( \frac{r}{r'} \right)}^{4}}\times R $
So, $ R'={{\left( \frac{\frac{r}{r}}{2} \right)}^{4}}R=16R $ $ =16\times 100=1600\Omega $