A uniform wire of length `1` and radius `r` has resistance `100 Omega`. it is recast into a thin wire of (i) length `2l` (ii) radius `r//2`. Calculate the resistance of new wire in each case.

Answer

(i) `400 Omega`,(ii)`1600 ohm`

Explanation

Let A, `l` be the original area of cross section and length of the given wire and `A_(1),l_(1)` be the respectively new volume when the wire is drawn to twice its original length. Then `l_(1) = 2l`. Using the relation `A l = A_(1)l_(1)`, we have `A_(1)=Al//l_(1)=Al//2l=A//2` Therefore,
`R_(1) =rhol_(1)//A_(1) = rho 2 //(A//2)`
`= 4 rho l//A = 4R = 4 xx 100 = 400 Omega`
(ii) when wire is drawn in half its radius new area becomes `1//4th` length would become 4 times, `:. R_(2)= 16R = 16 xx 100 = 1600` ohm