Two metallic wires A and B are connected in parallel, wire A has length L and radius r wire B has a length 2L and radius 2r. Compute the ratio of the total resistance of the parallel combination and resistance of wire A.

Explanation

(1) For metal wire A
Length = L
Radius = r
Area = `pi r^(2)`
Resistance, `R_(A) = (rho_(A) L)/(pi r^(2))`
where `rho_(A)` is specific resistance
For metal wire B
Length = 2L
Radius = 2 r
Area = `pi (2r)^(2) = 4 pi r^(2)`
Resistance, `R_(B) = (rho_(B)(2L))/(4 pi r^(2)) = (rho_(B) L)/(2 pi r^(2))`
Where `rho_(B)` is specific resistance.
(2) Total resistance of wire A and wire B in parallel combination is given by
`(1)/(R_(P)) = (1)/(R_(A)) + (1)/(R_(B))`
`R_(P) = (R_(A) R_(B))/(R_(A) + R_(B)) = ((rho_(A) L)/(pi r^(2))xx (P_(B)L)/(2 pi r^(2)))/((rho_(A)L)/(pi r^(2)) + (rho_(B) L)/(2 pi r^(2))) = ((L)/(pi r^(2))((rho_(A) rho_(B))/(2)))/((L)/(pi r^(2))(rho_(A) + (rho_(B))/(2)))`
`:. R_(P) (rho_(A) rho_(B))/([2rho_(A) + rho_(B)])`
(4) The ratio of the total resistance parallel combination to resistance of wire A, is given by
`(R_(P))/(R_(A)) = ((rho_(A) rho_(B))/(2P_(A) + rho_(b)))/((rho_(A) L)/(pi r^(2)))`
(5) `:. (R_(P))/(R_(A)) = (rho_(B) pi r^(2))/(L (2rho_(A) + rho_(B)))`