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Two conductors arc made of the same mate...

Two conductors arc made of the same material and have the same length. Conductor A is a solid wire of diameter 1 mm. Conductor B is a holJow tube of outer diameter 2 mm and inner diameter lrrun. Find the ratio of resistance ` R_(A)" to " R_(B)`.

Text Solution

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Resistance of A,
`R_(A) = (rho l )/(A) = (rho l)/(pi r^(2))`
`therefore R_(A) = (rho l )/(pi xx ((1 xx 10^(-3))/(2) )^(2) ) = (4 rho l )/(pi xx 10^(-6)) " " `... (1)
Resistance of B,
`R_(B) = (rho l )/(pi [ ((2 xx 10^(-3))/(2) ) - ((1 xx 10^(-3))/(2) )^(2) ] )`
`= (4 rho l)/( pi [ 4 xx 10^(-6) - 1 xx 10^(-6) ] ) `
`therefore R_(B) = (4 rho l)/(pi xx 3 xx 10^(-6))`
`rArr (R_(A))/(R_(B)) = (4 rho l)/(pi xx 10^(-6)) xx (pi xx 3 xx 10^(-6))/(4 rho l)`
`therefore (R_(A))/(R_(B)) = (3)/(1)`
`therefore R_(A) : R_(B) = 3 : 1 `
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