Two metal wires of identical dimensions are connected in series. If `sigma_(1) and sigma_(2)` are the conductivities of the metal wires respectively, the effective conductivity of the combination is

Options

A. `(sigma_(1) sigma_(2))/(sigma_(1) + sigma_(2))`
B. `(2 sigma_(1) sigma_(2))/(sigma_(1) + sigma_(2))`
C. `(sigma_(1) + sigma_(2))/(2 sigma_(1) sigma_(2))`
D. `(sigma_(1)+ sigma_(2))/(sigma_(1) sigma_(2))`

Answer

B

Explanation

`(2 sigma_(1) sigma_(2))/(sigma_(1) + sigma_(2))`
Effective resistor R `R_(1) + R_(2) `....(1)
But `R_(1) = (l)/(sigma_(1)A) and R_(2) = (l)/(sigma_(2)A)`
`[ therefore R = (rho l)/(A) and sigma = (1)/(gamma) ]`
From eq. (1) `(2l)/(sigma A) = (l)/(sigma_(1)A) + (l)/(sigma_(2)A) [ therefore R = (2l)/(sigma A)] `
`(2)/(sigma) = (1)/(sigma_(1)) + (1)/(sigma_(2)) = (sigma_(1) + sigma_(2))/(sigma_(1) sigma_(2))` ,
`therefore sigma = (2 sigma_(1) sigma_(2))/(sigma_(1) + sigma_(2))`