A cell of constant e.m.f. first connected to a resistance `R_(1)` and then connected to a resistance `R_(2)`. If power delivered in both cases is then the internal resistance o f the cell is

Options

A. `sqrt(R_(1) + R_(2))`
B. `sqrt((R_(1))/(R_(2)))`
C. `(R_(1) - R_(2))/(2)`
D. `(R_(1) + R_(2))/(2)`

Answer

A

Explanation

(a) Power dissipated `= i^(2) R = ((E)/(R + r))^(2) R`
`:. ((E)/(R_(1) + r))^(2) R_(1) = ((E)/(R_(2) + r))^(2) R_(2)`
`implies R_(1) (R_(1)^(2) + r_(2) + 2 R_(2) r) = R_(2) (R_(1)^(2) + r^(2) + 2 R_(1) r)`
`implies R_(2)^(2) R_(1) + R_(1) r^(2) + 2 R_(2) r = R_(1)^(2) R_(2) + R_(2) r^(2) + 2 R_(1) R_(2) r`
`implies (R_(1) - R_(2)) r^(2) = (R_(1) - R_(2)) r^(2) = (R_(1) - R_(2)) R_(1) R_(2)`
`implies r = sqrt(R_(1) R_(2))`