If a cell of constant `E.M.F.` produces the same amount of the heat during the same time in two independent resistors `R_1` and `R_2`, when they are separately connected across the terminals of the cell, one after the another, find the internal resistance of the cell.

Answer

`sqrt(R_(1)R_(2))`

Explanation

`I_(1)^(2)R_(1)=I_(2)^(2)R_(2)`
`(E/(R_(1)+r))^(2)xxR_(1)=(E/(R_(2)+r))^(2)xxR_(2)`
`R_(1)(R_(2)+r)^(2)=R_(2)(R_(1)+r)^(2)`
`R_(1)(R_(2)^(2)+r^(2)+2R_(2)r)=R_(2)(R_(1)^(2)+r^(2)+2R_(1)r)`
`R_(1)R^(2)_(2)+R_(1)r^(2)+2R_(1)R_(2)r=R_(2)R_(1)^(2)+R_(2)r^(2)+2R_(1)R_(2)r`
`R_(1)R_(2)(R_(2)-R_(1))=r^(2)(R_(2)-R_(1))`
`r=sqrt(R_(1)R_(2))`