Two unequal resistances `R_1 and R_2` ae connected across two identical batteries of `emf epsilon` and internal resistance `r (figure)`. Can the thermal energies developed in `R_1 and R_2` be equal in a given time. If yes, what will be the condition? (Figure Question)

Explanation

In time t, time energy developend in `R_(1) ` and `R_(2)` are
`H_(1) i_(1)^(2)R_(1)t = ((epsi)/(R_(1) + r))^(2) R_(1)t`
and `H_(2) = i_(1)^(2)R_(2) t = ((epsi)/(R_(2) + r))^(3) R_(2) t`
If `H_(1) = H_(2) ` then, `(R_(1))/((R_(1) + r)^(2)) = (R_(2))/((R_(2) + r)^(2))`
or `((R_(1) + r)/(R_(2) + r))^(2) = (R_(1))/(R_(2)) or , (R_(1) + r)/(R_(2) + r) = sqrt((R_(1))/(R_(2))`
`or R_(1) sqrt(R_(2)) + rsqrt(R_(2)) = R_(2) sqrt(R_(1)) + rsqrt(R_(1))`
or `r(sqrt(R_(2)) - sqrt(R_(1)) = sqrt(R_(1)R_(2)) ( sqrt(R_(2) - sqrt(R_(1)))`
or , ` r = sqrt(R_(1)R_(2))`
Thus , the thermal energies developend in `R_(1) ` and `R_(2)` will be equal when ` r = sqrt(R_(1)R_(2))`