Make sure that the current distribution over two resistances `R_(1)` and `R_(2)` connnected in parallel corresponds to the minimum thermal power generated in this circuit.

Explanation

We assume current conservation but not Kithoff's secound law. Then thermal power dissipated is
`P(i_(1)) = i_(1)^(2) R_(1) + (i -i_(1))^(2) R_(2)`
`= i_(1)^(2) (R_(1) + R_(2)) - 2ii_(1) R_(2) + i^(2) R_(2)`
`= [R_(1) + R_(2)] [i_(1) - (R_(2))/(R_(1) + R_(2)) i]^(2) + i^(2) (R_(1) R_(2))/(R_(1) + R_(2))`
The resistances being positive we see that the power dissipated is minimum when
`i_(2) = i (R_(2))/(R_(1) + R_(2))`
This correponds to usual distribution of currents over resistance joined is parallel.