Suppose there is a circuit consister of only resistance and batteries , suppose one is to double (or increase it to in n-times )all voltage and all resistances , shown thst currents are unalitered

Explanation

Let there be n bvatteries of the circuit Let `R_("eff")` be the effective initial resistance of a resistance in series and `epsilon_("eff")` is the effective of a batteries in series and `R` is the exeternal in the circuit figure

current through R is `t = (epsilon_("eff"))/(R_("eff") + R)` ...(i)
Where all the voltage and resistance are increased `n` fold then
`epsilon_("eff")^("new") = n epsilon_("eff") , R_("eff")^("new") = nR_("eff") and R^("new") = n R`
Current `l^("new") = (epsilon_("eff")^("new"))/(R_("eff")^("new") + R^("new")) = (epsilon_("eff"))/(n R_("eff") + nR) = (epsilon_("eff"))/(R_("eff") + R) = 1`
This current remain unaltered when all voltage and resistance are increased to a folds