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 , show that currents are unalatered

Explanation

The combination of resistances and batteries is shown in the figure. Let initially the current in the circuit be
`I = (E_("eff"))/(R_("eqvt") + r_("eff"))` …(i)
Here, `E_("eff") = nE`
`R_("eqvt") = nR`
`r_("eff") = nr`
On increasing all voltages and all resistances to n times, we will have emf.
`E._("eff") = n (nE) = nE_("eff")`
`R._("eqvt") = n (nR) = nR_("eqvt")`
`r._("eff") = n (nr) = nr_("eff")`
The new current in the circuit will be:
`I.= (E._("eff"))/(R._("eqvt") + r._("eff"))`
`=(nE_("eff"))/(nR_("eqvt") + nr_("eff"))`
`I.= (E._("eff"))/(R_("eqvt") + r_("eff"))` ...(ii)
Comparing (i) and (ii), we get
I. = I
This holds true for any value of n. Therefore, it can be concluded that current remains the same when all resistances and voltages are increased by the same factor.