Suppose there is a circull consisting of only resistances and batteries. Suppose one is to double (or increase it to n-times) all voltages and all resistances. Show that currents are unaltered.

Explanation

Let equivalent internal resistance of given cells be `R_(eq). ` Equivalent voltage be `V_(eq)` external resistance be R.
From Ohm.s law,
`I_(1) = (V_(eq))/(R_(eq) + R) " " `.... (1)
When value of resistors and all batteries made n times initial value then,
`therefore V_(eq) = nV_(eq), R_(eq) = nR_(eq)`
and new external resistor R = nR
Current in new (modified ) circuit,
`I_(1) = (n V_(eq))/(nR_(eq) + nR) = (n V_(eq))/(n (R_(eq) + R)) `
` = (V_(eq))/(R_(eq) + R)`
`therefore ` Comparing (1) and (2)
`I_(1) = I_(1)`
thus current do not change.