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Question
Suppose there is a circuit consisting of only resistances and batteries and we have to double (or increase it to n-times) all voltages and all resistances. Show that currents are unaltered. Do this for circuit of Example 3.7 in the NCERT Text Book for Class XII.
Solution
Let us first assume the equivalent internal resistance of the battery is Reff, the equivalent external resistance R and the equivalent voltage of the battery is Veff.
Now by applying Ohm's law,
Then current through R is given by `I = (V_"eff")/(R_"eff" + R)`
Now according to the question if all the resistances and the effective voltage are increased n-times, then we have
`V_("eff")^("new") = nV_("eff"), R_("eff")^("new") = nR_("eff")`
And Rnew = nR
Then, the new current is given by
`I^' = (nV_("eff"))/(nR_("eff") + nR) = (n(V_("eff")))/(n(R_("eff") + R)) = ((V_("eff")))/((R_("eff") + R)) = I`
The last result of two equations is same, so we can say that current remains the same.
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