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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.
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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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