question_answer

Current provided by a battery is maximum when:

A) internal resistance is equal to external resistance

B) internal resistance is greater than external resistance

C) internal resistance is less than external resistance

D) none of the above

Correct Answer: A

Solution :

Let n cells are connected in series, and such m series are connected in parallel. Let the emf of each cell be E and the internal resistance be r. This battery of cells is sending current in an external resistance R. Then current in the external circuit be \[i,\]then \[i=\frac{mnE}{nr+mR}\] It is clear from this equation, that for current to be maximum the value of (nr + mR) should be minimum. i.e., \[nr+mR={{[\sqrt{nr}-\sqrt{mr}]}^{2}}+2\sqrt{mnRr}\] If?s minimum value is zero. i.e.,                \[{{[\sqrt{nr}-\sqrt{mR}]}^{2}}=0\]                     \[\sqrt{nr}-\sqrt{mR}=0\]                         \[nr=mR\]                   \[\Rightarrow \]              \[R=\frac{nr}{m}\] but \[\frac{nr}{m}\] is the internal resistance of the whole battery. Thus, current is maximum when the internal resistance of battery is equal to external resistance.