Question

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

Answer 1

Answer: (A)

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., $nr+mR=[\sqrt{nr}-\sqrt{mR}{]}^2+2\sqrt{mnRr}$
It's minimum value is zero.
i.e., $[\sqrt{nr}-\sqrt{mR}{]}^2=0$
$\sqrt{nr}-\sqrt{mR}=0$
$nr=mR$
$R=\frac{nr}{m}$
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.