Question

When two identical batteries of internal resistance 1$\Omega$ each are connected in series across a resistor R, the rate of heat produced in R is $J_1$. When the same batteries are connected in parallel across R, the rate is $J_2$. If $J_1$ = 2.25 $J_2$, then the value of R (In $\Omega$ ) is ?

Answer 1

$\text{In case 1},J_1={\left(\frac{2\epsilon }{R+2r}\right)}^{2}R$

$={\left(\frac{2\epsilon }{R+2}\right)}^{2}R[r=1\Omega ]$

$\text{In case 2,}{J}_2={\left(\frac{\epsilon }{R+\frac{r}{2}}\right)}^2,R={\left(\frac{2\epsilon }{2R+1}\right)}^{2}R$

$\frac{J_1}{J_2}={\left(\frac{2R+1}{R+2}\right)}^2=2.25=\frac{9}{4}$

or $\frac{2R+1}{R+2}=\frac{3}{2}$

$\Rightarrow R=4\Omega$