Question

Switch $S$ is closed at time $t=0$. Which one of the following statements is correct ?

• A. Current in the resistance $R$ increases if $E_{1} r_{2} >E_{2}\left(R+r_{1}\right)$
• B. Current in the resistance $R$ increases if $E_{1} r_{2}< E_{2}\left(R+r_{1}\right)$
• C. Current in the resistance $R$ decreases if $E_{1} r_{2} >E_{2}\left(R+r_{1}\right)$
• D. Current in the resistance $R$ decreases if $E_{1} r_{2}=E_{2}\left(R+r_{1}\right)$

Answer 1

Answer: (B)

Initial current in circuit is given as
$i_{1}=\frac{E_{1}+E_{2}}{R+r_{1}+r_{2}}$
Final current, when second battery is short circuited is given as
$i_{2}=\frac{E_{1}}{R+r_{1}}$
We can check that $i_{2}>i_{1}$ if
$\frac{E_{1}}{R+r_{1}}>\frac{E_{1}+E_{2}}{R+r_{1}+r_{2}} $
$\Rightarrow E_{1} R+E_{1} r_{1}+E_{1} r_{2}>E_{1} R+E_{1} r_{1}+E_{2} R+E_{2} r_{1} $
$\Rightarrow E_{1} r_{2}>E_{2}\left(R+r_{1}\right)$