Question

In the circuit shown in figure, $E_1=3V,E_2=2V,E_3=1V$ and $r_1=r_2=r_3=1ohm$
(a) Find the potential difference between the points $A$ and $B$ and the currents through each branch.
(b) If $r_2$ is short circuited and the point $A$ is connected to point $B$ through a resistance $R$, find the currents through $E_1,E_2,E_3$ and the resistor $R$.

Answer 1

(a) Applying Kirchoff's loop law to mesh PLMQP and PLMQONP in the figure shown below, we have
$i_1{r}_1+i_1{r}_2=E_1-E_{2}or{i}_2+i_2=\mathrm{1.....}\left(i\right)$
$i_1{r}_1+i_3{r}_3=E_1-E_{3}or{i}_1+i_3=\mathrm{1.....}\left(ii\right)$
At $i_2+i_3=i_1.....\left(iii\right)$
On solving (i), (ii), (iii)
$i_1=1amp;i_2=0amp;i_3=1amp$
Since no current is drawn along the branck AP
$\therefore V_{AB}=V_{PQ}$
Potential difference across PQ
$V_{PQ}=E_1-i_1{r}_1=2volt$
(b) The figure shows the circuit when point A is connected to point B and $r_2$ is short-circuited.
Applying Kirchoff's junction rule at $P$, we get
$i=i_1+i_2+i_3....\left(iv\right)$
Applying Kirchoff's law to mesh ABMLA
$i_1{r}_1=E_1-E_2;i_1=1amp$
Applying Kirchoff's law to mesh ANOQML
$i_1{r}_1+i_3{r}_3=E_1-E_{3}or{i}_1-i_3=\mathrm{2....}\left(v\right)$
From above equations
$i_1=1amp,$

$i_2=2amp$

$i_3=-1amp$ (direction of current is opposite)
So, current through resistor $R$ will be $I=I_1+I_2+I_3=2amp$