Three resistors having resistances and are connected as shown in the given circuit. The ratio of currents in terms of resistances used in the circuit is [2021]

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Q.
Three resistors having resistances $r_{1}, r_{2}$ and $r_{3}$ are connected as shown in the given circuit. The ratio $\frac{i_{3}}{i_{1}}$ of currents in terms of resistances used in the circuit is [2021]

1 $\frac{r_{2}}{r_{1} + r_{3}}$

2 $\frac{r_{1}}{r_{2} + r_{3}}$

3 $\frac{r_{2}}{r_{2} + r_{3}}$

4 $\frac{r_{1}}{r_{1} + r_{2}}$

Ans.
(3)

$i_{1} = i_{2} + i_{3}$ ...(i)

Voltage is same in $r_{2}$ and $r_{3}$ as they are in parallel

   $i_{2} r_{2} = i_{3} r_{3}$

   $i_{2} = \frac{i_{3} r_{3}}{r_{2}}$ ...(ii)

From eqn. (i) and (ii), $i_{1} = \frac{i_{3} r_{3}}{r_{2}} + i_{3}$

or $i_{1} = \frac{i_{3} r_{3} + i_{3} r_{2}}{r_{2}} = i_{3} (\frac{r_{2} + r_{3}}{r_{2}})$    $;$    $\frac{i_{3}}{i_{1}} = \frac{r_{2}}{r_{2} + r_{3}}$

Solution

Q.
Three resistors having resistances $r_{1}, r_{2}$ and $r_{3}$ are connected as shown in the given circuit. The ratio $\frac{i_{3}}{i_{1}}$ of currents in terms of resistances used in the circuit is [2021]

1 $\frac{r_{2}}{r_{1} + r_{3}}$

2 $\frac{r_{1}}{r_{2} + r_{3}}$

3 $\frac{r_{2}}{r_{2} + r_{3}}$

4 $\frac{r_{1}}{r_{1} + r_{2}}$

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Solution

Q. Three resistors having resistances r1,r2 and r3 are connected as shown in the given circuit. The ratio i3i1 of currents in terms of resistances used in the circuit is [2021]

1 r2r1+r3

2 r1r2+r3

3 r2r2+r3

4 r1r1+r2

Ans. (3) i1=i2+i3 ...(i) Voltage is same in r2 and r3 as they are in parallel i2r2=i3r3 i2=i3r3r2 ...(ii) From eqn. (i) and (ii), i1=i3r3r2+i3 or i1=i3r3+i3r2r2=i3(r2+r3r2) ; i3i1=r2r2+r3

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Q.
Three resistors having resistances $r_{1}, r_{2}$ and $r_{3}$ are connected as shown in the given circuit. The ratio $\frac{i_{3}}{i_{1}}$ of currents in terms of resistances used in the circuit is [2021]

1 $\frac{r_{2}}{r_{1} + r_{3}}$

2 $\frac{r_{1}}{r_{2} + r_{3}}$

3 $\frac{r_{2}}{r_{2} + r_{3}}$

4 $\frac{r_{1}}{r_{1} + r_{2}}$