Question

Three resistors having resistances r₁, r₂ and r₃ 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 :

(1) \(\frac{r_1}{r_2 +r_3}\)

(2) \(\frac{r_2}{r_2 +r_3}\)

(3) \(\frac{r_1}{r_1 +r_2}\)

(4) \(\frac{r_2}{r_1 +r_3}\)

Answer 1

In parallel combination of resistances r₂ and r₃, potential difference will be equal across both resistance.

So, \(i_2r_2 = i_3r_3\)

⇒ \(i_2 = \frac{i_3r_3}{r_2}\)   ......(1)

As per Kirchhoff's first law,

\(i_1 = i_2 + i_3\)

\(i_1 = (\frac{r_3}{r_2} + 1)i_3\)   (From eq. 1)

⇒ \(\frac{i_3}{i_1}= \frac{r_2}{r_2 + r_3}\)

Answer 2

Option : (2) \(\frac{r_2}{r_2 +r_3}\)

v = ir

i = \(\frac{v}{r}\)

i ∝ \(\frac{i}{r}\)

[ v is same for r₂ & r₃]