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

• A. \(\frac{r_2}{r_1+r_3}\)
• B. \(\frac{r_1}{r_2+r_3}\)
• C. \(\frac{r_2}{r_2+r_3}\)
• D. \(\frac{r_1}{r_1+r_2}\)

Answer 1

The Correct Option is C

To find the ratio \(\frac{i_3}{i_1}\) of currents in terms of resistances \(r_1\), \(r_2\), and \(r_3\), we need to analyze the circuit shown in the image.

This circuit consists of a series and parallel combination of resistors: 

• Resistor \(r_1\) is in series with the parallel combination of resistors \(r_2\) and \(r_3\).

Let's break down the steps:

1. First, find the total resistance of the parallel combination of \(r_2\) and \(r_3\): \(R_{\text{parallel}} = \frac{r_2 \times r_3}{r_2 + r_3}\)
2. The total series resistance \(R_{\text{total}}\) of the circuit is: \(R_{\text{total}} = r_1 + \frac{r_2 \times r_3}{r_2 + r_3}\)
3. The total current \(i_1\) flows through \(r_1\). At the junction, it splits into \(i_2\) and \(i_3\) across \(r_2\) and \(r_3\) respectively.
4. By the current division rule, the current \(i_3\) through resistor \(r_3\) is given by: \(i_3 = i_1 \times \frac{r_2}{r_2 + r_3}\)
5. Therefore, the ratio \(\frac{i_3}{i_1}\) is: \(\frac{i_3}{i_1} = \frac{r_2}{r_2 + r_3}\)

Thus, the correct answer is \(\frac{r_2}{r_2 + r_3}\).