Question

A cell of emf E and internal resistance r is connected to two external resistances R1 and R2 and a perfect ammeter. The current in the circuit is measured in four different situations:

• A. \( c < b < d < a \)
• B. \( a < b < d < c \)
• C. \( c < d < b < a \)
• D. \( a < d < b < c \)

Hint:

In circuits with resistances in series, the total resistance increases, which decreases the current. In circuits with resistances in parallel, the total resistance decreases, increasing the current.

Answer 1

The Correct Option is A

The current in a circuit can be found using Ohm's law, which is given by: \[ I = \frac{E}{R_{\text{total}}}, \] where: - \( E \) is the emf of the cell, - \( R_{\text{total}} \) is the total resistance of the circuit. Let's analyze the current in each case. Case (a): Without any external resistance In this case, the circuit consists of only the internal resistance \( r \) of the cell. Therefore, the total resistance in the circuit is: \[ R_{\text{total}} = r. \] Thus, the current is: \[ I_a = \frac{E}{r}. \] Case (b): With resistance \( R_1 \) only In this case, the total resistance is the sum of the internal resistance \( r \) and \( R_1 \): \[ R_{\text{total}} = r + R_1. \] Thus, the current is: \[ I_b = \frac{E}{r + R_1}. \] Case (c): With \( R_1 \) and \( R_2 \) in series combination In this case, the total resistance is: \[ R_{\text{total}} = r + R_1 + R_2. \] Thus, the current is: \[ I_c = \frac{E}{r + R_1 + R_2}. \] Case (d): With \( R_1 \) and \( R_2 \) in parallel combination In this case, the total resistance is: \[ R_{\text{total}} = r + \frac{R_1 R_2}{R_1 + R_2}. \] Thus, the current is: \[ I_d = \frac{E}{r + \frac{R_1 R_2}{R_1 + R_2}}. \]
We can now compare the values of \( I_a \), \( I_b \), \( I_c \), and \( I_d \). Since the total resistance in case (a) is the smallest, the current will be highest in that case. For the other cases, the total resistance increases, and thus the current decreases in the following order: \[ I_c < I_b < I_d < I_a. \] Thus, the currents measured in the four cases in ascending order are \( c < b < d < a \), which corresponds to option (A).