Question

The above battery sends a current \( I_1 \) when \( R = R_1 \) and a current \( I_2 \) when \( R = R_2 \). Obtain the internal resistance of the battery in terms of \( I_1 \), \( I_2 \), \( R_1 \), and \( R_2 \).

Hint:

By applying Ohm’s law to two different scenarios and setting the emf equal in both, we can solve for the internal resistance of the battery using the known values of currents and resistances.

Answer 1

We are given a battery with internal resistance \( r \). When a resistance \( R_1 \) is connected across the battery, a current \( I_1 \) flows, and when a resistance \( R_2 \) is connected across the battery, a current \( I_2 \) flows. We need to find the internal resistance \( r \) of the battery in terms of the known quantities \( I_1 \), \( I_2 \), \( R_1 \), and \( R_2 \). Let the emf of the battery be \( \mathcal{E} \). Step 1: Apply Ohm’s Law for both cases For the first case when the external resistance is \( R_1 \), the total resistance in the circuit is \( R_1 + r \), and the current is \( I_1 \). Using Ohm's law: \[ \mathcal{E} = I_1 (R_1 + r) \] For the second case when the external resistance is \( R_2 \), the total resistance in the circuit is \( R_2 + r \), and the current is \( I_2 \). Again, applying Ohm’s law: \[ \mathcal{E} = I_2 (R_2 + r) \] Step 2: Solve for \( \mathcal{E} \) in both equations From the first equation: \[ \mathcal{E} = I_1 (R_1 + r) \] From the second equation: \[ \mathcal{E} = I_2 (R_2 + r) \] Step 3: Set the two expressions for \( \mathcal{E} \) equal to each other Equating the two expressions for \( \mathcal{E} \): \[ I_1 (R_1 + r) = I_2 (R_2 + r) \] Step 4: Solve for \( r \) Expanding both sides: \[ I_1 R_1 + I_1 r = I_2 R_2 + I_2 r \] Now, move all terms involving \( r \) to one side and constants to the other side: \[ I_1 r - I_2 r = I_2 R_2 - I_1 R_1 \] Factor out \( r \): \[ r (I_1 - I_2) = I_2 R_2 - I_1 R_1 \] Finally, solve for \( r \): \[ r = \frac{I_2 R_2 - I_1 R_1}{I_1 - I_2} \] Final Answer: The internal resistance \( r \) of the battery is given by: \[ r = \frac{I_2 R_2 - I_1 R_1}{I_1 - I_2} \]