Question

Expression for internal resistance in terms of I1, I2, R1, R2:

Answer 1

Calculation of Internal Resistance \( r \)

Given:

• Current \( I_1 = \frac{E}{R_1 + r} \)
• Current \( I_2 = \frac{E}{R_2 + r} \)

Step 1: Cross Multiply the Equations

From the given current equations, we can cross-multiply:

\[ I_1 (R_1 + r) = E \quad \text{and} \quad I_2 (R_2 + r) = E \]

Step 2: Equating the Two Expressions

Equating the two equations for \( E \), we get:

\[ I_1 (R_1 + r) = I_2 (R_2 + r) \]

Step 3: Expanding and Simplifying

Expanding both sides:

\[ I_1 R_1 + I_1 r = I_2 R_2 + I_2 r \]

Rearranging the terms:

\[ I_1 r - I_2 r = I_2 R_2 - I_1 R_1 \]

Factor out \( r \):

\[ r (I_1 - I_2) = I_2 R_2 - I_1 R_1 \]

Step 4: Solving for \( r \)

Solving for \( r \), we get:

\[ r = \frac{I_2 R_2 - I_1 R_1}{I_1 - I_2} \]

Final Answer:

The internal resistance \( r \) is given by:

\[ r = \frac{I_2 R_2 - I_1 R_1}{I_1 - I_2} \]