Expression for internal resistance in terms of I1, I2, R1, R2:
Solution and Explanation
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:
\[ r = \frac{I_2 R_2 - I_1 R_1}{I_1 - I_2} \]
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