Current Divider Principle

In a parallel circuit, the total current divides inversely proportional to the resistance of each branch. The formula is:

\[ I_R = I_{\text{Total}} \cdot \frac{R_{\text{Total}}}{R_i} \]

Where:

• \(I_R\): Current through a specific resistor
• \(I_{\text{Total}}\): Total circuit current
• \(R_{\text{Total}}\): Total equivalent resistance
• \(R_i\): Resistance of the specific resistor

Example 1

Given:

• \(I_{\text{Total}} = 10 \, \text{mA}\)
• \(R_1 = 100 \, \Omega\), \(R_2 = 200 \, \Omega\)

Calculation:

Total resistance of the parallel network:

\[ R_{\text{Total}} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}} = \frac{1}{\frac{1}{100} + \frac{1}{200}} = 66.67 \, \Omega \]

Current through \(R_1\):

\[ I_1 = 10 \, \text{mA} \cdot \frac{66.67}{100} = 6.67 \, \text{mA} \]

Current through \(R_2\):

\[ I_2 = 10 \, \text{mA} \cdot \frac{66.67}{200} = 3.33 \, \text{mA} \]

Example 2

Given:

• \(I_{\text{Total}} = 5 \, \text{A}\)
• \(R_1 = 50 \, \Omega\), \(R_2 = 100 \, \Omega\), \(R_3 = 200 \, \Omega\)

Calculation:

Total resistance of the parallel network:

\[ R_{\text{Total}} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}} = \frac{1}{\frac{1}{50} + \frac{1}{100} + \frac{1}{200}} = 28.57 \, \Omega \]

Current through \(R_1\):

\[ I_1 = 5 \, \text{A} \cdot \frac{28.57}{50} = 2.86 \, \text{A} \]

Current through \(R_2\):

\[ I_2 = 5 \, \text{A} \cdot \frac{28.57}{100} = 1.43 \, \text{A} \]

Current through \(R_3\):

\[ I_3 = 5 \, \text{A} \cdot \frac{28.57}{200} = 0.71 \, \text{A} \]