Notes on Parallel Circuits

Rules for Parallel Circuits

• Equal Voltage: The voltage across all components connected in parallel is the same.
• Current Division: The total current entering a parallel circuit splits among the parallel components; the split is based on the resistance of each component. The lower the resistance, the greater the current.
• Total Resistance: The total resistance of a parallel circuit is always less than the resistance of the smallest resistor in the network.

Calculating Total Resistance

• For Two Resistors: If there are only two resistors in parallel, the total resistance (Rt) can be calculated using the formula: Rt = \frac{R1 \cdot R2}{R1 + R2}

• For Multiple Resistors: The general formula for calculating total resistance of multiple resistors in parallel is:
  \frac{1}{Rt} = \frac{1}{R1} + \frac{1}{R2} + \frac{1}{R3} + …

Example Calculation

• Given:
  • R_1 = 100\,\text{ohms}
  • R_2 = 10\,\text{ohms}

1. Calculate Total Resistance

   • Using the formula for two resistors:
     R_t = \frac{100 \cdot 10}{100 + 10} = \frac{1000}{110} \approx 9.09\,\text{ohms}

2. Calculate Total Current

   • Use Ohm’s Law to find the total current (It): It = \frac{V}{R_t}
   • For a power supply of 120V:
     I_t = \frac{120}{9.09} \approx 13.2\,\text{A}

3. Calculate Total Power

   • Use the Power Formula:
     P = V \cdot I
   • Here:
     P = 120 \cdot 13.2 \approx 1584\,\text{W}

4. Determine Individual Currents

   • Calculate the current through each resistor:
     • For R1: I1 = \frac{V}{R_1} = \frac{120}{100} = 1.2\,\text{A}
     • For R2: I2 = \frac{V}{R_2} = \frac{120}{10} = 12\,\text{A}

Summary

• Voltage across parallel components is constant, current depends on resistance, and total resistance decreases with additional resistors.
• Total resistance can be found using the product-sum formula for two resistors, or the reciprocal method for multiple resistors.
• Ohm's Law and Power formulas are essential for calculating current and power in the circuit.