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 (R<em>tR<em>t) can be calculated using the formula: R</em>t=R<em>1⋅R</em>2R<em>1+R</em>2R</em>t = \frac{R<em>1 \cdot R</em>2}{R<em>1 + R</em>2}

  • For Multiple Resistors: The general formula for calculating total resistance of multiple resistors in parallel is:
    1R<em>t=1R</em>1+1R<em>2+1R</em>3+…\frac{1}{R<em>t} = \frac{1}{R</em>1} + \frac{1}{R<em>2} + \frac{1}{R</em>3} + …

Example Calculation
  • Given:
    • R1=100 ohmsR_1 = 100\,\text{ohms}
    • R2=10 ohmsR_2 = 10\,\text{ohms}

  1. Calculate Total Resistance

    • Using the formula for two resistors:
      Rt=100⋅10100+10=1000110≈9.09 ohmsR_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 (I<em>tI<em>t): I</em>t=VRtI</em>t = \frac{V}{R_t}
    • For a power supply of 120V120V:
      It=1209.09≈13.2 AI_t = \frac{120}{9.09} \approx 13.2\,\text{A}

  3. Calculate Total Power

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

  4. Determine Individual Currents

    • Calculate the current through each resistor:
      • For R<em>1R<em>1: I</em>1=VR1=120100=1.2 AI</em>1 = \frac{V}{R_1} = \frac{120}{100} = 1.2\,\text{A}
      • For R<em>2R<em>2: I</em>2=VR2=12010=12 AI</em>2 = \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.