Determine the current through each resistor

Determine the current through each resistor in Figure 75; then calculate each voltage drop.

Determine the current through each resistor in both circuits of Figure 76; then calculate each voltage drop.

The Correct Answer and Explanation is :

I’m unable to view the specific figures you’ve referenced, but I can guide you through the general process of analyzing resistive circuits to determine the current through each resistor and calculate the voltage drops. Here’s a step-by-step approach:

1. Identify Series and Parallel Components:

• Series Connection: Resistors are in series if the same current flows through them. The total resistance is the sum of individual resistances:
  [ R_{\text{total}} = R_1 + R_2 + \dots + R_n ]
• Parallel Connection: Resistors are in parallel if they share the same voltage across them. The total resistance is given by:
  [ \frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n} ]

2. Simplify the Circuit:

• Combine series and parallel resistors step by step to reduce the circuit to a single equivalent resistance.

3. Apply Ohm’s Law:

• Once you have the total resistance, use Ohm’s Law to find the total current supplied by the source:
  [ I_{\text{total}} = \frac{V_{\text{source}}}{R_{\text{total}}} ]

4. Determine Individual Currents:

• For Series Circuits: The current is the same through all resistors.
• For Parallel Circuits: Use the current division rule to find the current through each resistor:
  [ I_n = I_{\text{total}} \times \frac{R_{\text{total}}}{R_n} ]

5. Calculate Voltage Drops:

• Use Ohm’s Law to find the voltage drop across each resistor:
  [ V_n = I_n \times R_n ]

Example:
Consider a circuit with a 12V battery connected to three resistors: R₁ = 2Ω, R₂ = 3Ω, and R₃ = 6Ω. If R₂ and R₃ are in parallel, and their combination is in series with R₁:

1. Combine R₂ and R₃ in Parallel:
   [ \frac{1}{R_{\text{23}}} = \frac{1}{3} + \frac{1}{6} = \frac{1}{2} \quad \Rightarrow \quad R_{\text{23}} = 2Ω ]
2. Total Resistance:
   [ R_{\text{total}} = R_1 + R_{\text{23}} = 2Ω + 2Ω = 4Ω ]
3. Total Current:
   [ I_{\text{total}} = \frac{12V}{4Ω} = 3A ]
4. Current Through R₂ and R₃:

• Since R₂ and R₃ are in parallel, the voltage across them is the same.
• The current through R₂:
  [ I_2 = I_{\text{total}} \times \frac{R_{\text{23}}}{R_2} = 3A \times \frac{2Ω}{3Ω} = 2A ]
• The current through R₃:
  [ I_3 = I_{\text{total}} \times \frac{R_{\text{23}}}{R_3} = 3A \times \frac{2Ω}{6Ω} = 1A ]

1. Voltage Drops:

• Across R₁:
  [ V_1 = I_{\text{total}} \times R_1 = 3A \times 2Ω = 6V ]
• Across R₂:
  [ V_2 = I_2 \times R_2 = 2A \times 3Ω = 6V ]
• Across R₃:
  [ V_3 = I_3 \times R_3 = 1A \times 6Ω = 6V ]

This example illustrates how to analyze a resistive circuit to determine currents and voltage drops. By systematically applying these steps, you can solve similar problems involving different resistor configurations.