Math Problem Statement

Consider the three resistors R1 = 21 Ω, R2 = 48 Ω, and R3 = 94 Ω in the configuration shown in the figure. A potential difference ΔV = 8.5 V is applied between A and B. Calculate the current and the voltage distribution across the resistors.

Solution

To solve this circuit problem, let's analyze the setup and compute the equivalent resistance and the current through the circuit.

Given Data:

  1. Resistors:

    • R1=21ΩR_1 = 21 \, \Omega
    • R2=48ΩR_2 = 48 \, \Omega
    • R3=94ΩR_3 = 94 \, \Omega
  2. Potential difference between points AA and BB:

    • ΔV=8.5V\Delta V = 8.5 \, \text{V}

Step 1: Determine Equivalent Resistance

The configuration shows R2R_2 and R3R_3 connected in parallel, and their combination in series with R1R_1.

  1. Parallel Combination of R2R_2 and R3R_3: 1R23=1R2+1R3\frac{1}{R_{23}} = \frac{1}{R_2} + \frac{1}{R_3} Substituting the values: 1R23=148+194\frac{1}{R_{23}} = \frac{1}{48} + \frac{1}{94} Calculating this gives: R2331.88ΩR_{23} \approx 31.88 \, \Omega

  2. Total Equivalent Resistance ReqR_{\text{eq}}: Since R23R_{23} is in series with R1R_1, the total resistance is: Req=R1+R23=21+31.88=52.88ΩR_{\text{eq}} = R_1 + R_{23} = 21 + 31.88 = 52.88 \, \Omega

Step 2: Calculate the Current in the Circuit

Using Ohm's law: I=ΔVReq=8.552.880.161AI = \frac{\Delta V}{R_{\text{eq}}} = \frac{8.5}{52.88} \approx 0.161 \, \text{A}

Step 3: Voltage Across R1R_1 and the Parallel Combination

  1. Voltage across R1R_1: VR1=IR1=0.161×213.38VV_{R_1} = I \cdot R_1 = 0.161 \times 21 \approx 3.38 \, \text{V}

  2. Voltage across the parallel combination R23R_{23}: VR23=ΔVVR1=8.53.385.12VV_{R_{23}} = \Delta V - V_{R_1} = 8.5 - 3.38 \approx 5.12 \, \text{V}

Step 4: Current through R2R_2 and R3R_3

Using the voltage VR23V_{R_{23}} across R2R_2 and R3R_3:

  1. Current through R2R_2: IR2=VR23R2=5.12480.107AI_{R_2} = \frac{V_{R_{23}}}{R_2} = \frac{5.12}{48} \approx 0.107 \, \text{A}

  2. Current through R3R_3: IR3=VR23R3=5.12940.0545AI_{R_3} = \frac{V_{R_{23}}}{R_3} = \frac{5.12}{94} \approx 0.0545 \, \text{A}

Summary of Results:

  • Total current in the circuit: I0.161AI \approx 0.161 \, \text{A}
  • Voltage across R1R_1: VR13.38VV_{R_1} \approx 3.38 \, \text{V}
  • Voltage across R23R_{23} (parallel part): VR235.12VV_{R_{23}} \approx 5.12 \, \text{V}
  • Current through R2R_2: IR20.107AI_{R_2} \approx 0.107 \, \text{A}
  • Current through R3R_3: IR30.0545AI_{R_3} \approx 0.0545 \, \text{A}

Would you like further details on any specific step?


Related Questions:

  1. How would the total current change if R1R_1 were increased?
  2. What would happen to the equivalent resistance if R2R_2 and R3R_3 were swapped?
  3. How is the power dissipated in each resistor calculated?
  4. How would adding another resistor in series with R1R_1 affect the circuit?
  5. How does parallel vs. series configuration impact voltage and current distribution?

Tip:

For resistors in parallel, the combined resistance is always less than the smallest resistor in the group, as it provides more paths for current to flow.

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Math Problem Analysis

Mathematical Concepts

Series and Parallel Circuits
Ohm's Law
Electric Circuit Analysis

Formulas

Ohm's Law: V = IR
Parallel Resistance: 1/R_eq = 1/R2 + 1/R3
Series Resistance: R_total = R1 + R_eq

Theorems

Ohm's Law
Parallel and Series Resistor Theorems

Suitable Grade Level

Grades 11-12 or Introductory College Level