Math Problem Statement

Determine the total resistance at points a and b of the circuit shown below.

Solution

The problem asks to determine the total resistance between points aa and bb in the given circuit. The circuit appears to be a combination of resistors in series and parallel. Here's how we can analyze it step by step:

Step-by-Step Solution:

  1. Identify Parallel and Series Combinations:

    • From the image, we see that the resistors are grouped into parallel and series.
    • The resistors 12Ω12 \Omega, 26Ω26 \Omega, and 40Ω40 \Omega are in parallel.
    • The resistors 60Ω60 \Omega, 30Ω30 \Omega, and 14Ω14 \Omega are also in parallel.
    • These two parallel combinations are then connected in series.
  2. Calculate Equivalent Resistance for the First Parallel Group (12Ω, 26Ω, 40Ω): The formula for resistors in parallel is: 1Req1=1R1+1R2+1R3\frac{1}{R_{\text{eq1}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} Substituting the values: 1Req1=112+126+140\frac{1}{R_{\text{eq1}}} = \frac{1}{12} + \frac{1}{26} + \frac{1}{40} Solving this gives Req1R_{\text{eq1}}.

  3. Calculate Equivalent Resistance for the Second Parallel Group (60Ω, 30Ω, 14Ω): Using the same formula for parallel resistors: 1Req2=160+130+114\frac{1}{R_{\text{eq2}}} = \frac{1}{60} + \frac{1}{30} + \frac{1}{14} Solving this gives Req2R_{\text{eq2}}.

  4. Add the Two Equivalent Resistances in Series: Once we have Req1R_{\text{eq1}} and Req2R_{\text{eq2}}, we can add them since they are in series: Rtotal=Req1+Req2R_{\text{total}} = R_{\text{eq1}} + R_{\text{eq2}}

I will now calculate these values for you.The calculations yield the following results:

  • The equivalent resistance of the first parallel group (12Ω, 26Ω, 40Ω) is approximately 6.81Ω6.81 \, \Omega.
  • The equivalent resistance of the second parallel group (60Ω, 30Ω, 14Ω) is approximately 8.24Ω8.24 \, \Omega.
  • Therefore, the total resistance between points aa and bb is the sum of these two, which is approximately 15.05Ω15.05 \, \Omega.

Let me know if you want any more details or explanations.

Here are some related questions to explore further:

  1. What is the formula for calculating resistance in parallel and series circuits?
  2. How does changing one resistor in a parallel circuit affect the overall resistance?
  3. What are the practical applications of parallel and series resistances in real-world circuits?
  4. How would adding an additional resistor in series to the current setup change the total resistance?
  5. How can we verify the total resistance in a real circuit using multimeters?

Tip: Always double-check whether resistors are in series or parallel before applying formulas to ensure correct calculations.

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Math Problem Analysis

Mathematical Concepts

Electrical Circuit Analysis
Series and Parallel Resistors
Ohm's Law

Formulas

1 / Req = 1 / R1 + 1 / R2 + ... + 1 / Rn (for parallel resistors)
Req = R1 + R2 + ... + Rn (for series resistors)

Theorems

Series and Parallel Resistance Theorem

Suitable Grade Level

College Level or Advanced High School (Physics, Electrical Engineering)