Resistor Combination Resistance Calculation
This problem involves finding the individual resistances of two resistors, let's call them \(R_A\) and \(R_B\), given their equivalent resistances in parallel and series combinations.
Parallel and Series Resistance Formulas
The formulas for equivalent resistance (\(R_{eq}\)) are:
- Series Combination: \(R_{eq} = R_A + R_B\)
- Parallel Combination: \(R_{eq} = \frac{R_A \times R_B}{R_A + R_B}\)
Setting Up the Equations
From the problem statement, we have:
- For series combination: \(R_A + R_B = 12\,\Omega\) (Equation 1)
- For parallel combination: \(\frac{R_A \times R_B}{R_A + R_B} = 3\,\Omega\) (Equation 2)
Solving for Individual Resistances
We can substitute Equation 1 into Equation 2:
\(\frac{R_A \times R_B}{12\,\Omega} = 3\,\Omega\)
Multiply both sides by \(12\,\Omega\) to find the product of the resistances:
\(R_A \times R_B = 3\,\Omega \times 12\,\Omega = 36\,\Omega^2\)
Now we have a system of two equations:
- \(R_A + R_B = 12\,\Omega\)
- \(R_A \times R_B = 36\,\Omega^2\)
Consider a quadratic equation whose roots are \(R_A\) and \(R_B\). The general form is \(x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0\). Using our values:
\(x^2 - (12\,\Omega)x + 36\,\Omega^2 = 0\)
This equation is a perfect square:
\((x - 6\,\Omega)^2 = 0\)
Solving for \(x\) gives \(x = 6\,\Omega\). This means both roots are \(6\,\Omega\). Therefore:
\(R_A = 6\,\Omega \quad \text{and} \quad R_B = 6\,\Omega\)
Verification
Let's check if these values satisfy the initial conditions:
- Series: \(R_A + R_B = 6\,\Omega + 6\,\Omega = 12\,\Omega\) (Correct)
- Parallel: \(\frac{R_A \times R_B}{R_A + R_B} = \frac{6\,\Omega \times 6\,\Omega}{6\,\Omega + 6\,\Omega} = \frac{36\,\Omega^2}{12\,\Omega} = 3\,\Omega\) (Correct)
The resistances of the two resistors A and B are \(6\,\Omega\) and \(6\,\Omega\), respectively.
Important Questions from Combination of Resistors — Series and Parallel
Consider two resistors, $R_1$ and $R_2$, connected in series to a DC voltage source. Which of the following statements accurately describes the distribution of current and voltage across these resistors?
A cell of negligible resistance and e.m.f 2 volt is connected to series combination of 2 ohm, 3 ohm and 5 ohm. The potential difference across the 3 ohm resistance is:
Two bulbs A, of (100w, 100v), and B of (60 w, 100v) are connected in series and across the series combination 200 v is applied. Which bulb will be fused?
3 resistors of 3 ohm each connected in series. What is the mean values of resistors?
The equivalent resistance of the resistances (two) joined in parallel is 6/5 Ω. When one of the resistance wire is broken, the effective resistance becomes 2Ω. The resistance of the wire that got broken was :
