Question

Find heat dissipated in \(6\Omega\) resistance in \(100\) seconds in the given circuit.

• A. \(31\,J\)
• B. \(35\,J\)
• C. \(40\,J\)
• D. \(28\,J\)

Answer 1

The Correct Option is A

Concept: Heat produced in a resistor is given by Joule's law: \[ H=\frac{V^2}{R}t \] To find voltage across \(6\Omega\), node voltage method is used. Step 1: Apply node voltage equation.} Let node potential be \(x\). \[ \frac{x-2}{2}+\frac{x-0}{4}+\frac{x-3}{6}=0 \] Multiply by \(12\): \[ 6x-12+3x+2x-6=0 \] \[ 11x-18=0 \] \[ x=\frac{18}{11}\text{ volt} \]
Step 2: Find voltage across \(6\Omega\).} \[ V_{6\Omega}=3-\frac{18}{11} \] \[ V_{6\Omega}=\frac{33-18}{11}=\frac{15}{11} \]
Step 3: Calculate heat dissipated.} \[ H=\frac{V^2}{R}t \] \[ H=\frac{\left(\frac{15}{11}\right)^2}{6}\times100 \] \[ H=30.99 \approx 31\,J \] Final Result \[ H=31\,J \]