Question

Find the heat produced in the external circuit \( (AB) \) in one second.

• A. \(1181.25\,\text{J}\)
• B. \(1311.25\,\text{J}\)
• C. \(1207.50\,\text{J}\)
• D. \(1410.50\,\text{J}\)

Hint:

For circuit problems: \begin{itemize} \item Reduce complex resistor networks stepwise \item Use \( P = I^2R \) for heat or power calculation \item Heat produced \( H = Pt \) \end{itemize}

Answer 1

The Correct Option is C

Given:

• The circuit consists of resistors \( 1 \, \Omega \), \( 2 \, \Omega \), and \( 1 \, \Omega \) arranged as shown in the image.
• The voltage across the circuit is \( 9 \, V \).
• We need to calculate the heat produced in the external circuit \( AB \) in one second.

Step 1: Simplify the Circuit

The given circuit contains resistors \( 1 \, \Omega \), \( 2 \, \Omega \), and \( 1 \, \Omega \) arranged in a combination of series and parallel. Let's simplify the circuit step by step. First, consider the two resistors \( 1 \, \Omega \) and \( 2 \, \Omega \) in series:

\(R_{\text{eq1}} = 1 \, \Omega + 2 \, \Omega = 3 \, \Omega.\)

Now, consider the resistor \( 1 \, \Omega \) in parallel with the equivalent resistance \( R_{\text{eq1}} = 3 \, \Omega \):

\(\frac{1}{R_{\text{eq2}}} = \frac{1}{1 \, \Omega} + \frac{1}{3 \, \Omega} = \frac{4}{3} \quad \Rightarrow \quad R_{\text{eq2}} = \frac{3}{4} \, \Omega.\)

Finally, the equivalent resistance \( R_{\text{eq2}} = 0.75 \, \Omega \) is in series with the other \( 1 \, \Omega \) resistor, giving the total resistance of the circuit:

\(R_{\text{total}} = 0.75 \, \Omega + 1 \, \Omega = 1.75 \, \Omega.\)

Step 2: Use Ohm's Law to Find the Current

From Ohm's Law, we know that:

\(I = \frac{V}{R_{\text{total}}}.\)

Substituting the given values:

\(I = \frac{9 \, \text{V}}{1.75 \, \Omega} = 5.14 \, \text{A}.\)

Step 3: Calculate the Heat Produced in One Second

The heat produced in the circuit is given by Joule's law:

\(H = I^2 R_{\text{total}} t.\)

Where:

• \( I \) is the current in the circuit (\( 5.14 \, \text{A} \)),
• \( R_{\text{total}} \) is the total resistance (\( 1.75 \, \Omega \)),
• \( t \) is the time (1 second).

Substituting the values:

\(H = (5.14 \, \text{A})^2 \times 1.75 \, \Omega \times 1 \, \text{s}.\)

Calculating:

\(H = 26.42 \, \text{W} \times 1 \, \text{s} = 26.42 \, \text{J}.\)

Step 4: Conclusion

The heat produced in the external circuit (AB) in one second is \( \boxed{26.42 \, \text{J}} \).