Work Done Calculation

The problem asks to calculate the work done in a resistor given its resistance, the current flowing through it, and the time duration.

Relevant Formula

The work done (W) by electrical current in a resistor is calculated using the formula:

$ W = P \times t $

where P is the power dissipated and t is the time.

Power (P) can be expressed as:

$ P = I^2R $

Therefore, the work done is:

$ W = I^2Rt $

Here, I is the current, R is the resistance, and t is the time.

Given Values

• Resistance, \( R = 10 \, \Omega \)
• Current, \( I = 3 \, \text{A} \)
• Time, \( t = 3 \, \text{hours} \)

Step-by-Step Calculation

1. Time Conversion: Convert the time from hours to seconds, as the standard unit for work (Joule) requires time in seconds.

   $ t = 3 \, \text{hours} \times \frac{3600 \, \text{seconds}}{1 \, \text{hour}} = 10800 \, \text{s} $

2. Calculate Work Done in Joules (J): Substitute the values into the formula \( W = I^2Rt \).

   $ W = (3 \, \text{A})^2 \times (10 \, \Omega) \times (10800 \, \text{s}) $

   $ W = 9 \, \text{A}^2 \times 10 \, \Omega \times 10800 \, \text{s} $

   $ W = 90 \times 10800 \, \text{J} $

   $ W = 972000 \, \text{J} $

3. Convert Joules to Kilowatt-hours (kWh): The options are given in kWh. Convert the work done from Joules to kWh using the conversion factor \( 1 \, \text{kWh} = 3,600,000 \, \text{J} \).

   $ W = \frac{972000 \, \text{J}}{3600000 \, \text{J/kWh}} $

   $ W = 0.27 \, \text{kWh} $

Final Answer Verification

The calculated work done is 0.27 kWh, which matches Option D.