Understanding Energy, Power, and Time Relationship

This question asks us to calculate the total energy radiated by a microwave transmitter given its power output and the duration of operation. The fundamental relationship connecting energy ($E$), power ($P$), and time ($t$) is: $E = P \times t$ We are given the power output and the time interval, and we need to find the total energy. It's crucial to ensure that the units are consistent before performing the calculation.

Converting Units for Calculation

Before calculating the energy, we need to convert the given power and time values into standard SI units (Watts for power and seconds for time).

• Power Conversion: The transmitter's power output is given as $250 \text{ MW}$. The prefix 'M' stands for Mega, which means $10^6$. So, we convert Megawatts (MW) to Watts (W): $P = 250 \text{ MW} = 250 \times 10^6 \text{ W}$ Alternatively, this can be written in scientific notation as: $P = 2.5 \times 10^8 \text{ W}$
• Time Conversion: The transmitter operates for $20 \text{ minutes}$. We need to convert this duration into seconds, knowing that 1 minute equals 60 seconds: $t = 20 \text{ minutes} \times 60 \frac{\text{seconds}}{\text{minute}} = 1200 \text{ s}$ In scientific notation, this is: $t = 1.2 \times 10^3 \text{ s}$

Step-by-Step Energy Calculation

Now, we can use the formula $E = P \times t$ with the converted values to find the total energy radiated.

1. Apply the formula: Substitute the values of power ($P$) in Watts and time ($t$) in seconds into the energy formula. $E = P \times t$
2. Substitute values: $E = (250 \times 10^6 \text{ W}) \times (1200 \text{ s})$
3. Calculate the result: $E = (2.5 \times 10^8 \text{ W}) \times (1.2 \times 10^3 \text{ s})$ $E = (2.5 \times 1.2) \times 10^{(8 + 3)} \text{ J}$ $E = 3.0 \times 10^{11} \text{ J}$

The total energy ideally radiated by the transmitter in $20 \text{ minutes}$ is $3 \times 10^{11} \text{ J}$.

Conclusion

Comparing our calculated result with the given options, we find that the total energy radiated is $3 \times 10^{11} \text{ J}$. This matches the second option.