How much mass has to be converted into energy to produce electric power of 500 MW for one hour ?

Options

A. `2xx10^(-5)` kg
B. `1xx10^(-5)` kg
C. `3xx10^(-5)` kg
D. `4xx10^(-5)` kg

Explanation

To find out how much mass needs to be converted into energy to produce electric power of 500 MW for one hour, we can follow these steps:

### Step 1: Understand the relationship between power, energy, and time
Power (P) is defined as the rate at which energy (E) is produced or consumed over time (t). The relationship is given by the formula:
\[ P = \frac{E}{t} \]
From this, we can express energy as:
\[ E = P \times t \]

### Step 2: Convert power to SI units
The power given is 500 MW (megawatts). We need to convert this to watts (W):
\[ 1 \text{ MW} = 10^6 \text{ W} \]
Thus,
\[ P = 500 \text{ MW} = 500 \times 10^6 \text{ W} = 5 \times 10^8 \text{ W} \]

### Step 3: Convert time to seconds
The time given is one hour. We need to convert this to seconds:
\[ 1 \text{ hour} = 60 \text{ minutes} \times 60 \text{ seconds/minute} = 3600 \text{ seconds} \]
So,
\[ t = 3600 \text{ seconds} \]

### Step 4: Calculate the energy produced
Now, we can calculate the total energy produced in one hour:
\[ E = P \times t = (5 \times 10^8 \text{ W}) \times (3600 \text{ s}) \]
Calculating this gives:
\[ E = 5 \times 10^8 \times 3600 = 1.8 \times 10^{12} \text{ J} \]

### Step 5: Use Einstein's mass-energy equivalence
According to Einstein's mass-energy equivalence principle, the energy (E) can also be expressed in terms of mass (m) as:
\[ E = mc^2 \]
From this, we can solve for mass:
\[ m = \frac{E}{c^2} \]
Where \( c \) (the speed of light) is approximately \( 3 \times 10^8 \text{ m/s} \).

### Step 6: Substitute the values and calculate mass
Now substituting the values we have:
\[ m = \frac{1.8 \times 10^{12} \text{ J}}{(3 \times 10^8 \text{ m/s})^2} \]
Calculating \( c^2 \):
\[ c^2 = (3 \times 10^8)^2 = 9 \times 10^{16} \text{ m}^2/\text{s}^2 \]
Now substituting this back into the equation for mass:
\[ m = \frac{1.8 \times 10^{12}}{9 \times 10^{16}} \]
\[ m = 0.2 \times 10^{-4} \text{ kg} = 2 \times 10^{-5} \text{ kg} \]

### Final Answer
The mass that needs to be converted into energy to produce electric power of 500 MW for one hour is:
\[ m = 2 \times 10^{-5} \text{ kg} \]

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