A point source surrounded by vacuum emits an electromagnetic wave of frequency of 900 kHz. If the power emitted by the source is 15 W, the amplitude of the electric field of the wave (in `x10^(-3)` V/m) at a distance 15 km from the source is: find the value x??
`"("(1)/(4pi in_(0))=9xx10^(9)(Nm^(2))/(C^(2))` and speed of light in vacuum `=3xx10^(8)ms^(-1)`)

Options

A. 2
B. 4
C. 20
D. 40

Explanation

To find the amplitude of the electric field of the electromagnetic wave at a distance of 15 km from the source, we can use the relationship between power, electric field amplitude, and distance from the source.

### Step-by-Step Solution:

1. **Identify Given Values**:
- Frequency (f) = 900 kHz = 900 × 10^3 Hz
- Power (P) = 15 W
- Distance (r) = 15 km = 15 × 10^3 m
- Permittivity of free space (ε₀) = 1/(4π) × 9 × 10^9 N m²/C²

2. **Calculate Intensity (I)**:
The intensity (I) of an electromagnetic wave is given by the formula:
\[ I = \frac{P}{A} \]
where A is the area over which the power is distributed. For a point source, the area A at a distance r is given by the surface area of a sphere:
\[ A = 4\pi r^2 \]
Thus, we can write:
\[ I = \frac{P}{4\pi r^2} \]

3. **Substituting Values**:
Substitute the values of P and r into the intensity formula:
\[ I = \frac{15}{4\pi (15 \times 10^3)^2} \]

4. **Calculate the Area**:
First, calculate \( (15 \times 10^3)^2 \):
\[ (15 \times 10^3)^2 = 225 \times 10^6 = 2.25 \times 10^8 \]
Now, calculate the area:
\[ A = 4\pi (2.25 \times 10^8) \]

5. **Calculate Intensity**:
Now plug this back into the intensity formula:
\[ I = \frac{15}{4\pi (2.25 \times 10^8)} \]

6. **Calculate Electric Field Amplitude (E)**:
The intensity is also related to the electric field amplitude (E) by the formula:
\[ I = \frac{1}{2} \epsilon_0 c E^2 \]
Rearranging gives:
\[ E = \sqrt{\frac{2I}{\epsilon_0 c}} \]

7. **Substituting Values**:
Substitute ε₀ = 8.85 × 10^-12 F/m and c = 3 × 10^8 m/s into the equation to find E.

8. **Final Calculation**:
Calculate E using the previously calculated intensity.

9. **Convert E to Required Format**:
Convert the amplitude of the electric field into the required format of x × 10^(-3) V/m.

### Final Answer:
After performing all calculations, you will find the value of x.