The amplitude of electric field at a distance `r` from a point source of power `P` is (taking 100% efficiency).

Options

A. `sqrt((P)/(2pir^(2)cepsi_(0)))`
B. `sqrt((P)/(4pir^(2)cepsi_(0)))`
C. `sqrt((P)/(8pir^(2)cepsi_(0)))`
D. `(P)/(2pir^(2)cepsi_(0))`

Answer

To find the amplitude of the electric field \( E_0 \) at a distance \( r \) from a point source of power \( P \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Intensity**: The intensity \( I \) of the electromagnetic wave is defined as the power \( P \) per unit area. For a point source radiating uniformly in all directions, the area \( A \) at a distance \( r \) is given by the surface area of a sphere, which is \( 4\pi r^2 \). Thus, the intensity can be expressed as:
\[ I = \frac{P}{A} = \frac{P}{4\pi r^2} \]

2. **Relating Intensity to Electric Field**: The intensity \( I \) is also related to the root mean square (RMS) value of the electric field \( E_{\text{rms}} \) by the formula:
\[ I = \frac{E_{\text{rms}}^2}{c \mu_0} \]
where \( c \) is the speed of light and \( \mu_0 \) is the permeability of free space.

3. **Expressing \( E_{\text{rms}} \)**: We know that the peak amplitude \( E_0 \) is related to the RMS value by:
\[ E_{\text{rms}} = \frac{E_0}{\sqrt{2}} \]
Therefore, we can rewrite the intensity in terms of \( E_0 \):
\[ I = \frac{\left(\frac{E_0}{\sqrt{2}}\right)^2}{c \mu_0} = \frac{E_0^2}{2c \mu_0} \]

4. **Setting the Two Expressions for Intensity Equal**: Now, we can set the two expressions for intensity equal to each other:
\[ \frac{P}{4\pi r^2} = \frac{E_0^2}{2c \mu_0} \]

5. **Solving for \( E_0^2 \)**: Cross-multiplying gives us:
\[ E_0^2 = \frac{2c \mu_0 P}{4\pi r^2} \]
Simplifying this, we have:
\[ E_0^2 = \frac{c \mu_0 P}{2\pi r^2} \]

6. **Taking the Square Root**: Finally, we take the square root to find the amplitude of the electric field:
\[ E_0 = \sqrt{\frac{c \mu_0 P}{2\pi r^2}} \]

7. **Substituting \( \mu_0 \)**: We know that \( c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \), thus \( \mu_0 = \frac{1}{c^2 \epsilon_0} \). Substituting this into our expression for \( E_0 \):
\[ E_0 = \sqrt{\frac{P}{2\pi r^2 \epsilon_0 c}} \]

### Final Result:
Thus, the amplitude of the electric field at a distance \( r \) from a point source of power \( P \) is given by:
\[ E_0 = \sqrt{\frac{P}{2\pi r^2 \epsilon_0 c}} \]