A lamp radiates power `P_(0)` uniformly in all directions, the amplitude of elctric field strength `E_(0)` at a distance `r` from it is

Options

A. `(P)/(pi c epsilon_(0)r^(2))`
B. `(P)/(2pi c epsilonr^(2))`
C. `sqrt((P)/(pi epsilon_(0)r^(2)c))`
D. `sqrt((P)/(pi epsilon_(0)cr^(2)))`

Answer

To solve the problem of finding the amplitude of electric field strength \( E_0 \) at a distance \( r \) from a lamp that radiates power \( P_0 \) uniformly in all directions, we can follow these steps:

### Step 1: Understand the concept of intensity
The intensity \( I \) of the radiation at a distance \( r \) from the source can be defined as the power per unit area. Since the lamp radiates uniformly in all directions, the area over which the power is distributed at a distance \( r \) is the surface area of a sphere, which is given by:
\[ \text{Area} = 4\pi r^2 \]
Thus, the intensity \( I \) can be expressed as:
\[ I = \frac{P_0}{4\pi r^2} \]

### Step 2: Relate intensity to electric field
The intensity of an electromagnetic wave can also be related to the amplitude of the electric field \( E_0 \) using the formula:
\[ I = \frac{1}{2} \epsilon_0 c E_0^2 \]
where \( \epsilon_0 \) is the permittivity of free space and \( c \) is the speed of light in vacuum.

### Step 3: Set the two expressions for intensity equal
Since both expressions represent the intensity, we can set them equal to each other:
\[ \frac{P_0}{4\pi r^2} = \frac{1}{2} \epsilon_0 c E_0^2 \]

### Step 4: Solve for \( E_0^2 \)
Rearranging the equation to solve for \( E_0^2 \):
\[ E_0^2 = \frac{P_0}{2\pi r^2 \epsilon_0 c} \]

### Step 5: Take the square root to find \( E_0 \)
Taking the square root of both sides gives us the amplitude of the electric field strength \( E_0 \):
\[ E_0 = \sqrt{\frac{P_0}{2\pi r^2 \epsilon_0 c}} \]

### Final Answer
Thus, the amplitude of electric field strength \( E_0 \) at a distance \( r \) from the lamp is:
\[ E_0 = \sqrt{\frac{P_0}{2\pi r^2 \epsilon_0 c}} \]