An `EM` wave radiates out wards from a dipole antenna with `E_(0)` as the amplitude of its electric filed vector. The electric field `E_(0)` which transports significant energy from the source falls off as

Options

A. `(1)/(r^(3))`
B. `(1)/(r^(2))`
C. `(1)/(r)`
D. remains constant

Explanation

To solve the problem of how the electric field amplitude \( E_0 \) of an electromagnetic (EM) wave radiating from a dipole antenna falls off with distance, we can follow these steps:

### Step 1: Understand the Concept of EM Waves
Electromagnetic waves consist of oscillating electric and magnetic fields that propagate through space. The energy carried by these waves is related to the amplitude of the electric field.

**Hint:** Recall that the energy transported by an EM wave is proportional to the square of the electric field amplitude.

### Step 2: Recognize the Source of EM Waves
In this case, the source of the EM wave is a dipole antenna. As the wave propagates outward from the antenna, the intensity of the wave decreases with distance.

**Hint:** Think about how energy spreads out over an increasing area as distance from the source increases.

### Step 3: Apply the Inverse Square Law
The intensity \( I \) of an EM wave is defined as the power per unit area. For a point source (like a dipole antenna), the intensity falls off as the inverse square of the distance from the source:
\[ I \propto \frac{1}{r^2} \]
where \( r \) is the distance from the antenna.

**Hint:** Remember that intensity is related to the amplitude of the electric field.

### Step 4: Relate Intensity to Electric Field Amplitude
The intensity of an electromagnetic wave is also related to the amplitude of the electric field:
\[ I \propto E_0^2 \]
Combining the two relationships, we can express the electric field amplitude in terms of distance:
\[ E_0^2 \propto \frac{1}{r^2} \]

**Hint:** Consider how you can express \( E_0 \) in terms of \( r \).

### Step 5: Derive the Relationship for Electric Field Amplitude
From the above relationship, we can take the square root to find the amplitude of the electric field:
\[ E_0 \propto \frac{1}{r} \]
This means that the electric field amplitude \( E_0 \) falls off inversely with distance from the dipole antenna.

**Hint:** Think about how this relationship indicates that as you move further from the source, the electric field strength decreases.

### Conclusion
Thus, the electric field \( E_0 \) which transports significant energy from the source falls off as:
\[ E_0 \propto \frac{1}{r} \]

### Final Answer
The correct option is that the electric field \( E_0 \) falls off inversely with the distance \( r \) from the dipole antenna.

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