Microwaves from a transmitter are directed normally toward a plane reflector. A detector moves along the normal to the reflector. Between positions of 14 successive maxima the detector travels a distance 0.14 m. The frequency of the transmitter is `(c = 3 xx 10^(8) m s^(-1))`.

Options

A. `1.5 xx 10^(10) Hz`
B. `3.0 xx 10^(10) Hz`
C. `1.5 xx 10^(9) Hz`
D. `3.0 xx 10^(9) Hz`

Explanation

To solve the problem, we need to determine the frequency of the microwaves emitted by the transmitter. Here’s a step-by-step breakdown of the solution:

### Step 1: Understanding the Problem
The detector moves along the normal to the reflector and detects maxima due to the interference of direct and reflected microwaves. The distance traveled by the detector between 14 successive maxima is given as 0.14 m.

### Step 2: Relating Distance to Wavelength
The distance between two successive maxima in an interference pattern is equal to half the wavelength (λ/2). Therefore, if the detector travels a distance of 0.14 m for 14 maxima, we can express this mathematically.

### Step 3: Setting Up the Equation
Since there are 14 maxima, the number of half wavelengths (λ/2) that fit into the distance of 0.14 m is 14. Thus, we can write:
\[ 14 \times \frac{\lambda}{2} = 0.14 \text{ m} \]

### Step 4: Solving for Wavelength (λ)
Now, we can solve for λ:
\[ \frac{14 \lambda}{2} = 0.14 \]
\[ 7\lambda = 0.14 \]
\[ \lambda = \frac{0.14}{7} = 0.02 \text{ m} \]

### Step 5: Using the Wave Equation
We know the relationship between the speed of light (c), frequency (n), and wavelength (λ) is given by:
\[ c = n \lambda \]
where:
- \( c = 3 \times 10^8 \text{ m/s} \)
- \( \lambda = 0.02 \text{ m} \)

### Step 6: Solving for Frequency (n)
Rearranging the equation to solve for frequency (n):
\[ n = \frac{c}{\lambda} \]
Substituting the known values:
\[ n = \frac{3 \times 10^8}{0.02} \]
Calculating this gives:
\[ n = 3 \times 10^8 \div 0.02 = 1.5 \times 10^{10} \text{ Hz} \]

### Final Answer
The frequency of the transmitter is:
\[ n = 1.5 \times 10^{10} \text{ Hz} \]