Plane microwaves from a transmitter are directed normally towards a plane reflector. A detector moves along the normal to the reflection. Between positions of 14 successive maxima, the detector travels a distance 0.14m. If the velocity of light is `3 xx 10^(8) m//s`, find the frequency of the transmitter.

Options

A. `1.5 xx 10^(10)Hz`
B. `10^(10)Hz`
C. `3 xx 10^(10)Hz`
D. `6 xx 10^(10)Hz`

Answer

To solve the problem step by step, we will follow the given information and apply the relevant formulas.

### Step 1: Understand the relationship between maxima and wavelength
The distance between two successive maxima in a wave pattern is given by:
\[ \text{Distance between two maxima} = \frac{\lambda}{2} \]
where \(\lambda\) is the wavelength.

### Step 2: Calculate the distance traveled by the detector
The detector travels a total distance of 0.14 m while passing through 14 successive maxima. Therefore, the total distance covered can be expressed as:
\[ \text{Total distance} = 14 \times \left(\frac{\lambda}{2}\right) \]

### Step 3: Set up the equation for wavelength
From the above relationship, we can write:
\[ 0.14 = 14 \times \left(\frac{\lambda}{2}\right) \]
This simplifies to:
\[ 0.14 = 7\lambda \]

### Step 4: Solve for the wavelength
To find \(\lambda\), we rearrange the equation:
\[ \lambda = \frac{0.14}{7} \]
Calculating this gives:
\[ \lambda = 0.02 \, \text{m} \]

### Step 5: Use the wavelength to find the frequency
The frequency \(f\) of the wave can be calculated using the formula:
\[ f = \frac{c}{\lambda} \]
where \(c\) is the speed of light, given as \(3 \times 10^8 \, \text{m/s}\).

### Step 6: Substitute the values into the frequency formula
Substituting the known values:
\[ f = \frac{3 \times 10^8}{0.02} \]

### Step 7: Calculate the frequency
Calculating this gives:
\[ f = 1.5 \times 10^{10} \, \text{Hz} \]

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

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