An observer is moving with half the speed of light towards a stationary microwave source emitting waves at frequency 10 GhHz. What is the frequency of the microwave measured by the observer? (speed of light=`3xx10^(8)ms`)

Options

A. `"tan"^(-1)("sin" theta_(c))`
B. `(1)/("tan"^(-1) ("sin"theta_(c)))`
C. `"cos"^(-1) ("sin"theta_(c))`
D. `(1)/("cos"^(-1)("sin"theta_(c)))`

Answer

To solve the problem of finding the frequency of the microwave measured by an observer moving towards a stationary microwave source, we can use the relativistic Doppler effect formula. Here’s the step-by-step solution:

### Step 1: Identify the given values
- Frequency of the source (f₀) = 10 GHz = \(10 \times 10^9\) Hz
- Speed of light (c) = \(3 \times 10^8\) m/s
- Speed of the observer (v) = \(c/2 = \frac{3 \times 10^8}{2} = 1.5 \times 10^8\) m/s

### Step 2: Use the relativistic Doppler effect formula
The formula for the apparent frequency (f') when the observer is moving towards a stationary source is given by:

\[ f' = f_0 \sqrt{\frac{c + v}{c - v}} \]

### Step 3: Substitute the values into the formula
Now we can substitute the values into the formula:

\[ f' = 10 \times 10^9 \sqrt{\frac{3 \times 10^8 + 1.5 \times 10^8}{3 \times 10^8 - 1.5 \times 10^8}} \]

### Step 4: Simplify the expression inside the square root
Calculating the numerator and denominator:

- Numerator: \(3 \times 10^8 + 1.5 \times 10^8 = 4.5 \times 10^8\)
- Denominator: \(3 \times 10^8 - 1.5 \times 10^8 = 1.5 \times 10^8\)

So we have:

\[ f' = 10 \times 10^9 \sqrt{\frac{4.5 \times 10^8}{1.5 \times 10^8}} \]

### Step 5: Simplify the fraction
The fraction simplifies to:

\[ \frac{4.5 \times 10^8}{1.5 \times 10^8} = 3 \]

Thus, we can rewrite the equation:

\[ f' = 10 \times 10^9 \sqrt{3} \]

### Step 6: Calculate the square root and the final frequency
Now, we know that \(\sqrt{3} \approx 1.732\):

\[ f' = 10 \times 10^9 \times 1.732 = 17.32 \times 10^9 \text{ Hz} \]

Converting this back to GHz:

\[ f' \approx 17.32 \text{ GHz} \]

### Final Answer
The frequency of the microwave measured by the observer is approximately **17.32 GHz**.

---