A micro-wave and an ultrasonic sound wave have the same wavelength. Their frequencies are in the ratio (approximately)

Options

A. `10^(2)`
B. `10^(4)`
C. `10^(6)`
D. `10^(8)`

Explanation

To solve the problem of finding the ratio of frequencies of a microwave and an ultrasonic sound wave that have the same wavelength, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between wavelength, frequency, and velocity**:
The fundamental relationship between the speed (V), frequency (f), and wavelength (λ) of a wave is given by the equation:
\[ V = f \lambda \]
This means that the frequency can be expressed as:
\[ f = \frac{V}{\lambda} \]

2. **Set up the equations for both waves**:
For the microwave, we can write:
\[ f_m = \frac{V_m}{\lambda} \]
For the ultrasonic sound wave, we can write:
\[ f_u = \frac{V_u}{\lambda} \]
Here, \(V_m\) is the velocity of the microwave, \(V_u\) is the velocity of the ultrasonic wave, and \(\lambda\) is the common wavelength.

3. **Equate the wavelengths**:
Since the problem states that both waves have the same wavelength, we have:
\[ \lambda_m = \lambda_u = \lambda \]

4. **Express the ratio of frequencies**:
From the equations for frequencies, we can express the ratio of the frequencies as:
\[ \frac{f_m}{f_u} = \frac{V_m}{V_u} \]

5. **Substitute the known velocities**:
The velocity of microwaves \(V_m\) is approximately \(3 \times 10^8 \, \text{m/s}\) (speed of light), and the velocity of ultrasonic sound waves \(V_u\) is approximately \(3 \times 10^2 \, \text{m/s}\). Therefore, we can substitute these values into the ratio:
\[ \frac{f_m}{f_u} = \frac{3 \times 10^8}{3 \times 10^2} \]

6. **Calculate the ratio**:
Simplifying the above expression gives:
\[ \frac{f_m}{f_u} = \frac{3 \times 10^8}{3 \times 10^2} = \frac{10^8}{10^2} = 10^{8-2} = 10^6 \]

7. **Conclusion**:
Thus, the ratio of the frequencies of the microwave to the ultrasonic sound wave is approximately:
\[ \frac{f_m}{f_u} \approx 10^6 \]

### Final Answer:
The frequencies of the microwave and ultrasonic sound wave are in the ratio of approximately \(10^6\).

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