Equation of a progressive wave is given by
`y=0.2 cos pi (0.04t + .02 x -(pi)/(6))`
The distance is expressed in cm and time in second. What will be the minimum distance between two particles having the phase difference of `pi //2`

Options

A. 4 cm
B. 8 cm
C. 25 cm
D. 12.5 cm

Answer

To find the minimum distance between two particles having a phase difference of \( \frac{\pi}{2} \) in the given progressive wave, we can follow these steps:

### Step 1: Identify the wave equation parameters
The given wave equation is:
\[ y = 0.2 \cos \left( \pi (0.04t + 0.02x - \frac{\pi}{6}) \right) \]
From this equation, we can identify:
- Angular frequency \( \omega = 0.04\pi \)
- Wave number \( k = 0.02\pi \)

### Step 2: Calculate the wavelength
The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]
Rearranging gives:
\[ \lambda = \frac{2\pi}{k} \]
Substituting \( k = 0.02\pi \):
\[ \lambda = \frac{2\pi}{0.02\pi} = \frac{2}{0.02} = 100 \text{ cm} \]

### Step 3: Relate phase difference to path difference
The phase difference \( \Delta \phi \) between two particles is given as \( \frac{\pi}{2} \). The relationship between phase difference and path difference \( \Delta x \) is given by:
\[ \Delta \phi = \frac{2\pi}{\lambda} \Delta x \]
Rearranging gives:
\[ \Delta x = \frac{\lambda}{2\pi} \Delta \phi \]

### Step 4: Substitute values to find path difference
Substituting \( \lambda = 100 \text{ cm} \) and \( \Delta \phi = \frac{\pi}{2} \):
\[ \Delta x = \frac{100 \text{ cm}}{2\pi} \cdot \frac{\pi}{2} \]
This simplifies to:
\[ \Delta x = \frac{100 \text{ cm}}{2} = 50 \text{ cm} \]

### Step 5: Calculate the minimum distance
Since we are looking for the minimum distance between two particles with a phase difference of \( \frac{\pi}{2} \), we need to consider that the path difference corresponds to half the wavelength:
\[ \Delta x = \frac{100 \text{ cm}}{4} = 25 \text{ cm} \]

Thus, the minimum distance between the two particles is:
\[ \Delta x = 25 \text{ cm} \]

### Final Answer
The minimum distance between two particles having a phase difference of \( \frac{\pi}{2} \) is \( 25 \text{ cm} \).

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