91Ó°ÊÓ

When two waves travel from different sources to a common point, the path length difference refers to the physical difference in the distances each wave has traveled. In the exercise given, points A and B are sources of waves that travel to point P. To find the path length difference, we subtract the distance from source A to point P, from the distance from source B to point P. The formula used here is:

• \( \Delta L = L_B - L_A \)
• where \( L_B = 5.24 \,\text{m} \) (distance from B to P), and \( L_A = 4.86 \,\text{m} \) (distance from A to P)

By calculating \( \Delta L = 5.24 \,\text{m} - 4.86 \,\text{m} = 0.38 \,\text{m} \), we find how much further one wave has traveled compared to the other. This path length difference can directly influence the phase when the waves meet, affecting whether they interfere constructively or destructively.

The phase difference between two waves is a measure of how aligned or out of sync they are when they reach a certain point. It is crucial in determining the nature of interference at that point. A phase difference of zero or multiples of \( 2\pi \) implies constructive interference, while a half-integer multiple (like \( \pi \), \( 3\pi \)) usually results in destructive interference.

To calculate phase difference \( \Delta \phi \), we use:

• \( \Delta \phi = \frac{2\pi}{\lambda} \times \Delta L \)
• where \( \lambda \) is the wavelength, and \( \Delta L \) is the path length difference

For the exercise, with a wavelength of \( 0.02 \,\text{m} \), substituting in the values:

• \( \Delta \phi = \frac{2\pi}{0.02} \times 0.38 = 37.6991 \,\text{ radians} \)

This tells us how much one wave is leading or lagging behind the other in radians, ultimately governing the resulting interference pattern.

Since the problem supplies the wavelength in centimeters, but all other measurements are in meters, converting wavelengths to consistent units is essential. This ensures calculations are accurate and avoids mistakes caused by using mismatched units.

Here's the conversion process:

• The given wavelength is \( 2.00 \,\text{cm} \).
• Converting into meters, we divide by 100 (since 1 meter = 100cm): \( \lambda = 2.00 \,\text{cm} = 0.02 \,\text{m} \).

Consistency in units is a crucial step in solving physics problems, allowing us to deftly apply formulas without the risk of errors due to improper unit conversion. This step might seem simple, but it is fundamental to determining the correct phase difference.