Question

The distance between two particles in a wave motion vibrating out of phase of \[\pi \] radians is:
A. \[\lambda /4\]
B. \[\lambda /2\]
C. \[3\lambda /4\]
D. \[\lambda \]

Answer 1

Hint:Use the formula for phase difference between the two particles in the wave motion. This formula gives the relation between the phase difference between the two particles, path difference or distance between the two particles and wavelength of the wave. Substitute the given value of phase difference in this equation and calculate the distance between the two particles.

Formula used:
The phase difference \[\Delta \phi \] between the two particles in the wave motion is given by \[\Delta \phi = \dfrac{{2\pi \Delta x}}{\lambda }\] …… (1)
Here, \[\Delta x\] is the path difference between the particles and \[\lambda \] is the wavelength of the wave.

Complete step by step answer:
We have given that the phase difference between the two particles vibrating out of phase is \[\pi \] radians.

\[\Delta \phi = \pi \,{\text{radian}}\]

We have asked to determine the distance between these two particles in the wave motion.
Rearrange the equation (1) for path difference \[\Delta x\] between the particles.
\[\Delta x = \dfrac{{\Delta \phi \lambda }}{{2\pi }}\]
Substitute \[\pi \,{\text{radian}}\] for \[\Delta \phi \] in the above equation.
\[\Delta x = \dfrac{{\left( {\pi \,{\text{radian}}} \right)\lambda }}{{2\pi }}\]
\[ \Rightarrow \Delta x = \dfrac{{\pi \lambda }}{{2\pi }}\]
\[ \Rightarrow \Delta x = \dfrac{\lambda }{2}\]
Thus, the path difference between the two particles in the wave motion is \[\dfrac{\lambda }{2}\].

Therefore, the distance between the two particles in the wave motion which are out of phase is \[\dfrac{\lambda }{2}\].

Hence, the correct option is B.

Additional information:
Two particles on a wave are said to be in phase if they have phase difference of zero.
Two particles on a wave are said to be out of phase if they have a non-zero phase difference.

Note:The students may think that the path difference between the two particles in the wave motion cannot be the distance between the two particles in the wave motion. But the students should keep in mind that the path difference between these two particles in the wave motion here is the distance between the two particles on the wave in the wave motion.