Question

$$100\pi$$ phase difference$=$_____ path difference
$\begin{array}{rl}& \text{A}\text{. 10}\lambda \\ & \text{B}\text{. 25}\lambda \\ & \text{C}\text{. 50}\lambda \\ & \text{D}\text{. 100}\lambda \end{array}$

Answer 1

Hint: A wave is represented by sinusoidal form. So at the time of measurement I.e. $t=0$ it does not pass through the origin and said to have a phase difference or phase shift. The phase difference or phase shift is the angle $\varphi$ in degrees or radians that the waveform has shifted from a certain reference point along the horizontal zero axis. Path difference is the difference between the path travelled of two waves at a certain point.

Formula used:
Relationship between the phase difference$\left(\mathrm{\Delta }\varphi \right)$ , path difference$\left(\mathrm{\Delta }x\right)$ and the wavelength $\left(\lambda \right)$of a wave is given by $\mathrm{\Delta }\varphi ={\displaystyle \frac{2\pi }{\lambda }}\times \mathrm{\Delta }x$

Complete answer:
Here the phase difference given is $100\pi$ i.e. $\mathrm{\Delta }\varphi =100\pi$
But the relationship between the phase difference$\left(\mathrm{\Delta }\varphi \right)$ , path difference$\left(\mathrm{\Delta }x\right)$ and the wavelength $\left(\lambda \right)$of a wave is given by
 $\begin{array}{rl}& \mathrm{\Delta }\varphi ={\displaystyle \frac{2\pi }{\lambda }}\times \mathrm{\Delta }x,\text{ so }\\ & \Rightarrow 100\pi ={\displaystyle \frac{2\pi }{\lambda }}\times \mathrm{\Delta }x\\ & \Rightarrow \mathrm{\Delta }x=50\lambda \end{array}$

So, the correct answer is “Option C”.

Additional Information:
The equation of wave motion is given by
$y(x,t)=A\sin (\omega t-kx+{\varphi }_0)$
where
$\begin{array}{rl}& y(x,t)=\text{ displacement}\\ & A=\text{ amplitude}\\ & \omega =\text{ angular frequency}\\ & t=\text{time}\\ & k=\text{angular wave number}\\ & x=\text{position}\\ & {\varphi }_0=\text{ initial phase}\end{array}$
Phase: The phase of a harmonic wave is a quantity that gives complete information of the wave at any time and any position.
If a wave is represented by $y(x,t)=A\sin (\omega t-kx+{\varphi }_0)$
Then phase of the wave at position $x$ and time$t$ is given by. $\varphi =(\omega t-kx+{\varphi }_0)$
So clearly the phase of a wave is periodic both in time and space. So at a given point $x$the phase changes with time and at a given time $t$ the phase changes with position $x$
Now, phase ,$\varphi =(\omega t-kx+{\varphi }_0)$
Taking $x$as a constant differentiate $\varphi$ with respect to $t$ then,
${\displaystyle \frac{\mathrm{\Delta }\varphi }{\mathrm{\Delta }t}}=\omega$
Thus the phase change at a given position $x$ in time $\mathrm{\Delta }t$ is given by
$\mathrm{\Delta }\varphi =\omega \mathrm{\Delta }t={\displaystyle \frac{2\pi }{T}}\mathrm{\Delta }t$
Where T= Time period of the wave.
Taking time as constant
$\begin{array}{rl}& {\displaystyle \frac{\mathrm{\Delta }\varphi }{\mathrm{\Delta }x}}=-k\\ & \Rightarrow \mathrm{\Delta }\varphi =-k\mathrm{\Delta }x=-{\displaystyle \frac{2\pi }{\lambda }}\mathrm{\Delta }x\end{array}$

Note:
Note that from the equation of wave motion you can calculate the particle velocity particle acceleration, particle energy. Generally waves are of three main types such as.
Mechanical wave-it is the wave which obeys Newton’s law and can only exist within material medium e.g. water wave, sound wave, seismic wave.
Electromagnetic wave- wave associated with a charged objects.e.g x-ray.
Matter waves-waves associated with fundamental particles like electron,proton and neutron.