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Relation Between Path Difference And Phase Difference | Relation Between Path Difference, Phase Difference & Time Difference

Relation Between Path Difference And Phase Difference | Relation Between Path Difference, Phase Difference & Time Difference :- When two waves travel through a medium, they may reach a given point at different times or after traveling different distances. This leads to differences in their path, phase, and time, which are closely related to each other. These quantities are very important in understanding interference, wave motion, and many optical phenomena.

1. Path Difference : The path difference is the difference in the distances traveled by two waves to reach the same point.

Path of first wave = x₁
ath of second wave = x₂

then path difference (Δx),

Δx = x₂ – x₁

It is usually measured in meters. Path difference determines how much ahead or behind one wave is compared to another in space.

2. Time Difference : The time difference is the difference in the times taken by two waves to reach the same point.

Time taken by first wave = t₁
Time taken by second wave = t₂

then time difference (Δt),

Δt = t₂ – t₁

3. Phase Difference : The phase difference tells how much one wave is ahead or behind another in terms of angular position in its cycle.

Phase difference is represented by and is measured in radians/degrees.

One complete cycle corresponds to :

2π radians = λ distance = T time

where
= wavelength
= time period

4. Relation Between Path Difference And Phase Difference

In wave motion, a wave travels a distance equal to one wavelength (λ) in one time period (T), and the phase changes by radians during this time. So, we observe that

Path difference of λ corresponds to a phase difference of 2π radians

⇒ Path difference of 1 corresponds to a phase difference of (2π/λ) radians

⇒ Path difference of Δx corresponds to a phase difference of (2π/λ) × Δx radians

$\displaystyle \Rightarrow \Delta \phi =\left( \frac{2\pi }{\lambda } \right)\Delta x$

$\displaystyle \frac{\Delta \phi }{2\pi }=\frac{\Delta x}{\lambda }$ …..(1)

Where Δx is in meters and

5. Relation Between Path Difference And Time Difference

Path difference of λ corresponds to a time difference of T seconds

⇒ Path difference of 1 corresponds to a time difference of (T/λ) seconds

⇒ Path difference of Δx corresponds to a time difference of (T/λ) × Δx seconds

$\displaystyle \Rightarrow \Delta t=\left( \frac{T}{\lambda } \right)\Delta x$

$\displaystyle \frac{\Delta t}{T}=\frac{\Delta x}{\lambda }$ …..(2)

6. Combined Relation

From equations (1) and (2),

$\displaystyle \frac{\Delta \phi }{2\pi }=\frac{\Delta x}{\lambda }=\frac{\Delta t}{T}$ …..(3)

$\displaystyle \frac{Phase\text{ d}ifference}{2\pi }=\frac{Path\text{ }difference}{\lambda }=\frac{Time\text{ }difference}{T}$

Example 1.

A progressive wave of frequency 500 Hz is travelling with a velocity of 360 m/s. How far apart are two points 60º out of phase.

Solution :

We know that for a wave

$\displaystyle \upsilon =\nu \lambda$

$\displaystyle \Rightarrow \lambda =\frac{\upsilon }{\nu }=\frac{300}{500}=0.72m$

Given

$\displaystyle \Delta \phi ={{60}^{\circ }}=\frac{\pi }{{{180}^{\circ }}}\times {{60}^{\circ }}=\frac{\pi }{3}radian$

We know that,

$\displaystyle \frac{\Delta \phi }{2\pi }=\frac{\Delta x}{\lambda }$

Hence path difference (Δx),

$\displaystyle \Delta x=\frac{\Delta \phi }{2\pi }\times \lambda =\frac{{}^{\pi }\!\!\diagup\!\!{}_{3}\;}{2\pi }\times 0.72=0.12m$

Relation Between Path Difference And Phase Difference | Relation Between Path Difference, Phase Difference & Time Difference

4. Relation Between Path Difference And Phase Difference

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