In fig., light of wavelength `lambda = 5000 Å` is incident on the slits (in a horizontally fixed place).
Here, `d = 1 mm` and `D = 1 m`
Take origin at O and XY plane as shown in the figure. The screen is released from rest from the initial position as shown

Velocity of 2nd maixma w.r.t central maxima at `t = 2`s is

Options

A. `8(cms^(-1))hati+20(ms^(-1))hatj`
B. `8(cms^(-1))hati`
C. `2(cms^(-1))hati`
D. `86(ms^(-1))hati`

Answer

C

Explanation

Path difference corresponding to P.
`Delta x = d sin theta = d tantheta , Delta x = (d xx x)/(D^(1))`
For `2^(nd)` maxima
`Delta x = 2lambda implies (xd)/(D^(1)) = 2lambda , x = (2lambda D^(1))/(d)`
Location of central maxima is `[O, D+(1)/(2)"gt"^(2)]`
Location of `2^(nd)` maxima is `[(2lambda D^(1))/(d) , D^(1)]`
Velocity of `2^(nd)` maxima wrt central maxima,
`vecV_(red) = (2lambda)/(d) [0+"gt"]hati = (2lambda)/(d)xx"gt" hati`
`V_(red) = (2xx5000xx10^(-10)xx10xx2hati)/(1xx10^(-3)) = 2xx10^(-2)ms^(-1)hati`
`vecV_(red) = 2(cms^(-1)).hati`