Question

A charged particle as deflected by two mutually perpendicular oscillation electeic field such that the displecement of the particle that in each one at then given by `x = A sin (omega t) and y = A sin (omega t + (pi)/(6))` respectively. The frequency following by the changed particle is
A. a circle with equation `x^(2) + y^(2) = A^(2)`
B. a straight line with equation ` y=sqrt(3)a`
C. an ellipse with equation `x^(2) + y^(2) - xy= (3)/(4)A^(2)`
D. an ellipse with equation `x^(2) + y^(2) - sqrt(3)xy = (1)/(4)A^(2)`

Answer 1

Correct Answer - D
When two `SHM` (which one perpendicular to end other ) are given by `x = omega t and y = A_(2) sin (omega t + delta)` and when they are combirted the trangectory of the particle is given by
`(x^(2))/(A_(1)^(2)) + (y^(2))/(A_(2)^(2)) - (2xy)/(A_(1)A_(2)) cos delta = sin^(2) delta`
(This is equation of ellipse)
in the given equation , phase different is `delta = (pi)/(6)` and `A_(1) = A_(2) = A`
`:.(x^(2))/(A_(1)^(2)) + (y^(2))/(A_(2)^(2)) - (2xy)/(A_(1)A_(2)).sqrt(3)/(2) = (1)/(4)`
`x^(2) + y^(2) - sqrt(3)xy = (1)/(4) A^(2)` and this in an elipse