Subject: Applied Physics 2

Topic: Motion Of Charged Particles In Electric And Magnetic Fields

Difficulty: Low

Lissajous figures are defined as “the displacement patterns traced by the electron beam acted upon by two mutually perpendicular sinusoidal signals”

Consider two sinusoidal waves of equal time period and a phase difference acting perpendicular to each other. These are given by

X = a sin (ωt +ϕ ) --------------------------(1)

Y = b sin ωt. ----------------------------------(2)

where a and b are amplitudes. Equation (1) can be expanded as

x / a = sinωtcosϕ +cosωtsinϕ

Applying equation (2) in it,then

x/a = (y / b)ϕ + cos ϕ $\sqrt{(1-y2 / b2)}$sinϕ

Simplifying the above equation, we get

$x^2$ / $a^2$ + $y^2$ / $b^2$ -(2xy/ab) cosϕ =$sin^2 $ϕ-----------------------------------------(3)

Which is the general equation o an ellipse. Hence the displacement of the electron beam traces out an ellipse on the screen.