Before television screens were all flat, they used to work by boilingoff electro

Question

Before television screens were all flat, they used to work by boilingoff electrons (“cathode rays”) from a hot filament and acceleratingthem towards a phosphor-coated screen. The acceleration wasperformed with a pair of charged parallel plates, as in a capacitor, withsmall holes to allow the beam to pass through. The beam was steeredhorizontally and vertically across the screen by applying much smallervoltages across pairs of deflecting plates. By varying these deflectingvoltages, the beam was scanned repeatedly across and down the screen,repainting the image on the phosphor many times per second.First let the accelerating voltage be ?Va, the deflecting voltage be ?Vd ,the length of the deflecting plates be d, and the distance from the end of the deflecting plates to the screen be ?x. By what distance ?y is the beam deflected vertically on the screen? Now suppose that ?Va = 30 kV, ?Vd = 150 V, d = 7.5 cm, and ?x = 55 cm; how does ?y work out numerically?

Please show all work and relevant formulae. Thank you, it helps my learning.

Explanation / Answer

the vertical distance traveled by the particle between the plates is,

y1 = (1/2)at^2 = (1/2)(F/m)(d/v)^2

the vertical distance traveled by the particle beyond the plates is,

y2 = [Fd/mv^2]dx

thus, the total distance is,

  y = y1+y2

= (1/2)(F/m)(d/v)^2 +  [Fd/mv^2]x

= [Fdx /mv^2][1+ (d/x)]

let K = (1/2)mv^2 = qV then mv^2 = 2qV

hence,

   y = [Fdx / 2qV][1+ (d/x)]

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