Cathode ray tube: In the cathode ray tube shown in figure 2, a beam of electrons

Question

Cathode ray tube: In the cathode ray tube shown in figure 2, a beam of electrons is injected horizontally into the exact center of the parallel-plate region. Here L = 3.0 cm and d = 0.20 cm. In the region between the plates there is a uniform electric field E that points vertically downward. The magnitude of this electric field is E = 1.7 times 10^4 N/C. What minimum speed upsilon_0 should the electrons have in order to ensure that they do not strike the upper plate? (Source: Adapted from Ohanian and Market, Physics for Engineers and Scientists).

Explanation / Answer

Given:

Length of plate = L = 3x 10^-2 m

time taken by electron = L/vo

separation of plates = d0= 0.2 cm= 0.2 x 10^-2 m

vertical distance the electron should travel in order to not to strike upper plate is less than d0/2=0.2/2=0.1 x 10^-2 m

electric field between plates = E= 1.7 x 10^4 N/C

Force on electron = qE

= 1.6 x 10^-19 x 1.7 x 10^4

acceleration of electron =a = force/mass of electron

= 1.6 x 10^-19 x 1.7 x 10^4/9.11 x 10^-31

as initial vertical velocity is zero, applying equation of motion

y= 1/2at^2

so, 0.1 x 10^-2 = 1/2 x (1.6 x 10^-19 x 1.7 x 10^4/9.11 x 10^-31) x ( 3x 10^-2/vo)^2

so vo= 3.66x 10^7 m/s

the minimum velocity should be greater than 3.66 x10^7 m/s

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