A proton enters the gap between a pair of metal plates (an electrostatic separat

Question

A proton enters the gap between a pair of metal plates (an electrostatic separator) that produces a uniform, vertical electric field between them. Ignore the effect of gravity on the proton.

(a) Assuming that the length of the plates is 15.8 cm, and that the proton will approach the plates at a speed of 15.9 km/s, what electric field strength should the plates be designed to provide, if the proton must be deflected vertically by 1.41 10-3 rad?

(b) What speed does the proton have after exiting the electric field?

Explanation / Answer

tan ? = y / x, ? = 1.41 * 10-3 rad
y = x * tan ?
= 15.8 * tan (1.41 * 10-3)
= 3.88 * 10-4 m
time taken to traverse the plate length t = x / v
= 15.8 cm / 15.9 km/s
= 15.8 * 10-2 / 15.9 * 10^3
= 9.937 * 10-6 s
in same time, the proton must cover distance y vertically. Hence by second equation of motion
y = uy * t + (1/2) * a * t2
since initial vertical velocity uy = 0
3.88 * 10-4 = 0 * t + 0.5 * a * (9.937 * 10-6)2
acceleration a = 2 * 3.88 * 10-4 / (9.937 * 10-6)2
= 7.862 * 10^6 m/s2
F = m * a = q * E
hence the electric field E = m * a / q
= 1.66 * 10-27 * 7.862 * 106 / 1.6 * 10-19

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