Show that an electron (mass m) will pick-up an amount of kinetic energy equal to

Question

Show that an electron (mass m) will pick-up an amount of kinetic energy equal to eV when accelerated through a voltage difference V (say, in a vacuum from one electrode plate to the other), where e is the charge of the electron (1.6x10-19 coulombs). Do this by starting with the equation F=ma, where m is the mass of the electron and F is the force on the electron equal to eE. E is the electric field resulting from the voltage, V (that is, E = dV/dx).What is an equation for how long will it take the electron to pick up this energy? If V=1000 V and h=1 cm, what is the time?

Explanation / Answer

if V is electric potential

electric field E = dV/dx i

force F = E e

where e is charge

for ion , KE = eV

0.5 mv^2 = eV

velocity v = sqrt(2eV/m)

aslo from F = ma = Ee

a = Ee/m

Vtotal = integration dV = Integration Ee/m) dt

time to = (m Vx d)/Ee = sqrt(2md/Ve)

so here d = 1 cm

V = 1000 V

t = sqrt(2* 9.11 e-31 * 0.01/(1000* 1.6 e -19)

t = 10.67 ns
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