Consider two large flat metallic discs a distance d apart that form a parallel p

Question

Consider two large flat metallic discs a distance d apart that form a parallel plate capacitor. Each plate has an area A. A large potential difference V is applied between the plates. An electron and a proton are simultaneously released from the negative and positive plates and accelerate in the electric field of the capacitor. The two particles are initially at different vertical and horizontal positions, so they will not hit each other, and are sufficiently far apart that we can neglect any interactions between them in comparison to the electric force exerted by the capacitor plates. At some time the two particles will reach the same distance from the negative plate. What is this distance in terms of the total plate separation d and the masses of
the electron and proton, m_e and m_p respectively? (Hint: take care to set up a sensible
coordinate system, with the x axis running from one plate to the other, and the origin at one of
the plates.)

Explanation / Answer

Suppose that negative plate is at x=0 and positive plate is at x=d.

E = V/d

F = q E = qV/d

acceleration of the electron = a_e = qV/(d m_e)

acceleration of the proton = a_p = qV/(d m_p)

d_e = 0.5 a_e t^2 = 0.5 qV/(d m_e) t^2

d_p = 0.5 a_p t^2 = 0.5 qV/(d m_p) t^2

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d_e + d_p = d

0.5 qV/(d m_e) t^2 + 0.5 qV/(d m_p) t^2 = d

(0.5 qV/d) t^2 (1/m_e + 1/m_p) = d

(0.5 qV/d) t^2 ((m_e + m_p)/(m_e * m_p)) = d

>>> t = [(2d2/qV)((m_e * m_p)/(m_e + m_p))]^0.5

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