An electron is projected horizontally exactly midway between two parallel horizontal metal plates. The length of the plates is 10 ~cm and the separation between them is 2 ~cm . A potential difference of 91 ~V is applied across the plates. The minimum initial horizontal velocity required for the electron to just emerge from the plates without hitting the upper plate is (Given: mass of electron = 9.1 10⁻³¹ ~kg , charge

Step-by-step solution

Let the initial horizontal velocity be v_x . The electric field between the plates is: E = V d = 91 2 10⁻² = 4550 ~V/m The vertical acceleration of the electron is: a_y = eE m = 1.6 10⁻¹⁹ 4550 9.1 10⁻³¹ = 8 10¹⁴ ~m/s^2 The time taken to cross the plates of length L = 10 ~cm is: t = L v_x = 0.1 v_x For the electron to just emerge without hitting the upper plate, its maximum vertical deflection must be half the plate separation: y = d 2 = 1 ~cm = 0.01 ~m Using the second equation of motion for the vertical direction