Question

An electric field E exists in the region between the two electrically charged parallel plates shown above. A beam of electrons of mass m, charge q, and velocity v enters the region through a small hole at position A. The electrons exit the region between the plates through a small hole at position B. Express your answers to the following questions in terms of the quantities m, q, E, θ, and v. Ignore the effects of gravity.

a. i. On the diagram of the parallel plates above, draw and label a vector to show the direction of the electric field E between the plates.

ii. On the following diagram, show the direction of the force(s) acting on an electron after it enters the region between the plates.

iii. On the diagram of the parallel plates above, show the trajectory of an electron that will exit through the small hole at position B.

b. Determine the magnitude of the acceleration of an electron after it has entered the region between the parallel plates.

c. Determine the total time that it takes the electrons to go from position A to position B.

d. Determine the distance d between positions A and B.

e. Now assume that the effects of gravity cannot be ignored in this problem. How would the distance where the electron exits the region between the plates change for an electron entering the region at A? Explain your reasoning.

Answer 1

  

b. F = ma = qE gives a = qE/m

c. The acceleration is downward and at the top of the path, v_y  = v_0y – at = 0 and v_0y = v sin θ which gives  t _top = v sin θ/a or t_total = 2t_top = 2v sin θ/a and substituting a from part b gives t = (2mv sin θ)/qE  

d. d = x_xt where v_x = v cos θ giving d = (2mv² sin θ cos θ)/qE

e. The distance would be less because gravity, acting downward, will increase the electron’s downward acceleration, decreasing the time spent in the field.