Practice Problems: Motion of a Charged Particle in an E-field

1. (easy) An electron is released (from rest) in a uniform E-field with a magnitude of 1.5x10³ N/C. Determine the acceleration of the electron due to the E-field.
F= qE = ma
1.6x10^-19(1.5x10³) = (9.1x10^-31)a
a = 2.6x10¹⁴m/s² 

2. (easy) A single proton is accelerated in a uniform E-field (directed eastward) at 3.2x10⁸ m/s². Find the magnitude of the field and direction of the acceleration.
F = qE = ma
(1.6x10^-19)E = (1.7x10^-27)(3.2x10⁸)
E = 3.4 N/C
The acceleration on a positive charge is in the direction of the field: east.

3. (moderate) Two charged particles, one (with a charge of +2μC and a mass m) located on the origin of an axis system and a second (with a charge of +3μC and a mass of 2m) located at x = 1 m are exerting a force on each other. Determine the magnitude of the force and then describe the trajectory each particle will undergo, including their velocities and accelerations.
The instantaneous force magnitude they both exert on each other is by Coulomb's Law. Thus, for the initial positions:
F = kq₁q₂/r²  
F = (9x10⁹)(2x10^-6)(3x10^-6)/1² = 0.054 N
The particles will accelerate away from each other on a straight line. The smaller particle will move along the -x axis, while the larger particle will move along the +x axis. They will both speed up as time goes on, but the smaller particle will speed up faster because, with a lower mass, it will have a greater acceleration due to the common force. As they move apart the accelerations on each will decrease because the force will decrease. At some point the accelerations will be so small as to approach zero, and the particles will essentially stop speeding up and simply move away from each other at a constant speed.

4. (moderate) A charged particle (-3.0C with a mass of 0.0002 kg) is injected into an E-field with an initial speed of 2000 m/s along the +z axis. The E-field is uniform in this region (500 N/C), and directed in the +y direction. Determine the acceleration components for all three directions (x,y, and z). Additionally, calculate the length of time needed to the particle to move 1x10⁸ m in the -y direction and the distance moved along the other two axes over that time frame. Assume that the initial position of the particle is at the origin of the axis system.
The only acceleration will be in the -y direction as the E-field acts on the negative particle in a direction opposite to its own orientation. The accelerations in the x and z directions is zero.
|F| = |q|E
|F| = (3)(500) = 1500 N (in the -y direction)
F_y= ma_y
-1500 = 0.0002(a_y)
a_y = -7.5x10⁶ m/s²
Δy = v_oyt + ½a_yt²
1x10⁸ = 0 + ½(-7.5x10⁶)t²
t = 5.2 s
Distance moved along z axis:
Δz = v_ozt + ½a_zt² = 2000(5.2) + 0 = 10400 m
Distance moved along x axis:
Δx = v_ozt + ½a_zt² = 0 + 0 = 0 m

5. (moderate) Charge q₁ (positive) is located at position (0, 0.50 m) and has a magnitude of 2.9x10^-6 C. Charge q₂ is located at the origin. Assume that these charges are identical and unable to move. A third charge (q₃ = +1.0x10^-9 C and m = 4.0x10^-25 kg) is located at (1.00 m, 0.25 m). Determine the force on and the acceleration of the charge in this position, and describe the trajectory the third charge would take when released in the field caused by the other two charges.
The distance, r, from either q₁ or q₂ to q₃:
r = [1.0² + (0.25)²]^1/2= 1.03 m
The E-field from q₁ and q₂ can be calculated separately, then superpositioned:
E₁ = kq₁/r² = k(2.9x10^-6)/(1.03)² = 2.5x10⁴ N/C (pointing along the line that connects q₁ and q₃, away from q₃ at 346⁰)
E₂ = kq₂/r² = k(2.9x10^-6)/(1.03)² = 2.5x10⁴ N/C (pointing along the line that connects q₂ and q₃, away from q₃ at 14⁰)
The y-component of the E-fields cancel out. The x-components add together to point in the +x direction.
E_x = (2.5x10⁴cos346) + (2.5x10⁴cos14) = 4.9x10⁴ N/C
F = ma
qE = ma
1.0x10^-9(4.9x10⁴) = (4.0x10^-25)a
a = 1.2x10²⁰ m/s²
Once q₃ begins to move it will get further from q₁ and q2 moving in a straight line in the + x direction. It will move faster as time goes on , but with a decreasing acceleration.

6. (moderate) Based on the information shown in the sketch below, determine the trajectory of the positively charged particle as it enters into the E-fields shown.