Consider a disk of radius 10 cm and positive surface charge density +3.7 mC/m2. A particle of charge -4.5 mC and mass 75. mg accelerates under the effects of the electric field caused by the disk, from a point at a perpendicular distance from the center of the disk.
The final speed of the particle is 1.0 m/s and the work done on the particle by the electric field is -3.0 mJ.
How fast and in what direction was the particle originally moving?
To determine how fast and in what direction the particle was originally moving, we can use the conservation of energy principle. The initial kinetic energy of the particle is given by:
K_initial = (1/2) * m * v_initial^2
where m is the mass of the particle and v_initial is its initial speed. Let's denote the initial kinetic energy as E_initial.
E_initial = (1/2) * m * v_initial^2
The final kinetic energy of the particle is given by:
K_final = (1/2) * m * v_final^2
where v_final is the final speed of the particle. Let's denote the final kinetic energy as E_final.
E_final = (1/2) * m * v_final^2
The work done on the particle by the electric field is equal to the change in kinetic energy:
W = E_final - E_initial
Given that the work done is -3.0 mJ, we can write the equation as:
-3.0 mJ = [(1/2) * m * v_final^2] - [(1/2) * m * v_initial^2]
Substituting the values we know:
-3.0 * 10^-3 J = (1/2) * (75 * 10^-6 kg) * (1.0 m/s)^2 - (1/2) * (75 * 10^-6 kg) * v_initial^2
Simplifying the equation:
-3.0 * 10^-3 J = 0.0375 * 0.5 - 0.0375 * v_initial^2
0.0375 * v_initial^2 = 0.0375 * 0.5 + 3.0 * 10^-3
v_initial^2 = (0.0375 * 0.5 + 3.0 * 10^-3) / 0.0375
v_initial^2 = 0.0375 + (3.0 * 10^-3 / 0.0375)
v_initial^2 = 1.0 + 0.08
v_initial^2 = 1.08
Taking the square root of both sides:
v_initial = sqrt(1.08)
v_initial ≈ 1.04 m/s
Therefore, the particle was originally moving at a speed of approximately 1.04 m/s.
Since the work done by the electric field is negative, it means that the electric field is doing negative work on the particle, which implies that the direction of the electric field is opposite to the direction of the particle's motion. Thus, the particle was initially moving in the opposite direction of the electric field.