The voltage between the two plates inside of the air cleaner is 500 V. A soot particle whose mass is 10-12 kg and carries a charge of -1.0 x 10-11 C of charge enters the air cleaner. What will be the velocity of the soot particle as it moves from the negative to the positive plate?
To determine the velocity of the soot particle as it moves from the negative to the positive plate, we can make use of the principles of electric fields and work-energy theorem. Here's how we can calculate it:
1. Determine the electric field between the two plates: The electric field (E) can be calculated using the formula E = V/d, where V is the voltage between the plates (500 V) and d is the distance between the plates. The value of d is not given in the question, so we'll assume a typical value of 0.01 meters (10 mm).
E = 500 V / 0.01 m
E = 50,000 N/C
2. Calculate the force experienced by the soot particle: The force experienced by a charged particle in an electric field is given by the formula F = q * E, where q is the charge of the particle (-1.0 x 10-11 C) and E is the electric field strength (50,000 N/C).
F = (-1.0 x 10-11 C) * (50,000 N/C)
F = -0.5 N
3. Calculate the work done on the particle: The work done on a particle is equal to the change in its kinetic energy. Since the soot particle starts from rest, the initial kinetic energy is zero. Therefore, the work done on the particle is equal to the final kinetic energy.
Work = Change in kinetic energy
Work = (1/2) * m * v^2, where m is the mass of the particle (10^-12 kg) and v is the velocity of the particle.
The work done on the particle is equal to the force multiplied by the distance it moves. Since the force and displacement are in the same direction, we can use the formula:
Work = F * d
-0.5 N * d = (1/2) * m * v^2
4. Solve for the velocity: Rearrange the equation to solve for the velocity:
v = sqrt((-2 * (-0.5 N * d)) / m)
Plug in the values and calculate the velocity:
v = sqrt((-2 * (-0.5 N * 0.01 m)) / 10^-12 kg)
v = sqrt(0.01 N*m / 10^-12 kg)
v = sqrt(10^9 m^2/s^2)
v = 10^4 m/s
Therefore, the velocity of the soot particle as it moves from the negative to the positive plate is 10,000 m/s.