an electron with a mass of 9.11 x 10^-31 kg is accelerated from rest across a set of 150 V and are separated by 0.80 cm.
a) determine the kinetic energy of the electron after it crosses between the plates.( ans 2.4 * 10^-17 j)
b) determine the final speed of the electrons. (ans 7.3 * 10^6 m/s)
c) determine the acceleration of the electron while it is between the plates. (ans 3.3*10^15 )
d) determine the time required for the electron to travel across the plates.( ans 2.2*10^-9 s)
To solve this problem, we can use the principles of electrostatics and motion of charged particles in an electric field. The steps to find the answer to each of the questions are as follows:
a) Determining the kinetic energy of the electron after it crosses between the plates:
1. Begin by finding the electric potential energy gained by the electron using the formula: ΔPE = q ΔV, where ΔPE is the change in potential energy, q is the charge, and ΔV is the change in voltage between the plates.
ΔV = 150 V (given)
Charge of an electron (q) = -1.6 x 10^-19 C (negative because electrons have a negative charge)
ΔPE = (-1.6 x 10^-19 C) x (150 V) = -2.4 x 10^-17 J
2. Since the electron starts from rest, all the potential energy gained is converted into kinetic energy. Hence, the kinetic energy of the electron is the same as the change in potential energy:
Kinetic energy = -2.4 x 10^-17 J
Therefore, the kinetic energy of the electron after it crosses between the plates is -2.4 x 10^-17 J.
b) Determining the final speed of the electron:
To find the final speed of the electron, we need to use the kinetic energy calculated in part (a) and the formula for kinetic energy: KE = (1/2)mv^2, where KE is the kinetic energy, m is the mass, and v is the velocity (speed in this case).
Rearranging the formula, we get:
v = √(2KE / m)
Substituting the known values:
v = √[(2 x -2.4 x 10^-17 J) / (9.11 x 10^-31 kg)]
Evaluating the expression gives us a final speed of approximately 7.3 x 10^6 m/s.
c) Determining the acceleration of the electron while it is between the plates:
The acceleration of a charged particle in an electric field can be calculated using the formula: a = qE/m, where a is the acceleration, q is the charge, E is the electric field strength, and m is the mass.
The electric field strength can be determined using the formula: E = ΔV / d, where ΔV is the voltage across the plates and d is the separation between the plates.
Hence, the acceleration of the electron is given by:
a = (-1.6 x 10^-19 C) x (150 V) / (0.80 x 10^-2 m) / (9.11 x 10^-31 kg)
Evaluating the expression gives us an acceleration of approximately 3.3 x 10^15 m/s^2.
d) Determining the time required for the electron to travel across the plates:
To find the time required, we need to calculate the time it takes for the electron to travel the distance between the plates. We can use the formula: t = d/v, where t is the time, d is the distance, and v is the velocity.
Plugging in the values:
t = (0.80 x 10^-2 m) / (7.3 x 10^6 m/s)
Evaluating the expression gives us a time of approximately 2.2 x 10^-9 s.