An electron is released from rest at the negative plate of a parallel plate capacitor and accelerates to the positive plate (see the drawing). The plates are separated by a distance of 1.4 cm, and the electric field within the capacitor has a magnitude of 3.5 106 V/m. What is the kinetic energy of the electron just as it reaches the positive plate?
The Voltage change between plates is
(Field)*(Plate separation)
= 3.5*10^6*0.014 = 49*10^3 Volts
Multiply that by the electron charge to get the final kinetic energy.
To find the kinetic energy of the electron just as it reaches the positive plate, we can use the equation for the work done by the electric force on a charged particle:
W = q * ΔV
where W is the work done, q is the charge of the electron, and ΔV is the change in potential energy.
We know that the work done by the electric force is equal to the change in kinetic energy of the electron. So, the kinetic energy just as the electron reaches the positive plate is equal to the work done by the electric force.
We can determine the change in potential energy by using the equation:
ΔV = -Ed
where ΔV is the change in potential energy, E is the electric field, and d is the distance moved by the electron.
Given:
E = 3.5 * 10^6 V/m
d = 1.4 cm = 0.014 m (converted to meters)
Substituting the given values into the equation, we have:
ΔV = -E * d
= -(3.5 * 10^6 V/m) * (0.014 m)
= -49,000 V
The charge of an electron is approximately equal to -1.6 * 10^(-19) C. Substituting this value into the equation for the work done, we can find the kinetic energy:
W = q * ΔV
= (-1.6 * 10^(-19) C) * (-49,000 V)
= 7.84 * 10^(-15) J
Therefore, the kinetic energy of the electron just as it reaches the positive plate is approximately 7.84 * 10^(-15) J.
To find the kinetic energy of the electron just as it reaches the positive plate, we can use the concept of work-energy theorem.
The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy.
In this scenario, the electron is moving against an electric field from the negative plate to the positive plate. So, the work done by the electric field is negative (as it opposes the motion of the electron).
The work done by the electric field can be calculated by the formula:
Work = (Force * distance) * cos(theta)
In this case, the force is the electric force acting on the electron, which can be calculated using the formula:
Force = charge * electric field
The distance is the separation between the plates and theta is the angle between the direction of the force and the direction of motion (which is 0 degrees in this case).
Substituting the given values:
Electric field = 3.5 x 10^6 V/m
Distance = 1.4 cm = 0.014 m
Charge of an electron = 1.6 x 10^-19 C
Force = (1.6 x 10^-19 C) * (3.5 x 10^6 V/m) = 5.6 x 10^-13 N
Work = (5.6 x 10^-13 N) * (0.014 m) * cos(0) = 7.84 x 10^-15 J (negative because the work done is against the motion)
According to the work-energy theorem, this work done by the electric field is equal to the change in kinetic energy of the electron. Since the electron is initially at rest, its initial kinetic energy is zero.
Therefore, the final kinetic energy of the electron just as it reaches the positive plate is: 7.84 x 10^-15 J.