An electron is released from rest at the negative plate of a parallel-plate capacitor
The Correct Answer and Explanation is :
When an electron is released from rest at the negative plate of a parallel-plate capacitor, it will accelerate toward the positive plate due to the electric field created between the plates. The motion of the electron can be analyzed using the concepts of electric field, force, and energy.
Explanation:
- Electric Field in a Capacitor:
A parallel-plate capacitor creates a uniform electric field ((E)) between its plates. The electric field can be expressed as:
[
E = \frac{V}{d}
]
where (V) is the potential difference between the plates, and (d) is the distance between them. The direction of the electric field is from the positive plate to the negative plate. - Force on the Electron:
The force ((F)) acting on the electron, which has a charge of (-e) (where (e \approx 1.6 \times 10^{-19} \, \text{C})), is given by:
[
F = qE = -eE
]
Because the charge of the electron is negative, the force acts in the opposite direction of the electric field, which means it accelerates toward the positive plate. - Acceleration:
The acceleration ((a)) of the electron can be determined using Newton’s second law:
[
F = ma \implies a = \frac{F}{m} = \frac{-eE}{m}
]
where (m) is the mass of the electron ((m \approx 9.11 \times 10^{-31} \, \text{kg})). - Kinetic Energy:
As the electron accelerates towards the positive plate, it converts electric potential energy into kinetic energy. The work done on the electron by the electric field can be expressed as:
[
W = qV = -eV
]
This work results in an increase in kinetic energy, which can be calculated using the work-energy theorem. - Conclusion:
The electron will accelerate towards the positive plate of the capacitor, gaining speed as it moves through the electric field. The faster it moves, the more kinetic energy it accumulates, which can be expressed in terms of its final speed when it reaches the positive plate. The principles of electromagnetism and mechanics combine to describe the motion and energy transformation of the electron in the capacitor’s electric field.