A thin, circular, conducting plate has a radius of 1.0 m and a charge of -2.0 μC. An electron is placed 1.0 cm above the center of the plate. What is the acceleration of the electron?
To find the acceleration of the electron, we can use the principle of electrostatics. The electric field due to a uniformly charged circular plate can be considered as if all the charge of the plate is concentrated at its center.
To calculate the electric field at the position of the electron, we can use Coulomb's law. Coulomb's law states that the electric field due to a point charge is given by:
E = k * (Q / r^2),
where E is the electric field, k is the electrostatic constant (9 x 10^9 N⋅m^2/C^2), Q is the charge, and r is the distance between the charge and the point where the electric field is measured.
In this case, the charge of the plate is -2.0 μC, and the distance between the electron and the center of the plate is 1.0 cm, which can be converted to meters by dividing by 100.
Plugging in the values, we have:
E = (9 x 10^9 N⋅m^2/C^2) * (-2.0 x 10^-6 C) / (0.01 m)^2
E = -1.8 x 10^6 N/C
This negative sign indicates that the electric field is pointing downwards. Since the electron carries a negative charge, it will experience a force in the opposite direction of the electric field. The force acting on the electron can be calculated using the equation:
F = q * E,
where F is the force, q is the charge of the electron, and E is the electric field.
The charge of an electron is -1.6 x 10^-19 C. Substituting the values, we get:
F = (-1.6 x 10^-19 C) * (-1.8 x 10^6 N/C)
F = 2.88 x 10^-13 N
Now, we can calculate the acceleration of the electron using Newton's second law:
F = m * a,
where F is the force, m is the mass of the electron, and a is the acceleration.
The mass of an electron is approximately 9.1 x 10^-31 kg.
Substituting the values, we have:
2.88 x 10^-13 N = (9.1 x 10^-31 kg) * a
Solving for a, we find:
a = (2.88 x 10^-13 N) / (9.1 x 10^-31 kg)
a ≈ 3.16 x 10^17 m/s^2
Therefore, the acceleration of the electron is approximately 3.16 x 10^17 m/s^2.