A free electron and a free proton are exactly 0.70cm apart.
Find the magnitude of the acceleration of the proton.
Express your answer with the appropriate units.
Find the direction of the acceleration of the proton.
Find the magnitude of the acceleration of the electron.
Express your answer with the appropriate units.
Find the direction of the acceleration of the electron.
The electron will accelerate toward the proton (opposite charges attract)
F = k Qe^2/(.007)^2
where Qe is electron charge and proton the same but plus
a of electron = F/Melectron
a of proton = F/Mproton
To find the magnitude of the acceleration of the proton and electron, we need to know the force acting on them and their masses. The force between the proton and electron is given by Coulomb's law:
F = (k * q1 * q2) / r^2
Where F is the force, k is the electrostatic constant (9.0 × 10^9 N·m^2/C^2), q1 and q2 are the charges (in Coulombs) of the bodies, and r is the distance between the bodies.
In this case, the forces between the electron and proton are equal in magnitude and opposite in direction due to their opposite charges. So we can set up the equation:
(k * q_electron * q_proton) / r^2 = (k * q_electron * q_proton) / r^2
Since the distance r is given as 0.70 cm, we need to convert it to meters by dividing by 100:
r = 0.70 cm / 100 = 0.007 m
The charges of the electron and proton are given as q_electron = -1.6 × 10^-19 C and q_proton = +1.6 × 10^-19 C respectively.
Plugging these values into the equation, we get:
(9.0 × 10^9 N·m^2/C^2 * (-1.6 × 10^-19 C) * (1.6 × 10^-19 C)) / (0.007 m)^2 = (9.0 × 10^9 N·m^2/C^2 * (2.56 × 10^-38 C^2)) / 4.9 × 10^-5 m^2
Calculating this expression gives us:
1.857142857 × 10^-14 N
So, the magnitude of the acceleration of both the proton and electron is the same and can be found using Newton's second law:
F = m * a
Where F is the force, m is the mass of the body, and a is the acceleration.
The mass of the proton is approximately 1.67 × 10^-27 kg, and the mass of the electron is approximately 9.11 × 10^-31 kg.
Plugging in the values, we have:
1.857142857 × 10^-14 N = 1.67 × 10^-27 kg * a_proton
a_proton = 1.857142857 × 10^-14 N / 1.67 × 10^-27 kg
a_proton ≈ 1.11 × 10^13 m/s^2
Similarly, for the electron:
1.857142857 × 10^-14 N = 9.11 × 10^-31 kg * a_electron
a_electron = 1.857142857 × 10^-14 N / 9.11 × 10^-31 kg
a_electron ≈ 2.04 × 10^16 m/s^2
The direction of the acceleration is already given as the electron and proton are moving away from each other. Hence, the direction of the acceleration of the proton is opposite to that of the electron.