An electron is a subatomic particle (m = 9.11 x 10-31 kg) that is subject to electric forces. An electron moving in the +x direction accelerates from an initial velocity of +5.11 x 105 m/s to a final velocity of 1.67 x 106 m/s while traveling a distance of 0.0654 m. The electron's acceleration is due to two electric forces parallel to the x axis: = 8.18 x 10-17 N, and , which points in the -x direction. Find the magnitudes of (a) the net force acting on the electron and (b) the electric force .
To find the magnitudes of the net force acting on the electron and the electric force, we can use Newton's second law of motion:
F_net = m * a
where:
F_net is the net force acting on the electron,
m is the mass of the electron,
a is the acceleration of the electron.
Given:
Mass of the electron (m) = 9.11 x 10^(-31) kg
Initial velocity (v_i) = +5.11 x 10^5 m/s
Final velocity (v_f) = 1.67 x 10^6 m/s
Distance traveled (d) = 0.0654 m
Electric force in the +x direction (F_1) = 8.18 x 10^(-17) N
First, let's calculate the acceleration of the electron:
a = (v_f - v_i) / t
To find the time (t), we can use the equation: s = ut + (1/2)at^2, where s is the distance traveled.
Since the initial velocity is given and the acceleration is constant, the equation simplifies to:
d = (v_i * t) + (1/2) * a * t^2
Rearranging the equation gives us: (1/2) * a * t^2 + (v_i * t) - d = 0
We can solve this quadratic equation for t using the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac)) / (2a)
Where:
a = (1/2) * a
b = v_i
c = -d
Once we calculate t, we can substitute it back into the acceleration equation to find the acceleration (a).
Now, we can calculate the net force (F_net):
F_net = m * a
Finally, we can find the magnitude of the electric force F_2 using the equation:
F_2 = F_net - F_1
Substituting the values we know into the equations will give us the desired answers.