An electron with a horizontal speed of 4.0 * 10^6 m/s and no vertical component of velocity passes through two horizontal parallel plates. The magnitude of the electric field between the plates is 150 n/c the plates are 6.0 long
A) calculate the vertical component of the electron's final velocity. (Ans 4.0 * 10^5)
B) calculate the final velocity of the electron. (Ana4.0 * 10^6)
Please with full solution
v(x)=4•10⁶ m/s,
E=150 N/C
L= 6 (????)
t= L/v(x)
F=eE,
F=ma,
ma=eE
a=eE/m
where e = 1.6•10⁻¹⁹ C,
m=9.1•10⁻³¹ kg,
v(y)=at =...
v=sqrt{v(x) ²+v(y) ²} =…
To solve this problem, we can use the principles of projectile motion and the concept of electric fields. Let's break it down step by step:
Step 1: Calculate the time taken by the electron to travel between the plates.
- Since the plates are parallel, there is no horizontal force acting on the electron.
- Therefore, the horizontal speed remains constant throughout the motion.
- The distance between the plates is given as 6.0 m.
- Using the formula distance = speed * time, we can find the time taken (t) for the electron to pass between the plates:
distance = speed * time
6.0 m = 4.0 * 10^6 m/s * t
t = 6.0 m / (4.0 * 10^6 m/s)
t = 1.5 * 10^(-6) s
Step 2: Calculate the vertical displacement of the electron during its motion between the plates.
- The vertical displacement can be calculated using the formula: displacement = initial velocity * time + 0.5 * acceleration * time^2
- Since there is no initial vertical velocity, the initial velocity is 0.
- The acceleration acting on the electron is due to the electric field between the plates and is given by the formula: acceleration = electric field * charge / mass.
- The charge of an electron is -1.6 * 10^(-19) C, and the mass of an electron is 9.1 * 10^(-31) kg.
- Plugging in the values, we get:
acceleration = (150 N/C) * (-1.6 * 10^(-19) C) / (9.1 * 10^(-31) kg)
acceleration = -2.637 * 10^11 m/s^2
- Now, we can calculate the vertical displacement:
displacement = 0 * (1.5 * 10^(-6) s) + 0.5 * (-2.637 * 10^11 m/s^2) * (1.5 * 10^(-6) s)^2
displacement = -2.985 * 10^(-7) m
Step 3: Calculate the final vertical velocity of the electron.
- The final velocity can be calculated using the formula: final velocity = initial velocity + acceleration * time
- Since there was no initial vertical velocity, the initial velocity is 0.
- Plugging in the values, we get:
final velocity = 0 + (-2.637 * 10^11 m/s^2) * (1.5 * 10^(-6) s)
final velocity = -3.956 * 10^5 m/s
Step 4: Calculate the final velocity of the electron.
- Since there is no horizontal force acting on the electron, the final horizontal velocity remains the same as the initial horizontal velocity.
- Therefore, the final velocity is still 4.0 * 10^6 m/s.
So the answers to the given questions are:
A) The vertical component of the electron's final velocity is -4.0 * 10^5 m/s.
B) The final velocity of the electron is 4.0 * 10^6 m/s.
A) To calculate the vertical component of the electron's final velocity, we need to consider the force acting on the electron due to the electric field between the plates.
The force experienced by the electron due to the electric field is given by the equation F = qE, where F is the force, q is the charge of the electron, and E is the electric field strength.
The electric field strength between the plates is given as 150 N/C, and the charge of an electron is -1.6 x 10^-19 C.
We can use Newton's second law, F = ma, where m is the mass of the electron and a is its acceleration, to find the vertical acceleration of the electron.
Since the force acting on the electron is purely vertical, and no other forces are acting on it in the vertical direction, we have F = ma => qE = ma.
From here, we can rearrange the equation to find the acceleration: a = qE / m.
The mass of an electron is approximately 9.1 x 10^-31 kg.
Plugging in the values, we have a = (-1.6 x 10^-19 C) * (150 N/C) / (9.1 x 10^-31 kg).
Calculating this, we find the vertical acceleration to be approximately 2.63 x 10^11 m/s^2.
Since there is no initial vertical velocity, the final vertical velocity will be equal to the vertical acceleration multiplied by the time taken to pass through the plates.
Given that the length of the plates is 6.0 m, we can calculate the time taken by dividing the length by the horizontal velocity (since the velocity is constant).
Time taken = distance / velocity = 6.0 m / (4.0 x 10^6 m/s) = 1.5 x 10^-6 s.
Finally, we can find the vertical component of the electron's final velocity by multiplying the vertical acceleration by the time taken.
Vertical component of velocity = acceleration * time taken = (2.63 x 10^11 m/s^2) * (1.5 x 10^-6 s).
Calculating this, we find the vertical component of the electron's final velocity to be approximately 4.0 x 10^5 m/s.
B) To calculate the final velocity of the electron, we need to use vector addition to combine the initial horizontal velocity with the vertical component of the final velocity.
The initial horizontal velocity is given as 4.0 x 10^6 m/s.
The final velocity vector will have the same magnitude as the initial horizontal velocity but with a vertical component of 4.0 x 10^5 m/s.
Using the Pythagorean theorem, we can find the magnitude of the final velocity:
final velocity = √(initial horizontal velocity^2 + vertical component of velocity^2)
final velocity = √((4.0 x 10^6 m/s)^2 + (4.0 x 10^5 m/s)^2)
Calculating this, we find the magnitude of the final velocity of the electron to be approximately 4.0 x 10^6 m/s.
Therefore, the final velocity of the electron is 4.0 x 10^6 m/s.