A uniform electric field of 775 N/C is produced within a particle accelerator. Starting from rest, what are the following?
(a) the speed of an electron placed in this field after 25.4 ns have elapsed m/s
(b) the speed of a proton placed in this field after 25.4 ns have elapsed m/s
To find the speed of an electron and a proton after 25.4 ns have elapsed in a uniform electric field of 775 N/C, we can use the equations of motion for charged particles in an electric field.
(a) Speed of an Electron:
Given:
Electric field strength (E) = 775 N/C
Time (t) = 25.4 ns
The equations of motion for a charged particle in an electric field are:
v = u + at
s = ut + 0.5at^2
Where:
v = final velocity
u = initial velocity (0 in this case as the electron starts from rest)
a = acceleration
t = time
s = distance traveled
For an electron:
Charge (q) = -1.6 x 10^-19 C (negative due to the charge of an electron)
To find the acceleration (a), we use the equation:
a = qE/m
Where:
m = mass of the electron = 9.1 x 10^-31 kg
Substituting the given values into the equation, we can find the acceleration:
a = (-1.6 x 10^-19 C)(775 N/C) / (9.1 x 10^-31 kg)
a ≈ -1.37 x 10^12 m/s^2 (since the electron is negatively charged, the acceleration is negative)
Using the equation v = u + at, and substituting the values:
v = 0 + (-1.37 x 10^12 m/s^2)(25.4 x 10^-9 s)
v ≈ -3.48 x 10^4 m/s
However, since speed is the magnitude of velocity, we take the absolute value:
Speed of an electron after 25.4 ns ≈ 3.48 x 10^4 m/s
(b) Speed of a Proton:
For a proton, the charge (q) is positive:
Charge (q) = +1.6 x 10^-19 C
Substituting the given values into the equation for acceleration:
a = (+1.6 x 10^-19 C)(775 N/C) / (1.67 x 10^-27 kg)
a ≈ 9.27 x 10^5 m/s^2
Using the equation v = u + at, and substituting the values:
v = 0 + (9.27 x 10^5 m/s^2)(25.4 x 10^-9 s)
v ≈ 2.35 x 10^4 m/s
Again, taking the absolute value, the speed of a proton after 25.4 ns ≈ 2.35 x 10^4 m/s.
To solve this problem, we need to use the equations of motion for uniformly accelerated linear motion. We'll start by using the equation:
v = u + at
where:
v = final velocity
u = initial velocity
a = acceleration
t = time
Let's find the answers to each part:
(a) Speed of an electron:
The electron has a charge of -1.6 x 10^-19 C and a mass of 9.11 x 10^-31 kg.
Given:
Electric field (E) = 775 N/C
Time (t) = 25.4 ns = 25.4 x 10^-9 s
Initial velocity (u) = 0 m/s (starting from rest)
Acceleration (a) = eE / m (where e is the charge of the electron and m is its mass)
Using the formula:
a = eE / m
Substituting the values:
a = (-1.6 x 10^-19 C)(775 N/C) / (9.11 x 10^-31 kg)
Now, we can find the acceleration.
Next, use the equation:
v = u + at
Substituting the known values:
v = 0 m/s + (acceleration)(time)
Calculate the final velocity of the electron.
(b) Speed of a proton:
The proton has a charge of +1.6 x 10^-19 C and a mass of 1.67 x 10^-27 kg.
Using the same approach as before, calculate the acceleration and then find the final velocity of the proton.
Remember to be careful with the signs of the charges when calculating the acceleration and final velocity.
By following these steps, you should be able to find the answers to both parts (a) and (b) of the question.
force = 775 * particle charge in Coulombs
a = force / mass of electron or proton in kg
then v = a t