Calculating Kinetic Energy & Velocity of Electrons/Protons

[reality99]

Homework Statement

Electrons are allowed to be accelerated through a 300V potential difference.
a) Determine the electron's kinetic energy, total energy, and velocity.
b) Repeat for a proton instead of electron
c) Assume a potential difference of 1MV instead of 300V, repeat the above calculations

Homework Equations

W=qdV
KE=(1/2)mv^2
KE=(mc^2)/(sqrt(1-(v/c)^2))-mc^2

The Attempt at a Solution

It's been 2 years since a quantum mechanics course or physics course for that matter so bare with me if I missed something obvious.

(Part A&B)
KE is the same for both the electron and proton, 300eV
This translates to 4.8*10^-17J (from 1eV=1.6*10^-19J)
***What is the total energy?? I know it is PE+KE+U, but I can't remember how to find PE for electrons or protons. As the proton or electron is accelerated through the 300V, the KE gained=PE lost, correct? If so does that mean my total energy=2*KE or just KE?

I then used this and the mass of electron/proton to find V.

I calculated, for electron V=1.0271*10^7m/s, and for proton V=2.398*10^5m/s (from classical expression) what is the "threshold" where you must use relativistic equation? I know v<<c uses classical but what is considered much greater than c?

(Part C)
KE=10^6eV=1.6*10^-13J from same method as above.

Here I had to use relativistic equation for electron to find V=2.8228*10^8m/s, but used classical eq for proton and V=1.384*10^7m/s.

Is this the correct use of classical vs. relativistic equation?

If you made it to the end, thanks so much in advanced for the help!

 

[collinsmark]

KE is the same for both the electron and proton, 300eV
This translates to 4.8*10^-17J (from 1eV=1.6*10^-19J)

'Looks good.

***What is the total energy?? I know it is PE+KE+U, but I can't remember how to find PE for electrons or protons. As the proton or electron is accelerated through the 300V, the KE gained=PE lost, correct? If so does that mean my total energy=2*KE or just KE?

Don't overthink it. . Yeah, the particles do have potential energy from the electric field before they are accelerated. But after the acceleration, that potential energy (from the electric field) gets converted to the particles' kinetic energy. So, sure, there's conservation of energy in that respect. But don't dwell on it. I'm pretty sure that you're not supposed to consider it quite that way (but you could if you really had to).

When the question asks for "total" energy, you can usually assume it's asking for KE + Mass energy. In other words,

E_total = KE + mc².​

It's possible that the question is asking for some other form of potential energy, but when involving special relativity, usually not. Regarding this problem, after the particle has been accelerated there are no other forms of potential energy anyway, so it's clear that the total energy is just the KE + mc².

I then used this and the mass of electron/proton to find V.

I calculated, for electron V=1.0271*10^7m/s, and for proton V=2.398*10^5m/s (from classical expression)

'Looks reasonable.

what is the "threshold" where you must use relativistic equation? I know v<<c uses classical but what is considered much greater than c?

Oooh. That's a loaded question.

The truth is you should ask your instructor. Later in life, when you don't have an instructor, it depends on how precise you want to be.

I've heard some specify that if the velocities involved are less than 0.2c, you're okay using the classical equations. But if you're looking for precise numbers, you might want to lower that to 0.1c.

In terms of kinetic energies, energy is proportional to the velocity squared, using classical equations. So if you have a velocity threshold of 0.1c before switching to relativistic equations, the kinetic energy threshold should be around 0.01c². So if the kinetic energy involved is less than two orders of magnitude compared to the rest mass energy of the particle involved, you're probably okay sticking with classical equations. If the kinetic energy is greater than two orders of magnitude below the rest mass energy, and you might consider moving to the relativistic equations.

(Part C)
KE=10^6eV=1.6*10^-13J from same method as above.

Here I had to use relativistic equation for electron to find V=2.8228*10^8m/s, but used classical eq for proton and V=1.384*10^7m/s.

'Looks reasonable to me.

Is this the correct use of classical vs. relativistic equation?

I think it's fine. But in truth it depends on how much precision you want or need. Try finding the proton's velocity in the 1 MV potential, using the relativistic equations. You'll find that your answer is quite close to what you found using the classical equations. But is it close enough? That's an answer for your and your instructor to decide.