Energy for an electron in an accelerator

[arella]

Homework Statement

SLAC, the Stanford Linear Accelerator Collider, located at Stanford University in Palo Alto, California, accelerates electrons through a vacuum tube two miles long (it can be seen from an overpass of the Junipero Serra freeway that goes right over the accelerator). Electrons which are initially at rest are subjected to a continuous force of 1.1×10^-12 N along the entire length of two miles (one mile is 1.609 kilometers) and reach speeds very near the speed of light.

a) Calculate the final energy of the electron.
b) Calculate the final momentum of the electron.
c) Calculate the time required for the electron to go the two-mile distance.

Relevant Equations

E= F x d
E= .5mv^2, maybe?

I honestly just have no idea where to start on parts c and b. I tried 1.1*10^-12 N x 3218 m for part a, which was right. But I'm lost on b and c. I'm also wondering if we have to include a gamma factor in any of this, but I'm unsure how or where. Any help would be amazing.

 

[mitochan]

I tried 1.1*10^-12 N x 3218 m, which I know if far from correct, but it's the only thing I knew to start with.

It looks good. Nm=J.

 

[arella]

It looks good. Nm=J.

Ahh wow turns out the reason I couldn't get a was a rounding error. Thank you for inspiring me to recheck lol. Do you have any insight on the next two parts?

 

[etotheipi]

For (b) the relation $E^2 = (mc^2)^2 + (pc)^2$ might come in useful. Note that $E$ here is the total energy (KE + rest).

You'll need to do a little more work for (c), in the form of an integral. I'm not sure how much help I can give without an attempt on your part, however considering the change in momentum seems like a good way in. Make sure to use relativistic formulas!

 

[mitochan]

From relativistic form of energy you will get v(x), velocity as a function of coordinate x.
Time to get through would be given as
\int_0^x \frac{dx}{v(x)}

 

[PeroK]

But I'm lost on b and c. I'm also wondering if we have to include a gamma factor in any of this, but I'm unsure how or where. Any help would be amazing.

This is clearly a relativistic problem, so you must use relativistic equations for everything. Note that for a constant force you still have $E = Fd$, which is why part a) came out right.

Everything else requires an application of special relativitistic concepts and formulas.