Magnetic Field required to maintain protons at 7Tev in LHC

[Steel_Panther]

Homework Statement

Piece of homework I am stuck on. I have to calculate the magnetic field required to maintain protons at 7Tev in the LHC which has a radius of 27 km. I also have to work out the cyclotron frequency as well.

Homework Equations

7 x 10^15 = mc^2/√(1-v^2/c^2) - mc^2 ----(1)

R = mv/qB ----(2) Where R is radius, q is charge and B is the magnetic field

The Attempt at a Solution

I have tried to work out the velocity of the protons using the first equation as I am sure with that amount of energy I need to take relativity into account. I could then input that into equation 2, but with there being relativistic effects wouldn't these effects on m have to be taken into account? I am not sure if I am going the correct way about tackling this. Any thoughts? Thanks guys

 

[TSny]

Hi, and welcome to PF.

Yes, you will need to include relativistic effects for R. If you write R in the form R = p/qB where p is the momentum, then it will be valid for both relativistic and nonrelativistic cases.

There's a well-known equation that relates energy and momentum that will make it easy to find p.

 

[vela]

have tried to work out the velocity of the protons using the first equation as I am sure with that amount of energy I need to take relativity into account.

The energy given is typically the total energy, not just the kinetic energy. Also, it's best to avoid working with speeds if you can. Stick with energy and momentum.

I could then input that into equation 2, but with there being relativistic effects wouldn't these effects on m have to be taken into account? I'm not sure if I am going the correct way about tackling this. Any thoughts? Thanks guys

You should disabuse yourself of the notion that the mass of an object increases with its speed. The only difference between the relativistic mass, which is what you're referring to, and the energy is a factor of c², so you might as well just refer to the particle's energy. It's the energy that increases with speed, and the mass m is the invariant, or rest, mass, which is constant.

 

[TSny]

The energy given is typically the total energy, not just the kinetic energy.

Good point. I've seen "energy given" used both ways (KE or total). [EDIT: In a low energy situation it might seem a bit strange for the given energy to be the total energy; e.g., a "1-Gev proton"]. In this problem it won't make much difference.

 

[Dickfore]

At that energy, the protons are essentially moving at the speed of light ($E_k \gg m_p c^2$). The equation for perpendicular acceleration is:
$$a_{\bot} = \frac{c^2}{E} \, F_{\bot}$$

The Lorentz force is a perpendicular force. Do you know the expression for centripetal acceleration and Lorentz force?