# What is the larmor radius of an electron in the inner van allen belt?

1. Apr 18, 2012

### wyosteve

1. The problem statement, all variables and given/known data
What is the larmor radius of an electron in the inner van allen belt?
This is for a General Astr class so I feel like I must be over complicating it.

Electron Kinetic Energy in the inner VA belt: K≥30 [MeV] (much greater then its rest energy)

2. Relevant equations

where m is the mass of the particle, v is the component of its velocity perpendicular to the B field, q is the charge of the particle, and B is the B field magnitude at that point.

Earth's B field strength as a function of r: B(r)=(B0*R^3)/r^3
where B_o is the Earths B field strength at its surface, R is the radius of the earth, and r is the distance

Kinetic energy in relativistic terms: K=gm0c2-m0c2
where g is gamma, m is the particles rest mass and c is the speed of light

g=1/$\sqrt{1-(v^2/c^2)}$

3. The attempt at a solution

First I attempted to solve for B field strength in the inner VA belt

B(1.1R) = (3.1*10^-5)/(1.1^3) = 2.33*10^-5[T]
B(2.0R) = (3.1*10^-5)/(2.0^3) = 3.875*10^-6[T]
So the field strength is between 2.33*10^-5 and 3.875*10^-6 [T]

Next I used the electrons kinetic energy so solve for its velocity

v = $\sqrt{[1-(.511/30.511)^2]c^2}$
≈2.9996*10^8 [m/s]

now I found the max larmor radius by first assuming that all the electrons velocity is perpendicular (I have no idea how to determine how much of v is in fact perpendicular)
and using the weakest strength of B
r=(9.109*10^-31*2.9996*10^8)/(1.602*10^-19*3.875*10^-6)
≈440[m]
so the larmor radius must be ≤440[m] is my final conclusion

some obvious problems:
1:this doesnt really narrow down my answer very much
2:this course dosent assume any knowlage of relativity (so I must be overcomplicating it)

any input would be appriciated!!

2. Apr 19, 2012

### wyosteve

Never mind, that is the answer they were looking for.