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**1. Homework Statement**

ou're taking a class in space weather physics. Space weather deals with the dynamics of the far upper atmosphere and the magnetic regions surrounding the Earth. You're preparing a term paper on the van Allen belts, regions where high-energy particles are trapped in the Earth's magnetic field. Your textbook says the magnetic field strength at the belts is 0.1 G. To impress your professor, you calculate the radii of the spiral paths of 0.1-MeV, 1-MeV, and 10-MeV protons in the van Allen belts.

What values do you get for r(0.1 MeV), r(1 MeV), and r(10 MeV) in km?

**2. Homework Equations**

K = [tex]\frac{1}{2}[/tex]mv[tex]^{2}[/tex]

F[tex]_{B}[/tex] = m[tex]\vec{a}[/tex] = m[tex]\frac{v^{2}}{r}[/tex] = q[tex]\vec{v}[/tex] x [tex]\vec{B}[/tex]

**3. The Attempt at a Solution**

I keep getting the same values, but the mastering physics answers are different.

First I convert the MeV values into Joules which =

0.1 MeV = 100,000eV = 1.6*10[tex]^{-14}[/tex] J

1MeV = 1.6*10[tex]^{-13}[/tex] J

10MeV = 1.6*10[tex]^{-12}[/tex] J

then knowing the energy each proton has, we can find the velocities from:

K = [tex]\frac{1}{2}[/tex]mv[tex]^{2}[/tex]

v=[tex]\sqrt{\frac{2K}{m}}[/tex]

then knowing the velocities, we solve for the radius using Lorentz eq.:

r = [tex]\frac{mv}{qB}[/tex]

and from this I'm getting:

r(0.1 MeV) = 3.847 km

r(1 MeV) = 12.13 km

r(10 MeV) = 38.5 km

The mastering physics answers are:

4.6 km

14.0 km

46.0 km respectively

Is there something that I'm forgetting in my eq's? Should I be considering gravitational potential energy because the particle is at an altitude above the surface of the earth?

i actually tried this with some basic research on the altitude of the Van Allen belt where protons are, but the range I found was very large, and the velocities ended up being almost identical.

Any thoughts would be much appreciated.