Van Allen Belt
Estimating Energy Stored in the Geomagnetic Field
The magnetic field of the earth is complicated, not a simple dipole, but it can be approximated that way.
B_r =  2 B_0 \left( { R_E } \over r \right)^3 \cos \theta
B_{\theta} =  B_0 \left( { R_E } \over r \right)^3 \sin \theta
dE = { { {B_r}^2 + {B_{\theta}}^2 } \over { 2 {\mu}_0 } } ~ dV
dV = 2 \pi r^2 \sin \theta ~ dr ~ d\theta
dE = { { \left( 2 B_0 \left( R_E \over r \right)^3 \cos \theta \right)^2 + \left( B_0 \left( R_E \over r \right)^3 \sin \theta \right)^2 } \over { 2 {\mu}_0 } } ~ 2 \pi r^2 \sin \theta ~ dr ~ d\theta
dE = { { \pi {B_0}^2 {R_E}^6 } \over { { {\mu}_0 } r^4 } } ( 3 \cos^2 \theta + 1 ) \sin \theta ~ dr ~ d\theta
The total energy of the magnetic field is the double integral from \theta = 0 to \theta = \pi and from R = R_0 to infinity. Assuming the earth's mantle has a permeability of {\mu}_0 and the earth's iron core has an infinite permeability (and thus no appreciable magnetic energy storage), R_0 is the radius of the core, not the earth's surface, about 3500 km.
Since r and \theta and the scaling terms are independent, we separate the integrals:
E = \left( { \pi {B_0}^2 {R_E}^6 } \over { {\mu}_0 } \right) ~ \left( \int^{\infty}_{R_0} { dr \over { r^4 } } \right) ~ \left( \int_0^{\pi} ( 3 \cos^2 \theta + 1 ) \sin \theta ~ d\theta \right)
E = \left( { \pi {B_0}^2 {R_E}^6 } \over { {\mu}_0 } \right) ~ \left( 1 \over { 3 {R_0}^3 } \right) ~ \left( 3 \times { 2 \over 3 } + 2 \right)
E = { { 4 \pi {B_0}^2 {R_E}^6 } \over { 3 {\mu}_0 {R_0}^3 } } = 2 \left( { 1 \over { 2 {\mu}_0 } } \left( B_0 \left( {R_E} \over {R_0} \right)^3 \right)^2 \right) \left( { { 4 \pi } \over 3 } {R_0}^3 \right)
The B field at the surface of the core is B_0 \left( {R_E} / {R_0} \right)^3 . The total energy is twice the energy density at the surface of the core { B^2 / { 2 {\mu}_0 } } times the volume of the core { { 4 \pi } \over 3 } {R_0}^3 .
B_0 = 3.12E5 Tesla
R_E = 6378000 meters
R_0 = 3500000 meters
\mu_0 = 4e7 π meter^{3} Tesla^{2} Joule^{1}
Above the core at 3500km, E = 5.1E18 Joules = 160 gigawattyears. Above the earth's surface, E = 8.4E17 Joules = 26 gigawattyears.
The earth's magnetic field is decreasing at about 6% per century, so the energy is decreasing 0.12% per year, about 200 megawatts above the core.
In 1960, Dessler and Vestine estimated that the maximum stored energy of particles in the van Allen belt must be less than 6e15 Joules. If this amount of energy was expended filling and emptying the van Allen belt once per year, that would also be about 200 megawatts. It is interesting that these are about the same magnitude. Both are 1.6 parts per billion of the solar energy absorbed and reradiated by the earth, and about 1/3 of the power of the solar wind times the earth's area.
I would like to learn the actual numbers for the stored energy and the particle power flux of the van Allen belt.
Gyration Frequency at the Equator
B =  B_0 \left( { R_E } \over r \right)^3
freq = B q / 2 \pi m . . . . . for v << c
freq = B q / 2 \pi { m_0 } ( 1 + { E_k } / {E_0} ) . . . . . E_k is the particle energy in electron volts (not sure about energy scaling)
The rest energy E_0 of a proton is 938 MeV and particle energies range from 100 keV to over 400 MeV. The rest energy of an electrom is 511 keV with particle energies up to 10 MeV.
Equatorial Gyration Frequency (Hz) 

L Shell 
Electron Energy 
Proton Energy 


0 
100 keV 
1 MeV 
10 MeV 
0 
10 MeV 
100 MeV 
400 MeV 
1.0 
873K 
730K 
295K 
42K 
476 
471 
430 
230 
1.5 
259K 
216K 
88K 
13K 
141 
139 
127 
68.2 
2.0 
109K 
91K 
37K 
5307 
59.5 
58.8 
53.7 
28.8 
2.5 
56K 
47K 
19K 
2717 
30.4 
30.1 
27.5 
14.7 
3.0 
32K 
27K 
11K 
1573 
17.6 
17.4 
15.9 
8.53 
3.5 
20K 
17K 
6889 
990 
11.1 
11.0 
10.0 
5.37 
4.0 
14K 
11K 
4615 
663 
7.43 
7.36 
6.72 
3.60 
4.5 
9584 
8015 
3241 
466 
5.22 
5.17 
4.72 
2.53 
5.0 
6987 
5843 
2363 
340 
3.81 
3.77 
3.44 
1.84 
5.5 
5249 
4390 
1775 
255 
2.86 
2.83 
2.58 
1.38 
6.0 
4043 
3382 
1367 
197 
2.20 
2.18 
1.99 
1.07 
6.5 
3180 
2660 
1075 
155 
1.73 
1.71 
1.57 
0.84 
7.0 
2546 
2129 
861 
124 
1.39 
1.37 
1.25 
0.67 