This is a hard problem as magentic field energy is hard to contain and often diffuses out and is not solely contained.   What mechanism do you propose for this increase in strength over just 10^7 years?  What is creating this particular kind of field?

The energy of a charged particle sitting on n-th Landay level is

\hbar \omega_c (n + 1/2) plus the part that does not depend on the field. 

You have a change of the cyclotron frequency. Take initial distribution over n and calculate. The initial distribution might be Boltzmann. You'll have to know the temperature. 

Sorry. That was a diamagnetic part. There is yet the papamagnetic part, connected with the spin.

I think Landau Lifshitz Statistical mechanics has a complete solution. Statsum and everything.

Statistical sum has nothing to do here -- the system is out of thermal equlibrium (no collisions over all the time of the field increase, initial velocities perpendicular to the field). This sounds like a homework problem, hence somewhat idealistic formulation, omitting mentioning how large is that region of uniform B, what happens at its boundaries, etc.

With this -- it's a fair bet that this problem was for exercizing with the first adiabatic invariant \mu= m(v_\perp)^2/(2B)=~const (when B changes slowly, compared to the gyration period). As B increases by a factor of 10, so will m(v_\perp)^2 in this setup, and this in turn is the kinetic energy of the particle (as the particles are non-relativistic). So the answer is -- the energies of electrons and protons will be ten times their initial energies.

As a check -- \omega_gyr = eB/m , gyration period =2\pi/\omega_gyr=~0.35 s at B=10^{-6} G for electrons, some 700 s for protons. These periods are short compared with the 10^7 yrs time on which B changes by an order of magnitude, hence the adiabatic invariant indeed stays nearly constant. 

Further caveats are needed if the energy becomes relativistic or if the initial velocity is so large, that gyroradius exceeds the size of the domain where the magnetic field is nearly uniform.

Suppose that m and q are the mass and charge of particle (electron, proton) moving with v(t) speed in uniform B(t) magnetic field depending on time (t), and v is perpendicular to B. According to initial condition, we deal with galactic magnetic field (B~1-10 mkG) and an astronomic large time scale. I omitted bellow the signs and vector notations for simplicity.

Then, the change of magnetic flux, according to Faraday's law of induction, induces the electric field, E=(dB/dt)R/2 (see link below), where R(t)=mv/qB is a corresponding cyclotron radius. Replacing E=(dv/dt)m/q above, gives dLn(v)/dt=(1/2)dLn(B)/dt or v1/v0= sqrt(B1/B0). The corresponding change of kinetic energy for your case is Ek1/Ek0=B1/B0=10 regardless of the acceleration time, charge (|q|>0) and mass of particle.

Andrey Zhiglo obtained the same result above using directly adiabatic invariant pt2/B.

http://teachers.web.cern.ch/teachers/archiv/hst2001/accelerators/teachers%20notes/betatron.htm