Answer to Question #112829 in Electricity and Magnetism for Joel

Question

. The interstellar space in the Milky Way has magnetic field of about 3 micro G. This space also
has approximately one Hydrogen atom per cm3
, which have thermal velocity of about 103m/s.
Compare the magnetic field energy stored in a volume of interstellar space with the kinetic
energy of thermal motion.

Answer 1

The energy of the magnetic field per unit volume is

$$u_B=\frac{B^2}{2\mu_0}=\frac{(3\cdot10^{-6}\cdot10^{-4})^2}{2\cdot4\pi\cdot10^{-7}}=3.58\cdot10^{-14}\text{ J/m}^3.$$

1 cubic centimetre contains 1 hydrogen atom, which means that 1 cubic meter contains 1 000 000 atoms. Therefore, the kinetic energy per unit volume will be

$$E_K=\frac{1}{2}Nm_\text{H}\overline{v^2}=\\=\frac{1}{2}10^6(1.66\cdot10^{-27})(10^3)^2=8.3\cdot10^{-16}\text{ J/m}^3.$$