Question

In a proton accelerator used in elementary particle physics experiments, the trajectories of protons are controlled by bending magnets that produce a magnetic field of 4.80 T. What is the magnetic-field energy in a 10.0−cm3 volume of space where B = 4.80 T?

Answer 1

To solve this problem it is necessary to apply the concepts related to the energy density in the magnetic fields. Mathematically the expression that determines the relationship between the magnetic field, the permeability constant and the energy density is given by

$u = \frac{B^2}{2\mu_0}$

Where,

B = Magnetic Field

u = Energy density in magnetic field

$\mu_0$= Permeability constant

At the same time the energy of a given volume of space is given as

E = uV

Where

E = Magnetic field energy

V = Volume

Our values are given as

$\mu_0 = 4\pi*10^{-7}Tm/A$

$B = 4.8T$

$V = 10cm^3 = 10*10^{-6}m^3$

Replacing in the first equation we can find the energy density

$u = \frac{B^2}{2\mu_0}$

$u = \frac{4.8^2}{2(4\pi*10^{-7})}$

$u = 9.167*10^{6}J/m^3$

Now the net energy would be given by

$E = uV$

$E = (9.167*10^{6})(10*10^{-6})$

$E = 91.67J$

Therefore the Magnetic Field energy is 91.67J