Total Energy w/ Magnetic Field

jesuslovesu

1. The problem statement, all variables and given/known data
Assume that the magnitude of the magnetic field outside a sphere of radius R is B = B0 (R/r)^2. Determine the total energy stored in the magnetic field outside the sphere.


2. Relevant equations
I think it's necessary to use the energy density equation.
u = B^2/(2*u0)

total energy = u * volume.
3. The attempt at a solution
By just plugging in the given data, I come up with (2*pi*B0^2*R^3)/(u0*3). Assuming that r = R. Or 2*pi*B0^2*R^4/(3r*u0) if I don't assume that. My book's answer is similar except that the 3 in the denom. is not there. So I think I need to use calculus but I don't quite see how to set it up...

Andrew Mason

I don't understand your statement "Assuming that r = R". r is the distance from the centre. R is the radius of the sphere.

You have to integrate the energy density over volume from r=R to [itex]r = \infty[/itex]. All you have to know is that [itex]dV = 4\pi r^2 dr[/itex]

So:

[tex]E = \int_R^{\infty} U dV = \int_R^{\infty} U 4\pi r^2 dr [/tex]

where

[tex]U = \frac{B^2}{2\mu_0}[/tex] and

[tex]B = B_0\frac{R^2}{r^2}[/tex]

You should end up with U as a function of 1/r^2.

The answer I get is:

[tex]E = 2\pi B_0^2R^3/\mu_0[/tex]

AM