What Is the Total Energy Stored in a Magnetic Field Outside a Sphere?

[jesuslovesu]

Homework Statement

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.

Homework Equations

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

total energy = u * volume.

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]

Homework Statement

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.

Homework Equations

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

total energy = u * volume.

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...

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 $r = \infty$. All you have to know is that $dV = 4\pi r^2 dr$

So:

$$E = \int_R^{\infty} U dV = \int_R^{\infty} U 4\pi r^2 dr$$

where

$$U = \frac{B^2}{2\mu_0}$$ and

$$B = B_0\frac{R^2}{r^2}$$

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

The answer I get is:

$$E = 2\pi B_0^2R^3/\mu_0$$

AM