Total Energy w/ Magnetic Field

[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

 

[m2003]

I would approach this problem by first understanding the concept of energy stored in a magnetic field. The energy density equation, u = B^2/(2*u0), tells us that the energy stored in a magnetic field is directly proportional to the square of the magnetic field strength. This means that as the magnetic field increases, so does the energy stored in it.

Next, I would consider the given equation for the magnetic field outside the sphere, B = B0 (R/r)^2. This tells us that the magnetic field strength decreases as we move further away from the sphere, with r being the distance from the center of the sphere. This makes sense intuitively, as the magnetic field is strongest closer to the source (the sphere).

To find the total energy stored in the magnetic field outside the sphere, we can use the concept of energy density and integrate it over the volume outside the sphere. This can be expressed as:

Total energy = ∫ u dV

Since we are only interested in the energy outside the sphere, we can limit our integration to the volume outside the sphere, which can be expressed as:

V = 4/3 * π * (r^3 - R^3)

Substituting this into the integral, we get:

Total energy = ∫ u dV = ∫ u * 4/3 * π * (r^3 - R^3) dr

Now, we can substitute the given equation for the magnetic field, B = B0 (R/r)^2, into the energy density equation to get:

u = B^2/(2*u0) = (B0^2 * (R/r)^4)/(2*u0)

Substituting this into the integral, we get:

Total energy = ∫ u dV = ∫ (B0^2 * (R/r)^4)/(2*u0) * 4/3 * π * (r^3 - R^3) dr

This integral can be evaluated using standard techniques, and the result is:

Total energy = (2/3) * π * B0^2 * R^4 / u0

This matches with your attempt at a solution, but the 3 in the denominator is not necessary as it cancels out with the 3 in the numerator. Therefore, the final expression for the total energy stored in the magnetic field outside the sphere is:

Total energy = (2/3