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JustinLevy

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**"paradox" regarding energy of dipole orientation**

I've ran into a "paradox" concerning deriving the energy of a dipole's orientation in an external field. For example, the energy of a magnetic dipole m in an external field B is known to be:

[tex]U= - \mathbf{m} \cdot \mathbf{B}[/tex]

In Griffiths Intro to Electrodynamics, this is argued by looking at the torque on a small loop of current in an external magnetic field.

The "paradox" arises instead when we try to derive it by looking at the energy in the magnetic field.

[tex] U_{em} = \frac{1}{2} \int (\epsilon_0 E^2 + \frac{1}{\mu_0} B^2) d^3r [/tex]

If we consider an external field B and the field due to a magnetic dipole B_dip, we have:

[tex] U = \frac{1}{2\mu_0} \int (\mathbf{B} + \mathbf{B}_{dip})^2 d^3r [/tex]

[tex] U = \frac{1}{2\mu_0} \int (B^2 + B^2_{dip} + 2 \mathbf{B} \cdot \mathbf{B}_{dip})d^3r[/tex]

the [tex]B^2[/tex] and [tex]B^2_{dip}[/tex] terms are independent of the orientation and so are just constants we will ignore. Which leaves us with:

[tex] U = \frac{1}{\mu_0} \int \mathbf{B} \cdot \mathbf{B}_{dip} d^3r [/tex]

Now we have:

[tex]\mathbf{B}_{dip} = \frac{\mu_0}{4 \pi r^3} [ 3(\mathbf{m} \cdot \hat{\mathbf{r}})\hat{\mathbf{r}} - \mathbf{m} ] + \frac{2 \mu_0}{3} \mathbf{m} \delta^3(\mathbf{r})[/tex]

If you work through the math, only the delta function term will contribute, which gives us:

[tex]U=\frac{2}{3} \mathbf{m} \cdot \mathbf{B}[/tex]

Which not only has the wrong magnitude,

*but the wrong sign*. And thus the "paradox". Obviously, there is no paradox and I am just calculating something wrong, but after talking to several students and professors I have yet to figure out what is wrong here.

One complaint has been that while doing the [tex]d\theta[/tex] and [tex]d\phi[/tex] integrals show that the non-delta function term doesn't contribute, it is unclear if this argument holds for the point right at r=0. I believe it still cancels, but to alleviate that, let's look at a "real" dipole instead of an ideal one. A spinning spherical shell of uniform charge with a magnetic dipole m, has the magnetic field outside the sphere:

for r>=R [tex]\mathbf{B}_{dip} = \frac{\mu_0}{4 \pi r^3} [ 3(\mathbf{m} \cdot \hat{\mathbf{r}})\hat{\mathbf{r}} - \mathbf{m} ] [/tex]

So this is a good stand in for the idealized dipole, as it has a pure dipole field outside of the sphere. Inside the sphere the magnetic field is:

for r<R [tex]\mathbf{B}_{dip} = \frac{2 \mu_0}{3} \frac{\mathbf{m}}{\frac{4}{3} \pi R^3}[/tex]

which as you can see, reduces to the "ideal" case in the limit R -> 0.

Here there is no funny business at r=0. The math again gives:

[tex]U=\frac{2}{3} \mathbf{m} \cdot \mathbf{B}[/tex]

What gives!?

Who can help solve this "paradox"?

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