9. Properties of Magnetic Fields.
The previous section has covered the key points about magnetostatic fields. In particular we have found the relationship between the current (more strictly the current density) and the resultant magnetic field, \(\vec{B}\). We also intoduced the vector potential \(\vec{A}\) and showed its relationship to \(\vec{B}\) and \(\vec{J}\). As in the case of electrostatics there are a few further results that are useful when calculating magnetic fields. We introduce them in this section.
9.1 Boundary conditions.
As in the case of electrostatics there are some important properties of the magnetic field at boundaries of current. In the case of magnetic fields we are particularly interested in the properties of \(\vec{B}\) above and below a current carrying surface. In this case the appropriate quantity to use is the surface current density \(\vec{K}\) that we described in section. 7.5.1. The boundary conditions are then,
\(\displaystyle \vec{B}_{above}-\vec{B}_{below} = \mu_0 (\vec{K} \times \hat{\vec{n}})\)
\(\displaystyle \vec{A}_{above}=\vec{A}_{below}\)
\(\displaystyle \frac{\partial \vec{A}_{above}}{\partial n}- \frac{\partial \vec{A}_{below}}{\partial n}=-\mu_0\vec{K}\)
Exercise. What does the cross-product above imply about any change in \(\vec{B}\) parallel to the surface current? What are the boundary conditions for \(\vec{B}\) when we consider directions parallel and perpendicular to the vector \(\vec{K}\).
For a detailed discussion of these boundary conditions see Griffiths section 5.4.2.
9.2 Multipole expansions and the magnetic dipole.
As we saw for the case of electrostatics we might see that at large distances from the currents (closed loops) generating magnetic fields it should be possible to approximate the field in a series of higher moments. As we have no magnetic monopoles the first term in the series corresponds to the simple dipole. We will not look in detail at this expansion, formally we may write it, in terms of the vector potential, as,
\(\displaystyle \vec{A}(\vec{r})=\frac{\mu_0 I}{4\pi}\oint \frac{1}{|\vec{r}-\vec{r'}|} d\vec{l'}=\frac{\mu_0 I}{\ \pi}\sum_{n=0}^{\infty}\frac{1}{r^{n+1}}\oint (r')^n P_n(\cos \alpha)d\vec{l'}\)
noting each successive decays faster as \(1/r^n\). We will not concern ourselves about how to do such an expansion.
The first, non-zero, term of this expansion is the dipole term,
\(\displaystyle \vec{A}_{dip}(\vec{r})=\frac{\mu_0 I}{4\pi r^2}\oint r' \cos \alpha d\vec{l'}= \frac{\mu_0 I}{4\pi r^2}\oint (\hat{\vec{r}}\cdot \vec{r'})d\vec{l'}\)
With the appropriate vector identity the vector on the right may be written as,
\(\displaystyle \oint(\hat{\vec{r}}\cdot \vec{r'})d\vec{l'}= - \hat{\vec{r}}\times\int_A d\vec{a'}\)
that leads us to write the dipole field in terms of the vector potential as,
###where we have defined the magnetic dipole moment, \(\vec{m}\) as,
\(\displaystyle \vec{m}=I\int d\vec{a}\)
This may be thought of as a small loop of wire of area \(a\) carrying a current \(I\). The direction of the moment (given by \(\vec{a}\)) depends on the direction in which \(I\) flows around the loop. Note, that the result for the dipole field only applies (as in the case of the electric dipole) when we are far away from this current loop.
If we assume that \(\vec{m}\) is aligned along the \(z\) axis in spherical polar coordinates we can write,
\(\displaystyle \vec{A}_{dipole}(\vec{r})=\frac{\mu_0}{4\pi} \frac{m \sin\theta}{r^2}\hat{\phi}\)
so that,
\(\displaystyle \vec{B}_{dipole}=\nabla \times \vec{A}_{dipole}= \frac{\mu_0 m}{4\pi r^3}(2 \cos\theta\;\hat{\vec{r}}+\sin\theta\; \hat{\theta})\).
This is exactly the same as the field from an electric dipole except we have changed the constant \(p/\epsilon_0\) with the constant \(\mu_0 m\). The field lines look the same.
Similarly we can write the field from a magnetic dipole in coordinate indpendent form as,
###The magnetic dipole plays an important role in a first understanding of magnetic fields in matter that we will come to later on in these lectures.
9.3 Summary of magnetic fields
As we have considered magnetostatic fields we have noticed some similarities and differences from our results for electrostatic fields. In summary they are,
- There are no magnetic monopoles (\(\nabla \cdot \vec{B}=0\)),
- magnetic fields do no work,
- magnetic fields are not conservative (\(\nabla \times \vec{B}=\mu_0 \vec{J})\),
- we can’t define a universal magnetic scalar potential but we can defines a vector potential \(\vec{A}\),
- for magnetostatic fields we chose \(\nabla\cdot\vec{A}=0\) (the Coulomb gauge) to simplify our calculations.
9.4 Aside
You will note that we have not considered anything about the energy associated with the magnetic field. We leave this until we consider electrodynamics. Needless to say, as for the case of the electric field we can associate an energy with the magnetic field itself.