13.8 The auxiliary field \(\vec{H}\)
In the previous sections we showed how the formation of electric dipoles in matter (either permanent molecular dipoles or induced dipoles) modified the electric field by introducing the idea of bound charges that modified the electric field. In order to avoid finding these bound charges each time we use the \(\vec{D}\) field as a device for handling the electric fields in matter. We considered particularly the case of linear materials. What happens in the case of magnetic fields in materials? The origin of magnetism in materials is buried deep in the physics and quantum mechanics of the atom - namely in the spin and orbital angular momentum of the electrons. In a simple description we can image the electrons as producing tiny magnetic dipoles as if they are tiny current loops in the atom. We won’t go into a detailed description here but consider three different types of magnetic material:
- paramagnets in which these dipoles tend to line up in the external field,
- diamagnets in which the dipoles tend to line up opposite to the external field,
- ferromagnets in which the dipoles are aligned and remain aligned even in the absence of an external field.
More detail and desrciption of the a microscopic origin of these materials may be found in Griffiths. We will just assume all three are possible.
You are familiar with ferromagnets. A typical bar magnet shows this behaviour. In it the magnetic dipoles on the atoms (say for example Iron) can be aligned in an external magnetic field and remain when the field is removed. We are used to saying the Iron is magnetised.
In the same way as we defined Polarization we can define Magnetisation as,
\(\vec{M}\) is the magnetic moment per unit volume arising from the alignmment of the magnetic dipoles.
The effect of these aligned dipoles is to produce a macroscopic dipole - a bar magnet is an example (if the dipole alignment is permanent). The field lines from this magnet and a macroscopic electric dipole are the same that leads to interpretation of magnets in terms of north and south poles. It is possible to solve magnetic problems with the idea of these poles, it is known as the Gilbert model, but deep down we must talk about current loops to properly describe magnetization.
13.9 Bound currents.
A sensible point to start our description of magnetic materials is hence in terms of the additive effect of many microscopic magnetic dipoles. As we did with the Polarization we start with our basic description of the magnetic dipole in terms of the magnetic vector potential, i.e.
\(\displaystyle \vec{A}(\vec{r})=\frac{\mu_0}{4\pi}\frac{\vec{m}\times (\vec{r}-\vec{r'})}{(|\vec{r}-\vec{r'}|^3)} =\frac{\mu_0}{4\pi}\frac{\vec{m}\times \hat{\vec{r''}}}{r''^2}\)
where we have written \(\vec{r''}=\vec{r}-\vec{r'}\).
now if we write \(\vec{m}=\vec{M}d\tau'\) (the moment from a small volume \(d\tau'\) we can write the total vector potential (by superposition)
\(\displaystyle \vec{A}(\vec{r})=\frac{\mu_0}{4\pi}\int_{Vol}\frac{\vec{M}(\vec{r'})\times \hat{\vec{r''}}}{r''^2}d\tau'\)
Again writing,
\(\displaystyle \nabla'\left(\frac{1}{r''}\right)=\frac{\hat{\vec{r''}}}{r''^2}\)
so,
\(\displaystyle \vec{A}(\vec{r})=\frac{\mu_0}{4\pi}\int \left[ \vec{M}(\vec{r'})\times \nabla'\left(\frac{1}{r''}\right)\right]d\tau'\)
integrating by parts and applying Stoke’s law we get,
\(\displaystyle \vec{A}(\vec{r})=\frac{\mu_0}{4\pi}\left[\int\frac{1}{r''}\left(\nabla'\times\vec{M}(\vec{r'})\right)d\tau'-\int\nabla '\times \left[\frac{\vec{M}(\vec{r'})}{r''}\right]d\tau'\right]\)
\(\displaystyle \vec{A}(\vec{r})=\frac{\mu_0}{4\pi}\left[\int\frac{1}{r''}(\nabla'\times\vec{M}(\vec{r'}))d\tau'+\oint\frac{1}{r''}(\vec{M}(\vec{r'})\times d\vec{a'})\right]\)
or if we write,
###and
###\(\displaystyle \vec{A}(\vec{r})=\frac{\mu+0}{4\pi}\int_{Vol} \frac{\vec{J}_b(\vec{r'})}{r''}d\tau'+\frac{\mu_0}{4\pi}\oint \frac{\vec{K}_b(\vec{r'})}{r''}da'\)
\(\vec{J}_b\) and \(\vec{K}_b\) play a similar role to the bound surface and volume charge densities we found for dielectrics. We will show how we might visualise these in the lectures but our aim, as in the case of \(\rho_b\) and \(\sigma_b\) is to avoid the need to calculate them to determine \(\vec{B}\) (which needs to be calculated from both these currents and any free currents that are circulating in our system).
13.10 The auxiliary field \(\vec{H}\)( or just the \(\vec{H}\) field)
As in the case of the \(\vec{D}\) field our aim here is o find an analogous field for the magnetic field in which we can eliminate the need to calculate the bound current densities in our calculations. So by analogy we write,
\(\displaystyle \vec{J}=\vec{J}_b+\vec{J}_{free}\).
As in the case of \(\sigma_b\) we can incorporate our definition of \(\vec{K}_b\) into \(\vec{J}_b\) as \(\vec{J}\) goes to zero at the boundary. Now from Ampere’s law and considering all currents we have, by substituting \(\vec{J}_b=\nabla\times \vec{M}\)
\(\displaystyle \frac{1}{\mu_0}(\nabla\times\vec{B})=\vec{J}=\vec{J}_b+\vec{J}_{free}=\vec{J}_{free}+\nabla \times \vec{M}\)
so that,
\(\displaystyle \nabla\times \left(\frac{1}{\mu_0}\vec{B}-\vec{M}\right)=\vec{J}_{free}\),
from which we define,
###This can be compared with the similar result we found for electric fields, \(\vec{D}=\epsilon_0\vec{E}+\vec{P}\) and again shows, if we know the magnetization, we don’t need to calculate the bound current densities in order to calculate \(\vec{B}\).
We now also have the equaivalent of Ampere’s law for \(\vec{H}\),
###and its integral form
\(\displaystyle \oint\vec{H}\cdot d\vec{l}=I_{free}\)
We can use \(\vec{H}\) in an analagous way to the way we used \(\vec{D}\) in electrostatics. In this case we can use Ampere’s law and the free currents to calculate \(\vec{H}\).
As in the case of \(\vec{D}\) there is a sting in the tail that can lead to errors in use. We need to remember that the curl does not uniquely define a vector field. We also need to know the divergence. If we take the divergence of 13.6 we get
\(\displaystyle \nabla\cdot\vec{H}=-\nabla\cdot\vec{M}\)
as \(\nabla\cdot\vec{B}\) is always zero. Hence the caveat is that we can always use Ampere’s law to calculate \(\vec{H}\) provided \(\nabla\cdot\vec{M}=0\) (c.f. we needed \(\nabla\times\vec{P}=0\) for the case of the \(\vec{D}\) field). Again this normally works in cases of high symmetry. If in doubt check the divergence of \(\vec{M}\).
13.11 The boundary conditions for \(\vec{H}\).
As in the case of the \(\vec{D}\) field, the boundary conditions for \(\vec{H}\) are particularly useful for solving problems on the interface between tow materials. Again without proof I will state them here. The proofs may ve found in Griffiths 6.3.3.
###13.12 Linear magnetic materials.
As we did in the case of dielectric materials we can consider the simple case of induced mgnetization by a magnetic field. Logic might suggest that we should write \(\vec{M}\propto\vec{B}\) as we did for \(\vec{D}\) and \(\vec{E}\). However, convention has meant it is normally written as,
\(\displaystyle \vec{M}=\chi_m\vec{H}\)
where \(\chi_m\) is known as the magnetic susceptibility. Here we recognise it is possible to have two types of linear response - paramagnetic materials where \(\chi_m \gt 0\) and diamagnetic materials where \(\chi_m \lt 0\). From 13.6. we than have,
\(\displaystyle \vec{H}=(\frac{\vec{B}}{\mu_0}-\vec{M})=(\frac{\vec{B}}{\mu_0}-\chi_m \vec{H})\)
so
\(\displaystyle \vec{B}=\mu_0(1+\chi_m)\vec{H}\)
###where \(\mu=\mu_0(1+\chi_m)\) and \(\mu_r=(1+\chi_m)\) where \(\mu_r\) is the relative permeability.
We should note here that some common materials (Iron for instance) show a property called ferromagnetism. In this case the material can remain magnetized even when the external field is removed. In addition, at some point, the magnetic dipoles become pretty much oriented perfectly in line with the applied field. This is called saturation and the application of a higher magnetic field has little effect on the magnetization. This is often shown in a magnetization curve that also shows properties such as hysteresis. We don’t have time to discuss these much here but it is a rich area to explore and covered a little further in Griffiths if you wish to learn more.
13.13 Why use \(\vec{H}\)?.
In much the same way as we use \(\vec{D}\), \(\vec{H}\) can be useful when discussing the boundaries between materials. However, it possibly easier to see why we might use it compared to \(\vec{D}\). In experiments/technology we will create fields by winding coils etc. and passing currents through them. These currents are the free currents that we refer to. Hence, it is very easy to calculate \(\vec{H}\) and once we know the magnetization and especially in the case of linear materials, it is straightforward to calculate \(\vec{B}\) in the material for the the applied field.
13.14 Maxwell’s equations in matter.
We are almost there in our discussion of fields in matter but there remains the question about what happens in time dependent fields. So far, when we have talked about bound charges \(\rho_b\) and bound currents \(\vec{J_b}\) we have assumed static fields and used the results,
\(\displaystyle \rho_b=-\nabla\cdot\vec{P}\)
and
\(\displaystyle \vec{J_b}=\nabla\times \vec{M}\)
However, what happens if we have non-static fields? Well, any change in the lectric polarization would appear as a ‘flow’ of the bound charge (onto or off the surface for example). How might we imagine this? If we have a slab of polarised material, for example, something we represented as in section 13.3 we will have surface charges \(\pm \sigma_b\) at each end. If \(\vec{P}\) no increases (say by increasing the external field) there will be a consequent increase in the surface charge that appears as net current at the surface. We can write this as
\(\displaystyle dI=\frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partial t}da_{\perp}\)
(remembering \(\sigma\; da = Q\) and \(\sigma_B = \vec{P}\cdot\hat{\vec{n}}\) at the surface).
Now we should note that this has nothing to do with the free and bound currents that we discussed in the magnetization of materials. It is known as the polarisation current \(\vec{J}_p\) and may be written more generally as,
\(\displaystyle \vec{J}_p=\frac{\partial \vec{P}}{\partial t}\)
What about the continuity equations for \(\vec{J}_P\)? Well we have,
\(\displaystyle \nabla \cdot \vec{J}_p=\nabla\cdot \frac{\partial \vec{P}}{\partial t}=\frac{\partial}{\partial t}(\nabla\cdot\vec{P})=-\frac{\partial \rho_b}{\partial t}\)
i.e.
\(\displaystyle \nabla \cdot \vec{J}_p=-\frac{\partial \rho_b}{\partial t}\) as we’d expect. It is also required if you think about it to ensure the conservation of the bound charge. So, as before, we still have
\(\displaystyle \rho=\rho_f+\rho_b=\rho_f-\nabla\cdot\vec{P}\)
but we need to include the polarisation currnet in the total current density, i.e.,
\(\displaystyle \vec{J}=\vec{J_f}+\vec{J_b}+\vec{J_p}=\vec{J_f}+\nabla\times\vec{M}+\frac{\partial \vec{P}}{\partial t}\)
Now if we take these results for \(\rho\) and \(\vec{J}\) and apply them to Maxwells equations in turn we get.
\(\\displaystyle nabla\cdot\vec{E}=\frac{1}{\epsilon_0}(\rho_f-\nabla\cdot\vec{P})\) which gives,
\(\displaystyle \nabla\cdot\vec{D}=\rho_f\)
as before.
In \(\nabla\cdot\vec{B}=0\) there is nothing to change so we still have
\(\nabla\cdot\vec{B}=0\).
as before. Similarly in \(\nabla\times \vec{E}=-\partial \vec{B}/\partial t\) there is also nothing to change. so
\(\displaystyle \nabla\times \vec{E}=-\frac{\partial \vec{B}}{\partial t}\)
as before. However, including the polarisation current we now have,
\(\displaystyle \nabla\times \vec{B}=\mu_0(\vec{J}_f+\nabla\times\vec{M}+\frac{\partial \vec{P}}{\partial t})+\mu_0\epsilon_0\frac{\partial \vec{E}}{\partial t}\)
Now, from our definition of \(\vec{H}\) we have,
\(\displaystyle \nabla\times\vec{B}=\mu_0(\nabla\times \vec{H}+\nabla\times\vec{M})\)
and from our definition of \(\vec{D}\) we have,
\(\displaystyle \mu_0\frac{\partial}{\partial t}(\vec{P}+\epsilon_0\vec{E})=\mu_0\frac{\partial \vec{D}}{\partial t}\)
Putting all this together we get,
\(\displaystyle \nabla\times\vec{H}=\vec{J_f}+\frac{\partial \vec{D}}{\partial t}\)
13.15 Maxwell’s equations in matter.
So, we are finally able to write our Maxwell’s equations taking into account the presence of matter.In summary these are,
#These are true even for non-linear materials. For linear materials we can express \(\vec{D}\) and \(\vec{H}\) in terms of the susceptibilities and the strength of the external field. This is the end point of this course. In future lectures, especially in the treatment of light as an electromagnetic phenomemon, you will explore them more deeply.
When working with these equations the boundary conditions required for the fields are relatively straight forward ro write but it turns out that these are most easily remembered and used using all four fields (\(\vec{E}\),\(\vec{D}\),\(\vec{B}\) and \(\vec{H}\)). In summary they are,
###In problems you often see questions referring to the boundaries when there are no free charges and currents present (at the boundary). In this case all of these relations become equal to zero. Now it is easy to see that in the absence of free charges the perpendicular component of \(\vec{D}\) is continuous - you often see this used in solution of problems with dielectrics. You also see that the perpendicular component of the \(\vec{B}\) field is continuous which you again often see used to solve problems (the magnetic field in a small cap of a toroidal solenoid is a common example).
This concludes the key part of the lectures. Maxwell’s equations in this form and the Lorentz force law are the starting point for the study of more advanced electromagnetic phenomena.