Do volume integrals involve bounding surfaces?

[yucheng]

In Vanderlinde page 171-172, the author derives the vector potential for the magnetic dipole (and free currents)

\begin{align}
\vec{A}(\vec{r}) &=\frac{\mu_{0}}{4 \pi} \int_{\tau} \frac{\vec{J}\left(\vec{r}^{\prime}\right) d^{3} r^{\prime}}{\left|\vec{r}-\vec{r}^{\prime}\right|}+\frac{\mu_{0}}{4 \pi} \int_{\tau} \frac{\vec{M}\left(\vec{r}^{\prime}\right) \times\left(\vec{r}-\vec{r}^{\prime}\right)}{\left|\vec{r}-\vec{r}^{\prime}\right|^{3}} d^{3} r^{\prime} (1) \\
&= \frac{\mu_{0}}{4 \pi} \int_{\tau} \frac{\vec{J}\left(\vec{r}^{\prime}\right) d^{3} r^{\prime}}{\left|\vec{r}-\vec{r}^{\prime}\right|}+\frac{\mu_{0}}{4 \pi} \int_{\tau} \vec{M}\left(\vec{r}^{\prime}\right) \times \vec{\nabla}^{\prime}\left(\frac{1}{\left|\vec{r}-\vec{r}^{\prime}\right|}\right) d^{3} r^{\prime} \\
&= \frac{\mu_{0}}{4 \pi} \int \frac{\vec{J}\left(\vec{r}^{\prime}\right)+\vec{\nabla}^{\prime} \times \vec{M}\left(\vec{r}^{\prime}\right)}{\left|\vec{r}-\vec{r}^{\prime}\right|} d^{3} r^{\prime}\\
&= \frac{\mu_{0}}{4 \pi} \int_{\tau - S^{\prime}} \frac{\vec{J}\left(\vec{r}^{\prime}\right)+\vec{\nabla}^{\prime} \times \vec{M}\left(\vec{r}^{\prime}\right)}{\left|\vec{r}-\vec{r}^{\prime}\right|} d^{3} r^{\prime}+\frac{\mu_{0}}{4 \pi} \oint_{S^{\prime}} \frac{\vec{M}\left(\vec{r}^{\prime}\right) \times d \vec{S}^{\prime}}{\left|\vec{r}-\vec{r}^{\prime}\right|}
\end{align}

Using integration by parts, we see that the second term of (2) is
$$\int_{\tau} \vec{M}\left(\vec{r}^{\prime}\right) \times \vec{\nabla}^{\prime}\left(\frac{1}{\left|\vec{r}-\vec{r}^{\prime}\right|}\right) d^{3} r^{\prime} = \int_{\tau} \frac{\vec{\nabla}^{\prime} \times \vec{M}\left(\vec{r}^{\prime}\right)}{\left|\vec{r}-\vec{r}^{\prime}\right|} d^{3} r^{\prime}+\oint_{S^{\prime}} \frac{\vec{M}\left(\vec{r}^{\prime}\right)}{\left|\vec{r}-\vec{r}^{\prime}\right|} \times d \vec{S}^{\prime} \tag{5}$$

However, in (5) since the volume on the LHS is inclusive of the surface as well, isn't the volume on the RHS also inclusive of the surface?

In (3), the surface integral term drops out because we integrate over a larger volume such that the magnetization there is zero.

However, for (3) the author claims that the curl is undefined at the boundaries of the magnetized material, hence it is easier to integrate over the volume of the magnetized integral, (4): the volume integral excluding the surface $\tau - S^{'}$ where it is undefined, and the surface integral taking care of the surface $S^{'}$. However, why can we exclude the surface from the volume integral (now we have an 'open' volume)?

Does this mean that for

$$
\vec{E}(\vec{r})=\frac{1}{4 \pi \varepsilon_{0}} \int_{\tau} \frac{\left[-\vec{\nabla}^{\prime} \cdot \vec{P}\left(\vec{r}^{\prime}\right)\right]\left(\vec{r}-\vec{r}^{\prime}\right)}{\left|\vec{r}-\vec{r}^{\prime}\right|^{3}} d^{3} r^{\prime}+\frac{1}{4 \pi \varepsilon_{0}} \oint_{S} \frac{(\vec{P} \cdot \hat{n})\left(\vec{r}-\vec{r}^{\prime}\right)}{\left|\vec{r}-\vec{r}^{\prime}\right|^{3}} d S^{\prime}
$$
we ought to exclude the bounding surface from the volume integral as well?

Thanks in advance!

 

[vanhees71]

It depends a bit on how you define the involved derivatives.

As an example take a homogeneously magnetized sphere, i.e.,
$$\vec{M}(\vec{x})=M \vec{e}_3 \Theta(a-r), \quad r=|\vec{x}|.$$
Then you can calculate the equivalent current density
$$\vec{j}_M=\vec{\nabla} \times \vec{M}.$$
Of course that's defined in the usual sense only for $r \neq a$, and there $\vec{j}_M=0$. Physically it's clear that the magnetic field is not 0 everywhere. So the effective current is in fact to be understood as a surface current.

Again there are two possibilities to treat it:

(a) using generalized functions

You can read the rotation of the magnetization as an equation for generalized functions. Then you get
$$\vec{j}_M=\vec{\nabla} \times \vec{M}=\vec{\nabla} \times [M \vec{e}_3 \Theta(a-r)]=-m \vec{e}_3 \times \vec{\nabla} \Theta(a-r)=M \vec{e}_3 \times \frac{\vec{r}}{r} \delta(r-a)=\vec{k} \delta(r-a).$$
Here $\vec{k}$ is the said surface current. Now you can use this distribution-valued current density in Biot-Savart's Law for the vector potential (in Coulomb gauge),
$$\vec{A}(\vec{r})=\mu_0 \int_{\mathbb{R}^3} \mathrm{d}^3 r' \frac{\vec{j}_M(\vec{r}')}{4 \pi |\vec{r}-\vec{r}'|}. \qquad (*) $$

(b) using the "surface curl"

There you directly calculate the surface current density as
$$\text{Curl} \vec{M}=\vec{n} \times (\vec{M}_{>}-\vec{M}_{<})$$
along the surface, i.e., the spherical shell $r=a$. Here $\vec{n}$ is the surfas-normal vector pointing outwards, i.e., $\vec{n}=\vec{r}/r$,
$$\vec{k}=\text{Curl} \vec{M} = \vec{n} \times (-M \vec{e}_3) = M \vec{e}_3 \times \frac{\vec{r}}{a}$$
for $\vec{r}$ on the spherical shell. Again with Biot-Savart's Law
$$\vec{A}(\vec{r})=\mu_0 \int_{S_a} \mathrm{d}^2 f' \frac{\vec{k}(\vec{r}')}{4 \pi |\vec{r}-\vec{r}'|}.$$
This you get of course also by integrating out the $r$ integral in (*).

 

[yucheng]

Thanks @vanhees71 ! I was actually exploring the use of generalized functions to compute the curl and divergence of piecewise functions, especially those representable as step functions. I got really confused when I tried to identify bound volume/surface charges and currents.

P.S. textbooks should really include this example of using generalized functions

 

[yucheng]

Again there are two possibilities to treat it:

Definitely, I should choose either one, but not both! That is, either use volume currents throughout (by evaluating the curl at the surface using generalized functions), or I separate it into volume and surface currents, evaluating wherever the functions are well behaved (evaluating volume currents at the volume but excluding the surface where it's undefined), right?

 

[vanhees71]

Yes, of course you must use the equivalent surface current only once, either be it as a distribution-valued current density or as a surface-current density. Taking both would be double counting.

 

[jasonRF]

P.S. textbooks should really include this example of using generalized functions

There is at least one book that takes this approach - I do not know how good it is, though.

https://www.amazon.com/dp/1118034155/?tag=pfamazon01-20

If you have access to a library that has it then it might be worth a look. Otherwise, perhaps the author has written papers on the topic that are available online.

Jason

 

[vanhees71]

Another great, very modern book on classical E&M, making heavy use of the theory of generalized functions is

K. Lechner, Classical Electrodynamics, Springer International
Publishing AG, Cham (2018),
https://doi.org/10.1007/978-3-319-91809-9

 

[yucheng]

There is at least one book that takes this approach - I do not know how good it is, though.

It appears very specialized ;)

 

[vanhees71]

Of course you can use either approach: The traditional one, where the singularities are treated within the integrals or use the equivalent treatment in terms of generalized functions. Both are completely equivalent.

 

[yucheng]

There is at least one book that takes this approach - I do not know how good it is, though.

https://www.amazon.com/dp/1118034155/?tag=pfamazon01-20

Just curious, how did you come to know of this book?

 

[jasonRF]

Just curious, how did you come to know of this book?

If I recall correctly, I saw it in the library at work. I might have flipped through it, but have never read it.

 

[yucheng]

Another great, very modern book on classical E&M, making heavy use of the theory of generalized functions is
K. Lechner, Classical Electrodynamics, Springer International

It appears that Lechner's book mainly deals with singularities, renormalization, relativitstic EM!

 

[vanhees71]

Indeed, and that makes it a great addition to more traditional textbooks!

 

[yucheng]

Indeed, and that makes it a great addition to more traditional textbooks!

Are they not covered in the more traditional books? I mean, it's rare to see one that's mathematically rigorous (in the sense as to avoid singularities etc, literally, being careful)

 

[vanhees71]

I think the discussion of the radiation-reaction problem is treated in the most satisfying way in this book.