Unconvinced by a standard step in deriving Maxwell-Ampère's law from Biot and Savart's law

Question

I have an issue with a very standard step used in the derivation of Ampère's law from Biot and Savart's law.

It can be found in multiple questions already asked here or here for example on this site.

I am not too sure where it originates from in terms of literature. It can be found in Griffith's introduction to electrodynamics and also in Jackson's Classical Electrodynamics.

The step in question is when starting from the Biot and Savart's law

\begin{equation} \vec{B}(\vec{r}) = \int_{V}\:\frac{\mu_0 \:\vec{j}(\vec{r}\:')\times(\vec{r}-\vec{r}\:')\:d\tau'}{4\pi\:|\vec{r}-\vec{r}'|^3}, \end{equation}

where $d\tau'$ is the differential volume element and $V$ the volume of space we integrate the coordinates $\vec{r}\:'$ over, and then in the midst of the calculation of the curl of $\vec{B}(\vec{r})$, one ends-up getting a term of the form

\begin{equation} \frac{\mu_0}{4\pi}\int_V\:(\vec{j}(\vec{r}\:')\cdot \vec{\nabla})\frac{\vec{r}-\vec{r}\:'}{|\vec{r}-\vec{r}\:'|^3} \:d\tau' \end{equation}which is usually made quick work of by noting that $\vec{\nabla}\frac{\vec{r}-\vec{r}\:'}{|\vec{r}-\vec{r}\:'|^3}=-\vec{\nabla}'\frac{\vec{r}-\vec{r}\:'}{|\vec{r}-\vec{r}\:'|^3}$ and integrating by part to get

\begin{equation} -\frac{\mu_0}{4\pi}\int_V\:(\vec{j}(\vec{r}\:')\cdot \vec{\nabla}')\frac{\vec{r}-\vec{r}\:'}{|\vec{r}-\vec{r}\:'|^3}\:d\tau' = \frac{\mu_0}{4\pi}\int_V\:( \vec{\nabla}'\cdot \vec{j}(\vec{r}\:'))\frac{\vec{r}-\vec{r}\:'}{|\vec{r}-\vec{r}\:'|^3} \:d\tau'- \frac{\mu_0}{4\pi}\int_{\partial V}\: \frac{\vec{r}-\vec{r}\:'}{|\vec{r}-\vec{r}\:'|^3}\:\vec{j}(\vec{r}\:')\cdot ds' \end{equation}

The first term on the right hand side is zero for steady currents because of the continuity equation. My contention is with the second term. There is no physical reason for having it to be zero in general. The standard -- and only -- argument I have seen consists in saying that we must consider a volume $V$ that strictly encloses all currents and therefore the volume current on the surface of the volume $V$ will be strictly zero. This is the weaker variant of the argument. The other variant consists in having to choose the initial to be infinite and setting the current at infinity to be zero.

The problem I have with this argument is that the end seems to justify the means here.

• From the outset, the Biot and Savart law does not make any specific requirement about enclosing all possible currents in the volume being used. In fact, it can be used to determine the field generated by any portion of volume of current if one fancies it.
• The notion that the volume should enclose all possible currents is quite imprecise. Do we mean that if one wants to find the magnetic field from one wire acting onto another they need to use a volume that incorporates all currents, including that of the wire being acted upon? That cannot be correct, for the force being obtained would be incorrect (in addition to being possibly infinite).
• The Maxwell-Ampère's law is supposed to be local. It is physically unclear why/how the curl of a magnetic field at $\vec{r}$ should depend on the currents at infinity or remotely away from the point of interest.

Please do not be mistaken. I believe can follow the argument, parrot it at infinitum and even teach it, but it's just that is seems so contrived that I find it very unsatisfactory. It does not seem to be based on any physically sound statement aside from the requirement that it should be zero otherwise we don't retrieve the expected result.

My question is then two-fold

• Is there any other argument in the literature to justify that this term is zero?
• By any chance, would anyone know where did this argument appeared first (I suppose it's more a history of physics question though)?

I am of course happy if someone can convince me that is also a perfectly satisfactory step to be made.

Answer 1

First off, I don't see why you think integrating over all currents is artificial. Ampere's law is a statement about the total magnetic field, and the total magnetic field depends on all the currents. So we must take $V$ to contain all currents. We can guarantee this by taking $V$ to be all space.

Once we do this, we are still left with a boundary term that can spoil the calculation, due to currents extending all the way to infinity. But this isn't a violation of locality, it's a shortcoming of the original Biot-Savart expression. If you really had such currents (e.g. suppose $\mathbf{J}$ was uniform out to infinity), then your original Biot-Savart integral expression is not even well-defined; it is analogous to the integral $\int_{-\infty}^\infty x \, dx$, and all the subsequent manipulations are meaningless.

This is a common issue with textbook idealizations, which also appears in electrostatics and gravity. One solution is to demand that the current distribution has finite extent, as would be true for any real current, so that the unwanted term vanishes. An alternative solution is just to accept that the differential result (i.e. Ampere's law) is more fundamental. We can't prove Ampere's law from Biot-Savart in general; we can only motivate it in special cases.

As a sidenote, while you say that there's no restriction on the integration volume in the Biot-Savart law, that's not quite true. It is true that in the case of electrostatics, you can restrict integration volumes to find the electrostatic force of any part of a charge distribution on any other part. But you shouldn't do that in magnetostatics, because every current element has to connect to something else. Back in the 1800s, there was an extensive debate over the force between two current elements. However, it was eventually realized that all of the forms gave the exact same result for the total force between two current loops. Because of this history, textbooks tend to prefer to apply to Biot-Savart law only to closed current loops, or equivalently only to volumes $V$ for which there is no current passing through $\partial V$.

Answer 2

Not sure if this is relevant to your question, so let me know if I'm on the wrong track.

If we assume that $\nabla\cdot j = 0$ everywhere in $\mathbb R^3$ say, differential topology (e.g. the Poincar'e lemma) tells us that there exists a vector field $\gamma$ such that $\nabla\times \gamma = j$. This wouldn't be the case if $j$ were defined on a manifold with some nontrivial topology such as a 3-torus, in which case we would also need to worry about special divergence-free current distributions that nonetheless have non-zero net fluxes over certain closed and compact surfaces.

Assuming that there exists $\gamma$ with $\nabla\times\gamma = j$, you can partition $\gamma$ into smooth components $\gamma_i$ such that $\gamma(x) = \sum_i\gamma_i(x)$, and each $\gamma_i$ is compactly supported (a "partition of unity".) These components naturally induce compactly supported $j_i(x)$ functions that are divergence free (because $\nabla\cdot\nabla\times f(x) = 0$), and you can consider the Biot-Savart magnetic fields $B_i(x)$ generated by each part, and prove Amp`ere's law for each component.

In theory there's nothing stopping you from defining your own integral (e.g. a special sort of regularization) to make sense of fields produced by current distributions that don't necessarily decay at infinity, but you might be hard pressed to test its physical relevance (apart from, e.g., cosmology.)