The Relationship between \boldsymbol{\vec{B}} \boldsymbol{\vec{A}} and \boldsymbol{\vec{J}}

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Section 20.2 The Relationship between \(\boldsymbol{\vec{B}}\text{,}\) \(\boldsymbol{\vec{A}}\text{,}\) and \(\boldsymbol{\vec{J}}\)

Starting with the vector potential for a line current, we used the superposition principle in Section 17.3 to obtain an integral formula for the vector potential due to any current distribution, namely

\begin{equation} \AA(\rr) = {\mu_0\over 4\pi} \int {\JJ(\rrp)\,d\tau'\over|\rr-\rrp|}\tag{20.2.1} \end{equation}

as well as a similar formula for the magnetic field, the Biot–Savart Law, namely

\begin{equation} \BB(\rr) = {\mu_0\over 4\pi} \int {\JJ(\rrp)\times(\rr-\rrp)\,d\tau'\over|\rr-\rrp|^3}\tag{20.2.2} \end{equation}

These relationships correspond, respectively, to the arrows numbered 1 and 2 in Figure 20.2.1. Furthermore, as also discussed in Section 17.3, the magnetic field is just the curl of the vector potential,

\begin{equation} \BB = \grad\times\AA\tag{20.2.3} \end{equation}

and this relationship can (in principle) be inverted to obtain the vector potential as an integral of the magnetic field; these are arrows 5 and 3, respectively.

Figure 20.2.1. The relationships between \(\AA\text{,}\) \(\BB\text{,}\) and \(\JJ\text{.}\) Each numbered arrow is discussed in the text; moving down the diagram corresponds to differentiation. Compare Figure 20.1.1.

We have also seen in Section 18.5 that the current density can be recovered as the curl of the magnetic field, namely

\begin{equation} \mu_0\JJ = \grad\times\BB\tag{20.2.4} \end{equation}

which is arrow number 4. It remains to show how to recover the current density from the vector potential (arrow number 6), at which point we are able to compute any of the quantities \(\BB\text{,}\) \(\AA\text{,}\) and \(\JJ\) from any of the others.

But from (20.2.3) and (20.2.4) it is easy to compute

\begin{equation} \mu_0\JJ = \grad\times(\grad\times\AA)\tag{20.2.5} \end{equation}

which is the desired relation. These relationships are nicely summarized in the triangle in Griffiths in Figure 5.48 on page 240, or alternatively in Figure 20.2.1. (This vector second derivative can be rewritten using formulas on the inside front cover of Griffiths.) The corresponding relationships between \(\EE\text{,}\) \(V\text{,}\) and \(\rho\) are discussed in Section 20.1.