Section 18.2 Current in a wire
One of the most fundamental examples in electromagnetism is the magnetic field around a wire.
The magnetic field along an infinitely long straight wire along the \(z\) axis, carrying uniform current \(I\text{,}\) is given by
\begin{equation}
\BB
= \frac{\mu_0I}{2\pi} \frac{\phat}{r}
= \frac{\mu_0I}{2\pi} \frac{x\,\yhat-y\,\xhat}{x^2+y^2}\tag{18.2.1}
\end{equation}
where \(\mu_0\) and \(I\) are constant. Note that the first expression clearly indicates both the direction of \(\BB\) and its \(\frac{1}{r}\) fall-off behavior, while the second expression does neither. Given the magnetic field, Ampère’s Law allows one to determine the current flowing through (not around) any loop \(C\text{,}\) namely
\begin{equation}
\mu_0 I = \Lint \BB\cdot d\rr\tag{18.2.2}
\end{equation}
which is just Stokes’ Theorem. We verify this for the case of a circle around the \(z\)-axis.
What do we know? Our circle lies in the \(xy\)-plane; it is enough to use polar coordinates. We therefore start with the formula (6.3.1) and insert \(dr=0\text{,}\) so that \(d\rr=r\,d\phi\,\phat\text{,}\) and the integral becomes
\begin{equation}
\Lint \BB\cdot d\rr
= \int_0^{2\pi} \frac{\mu_0I}{2\pi} \,d\phi
= \mu_0 I\tag{18.2.3}
\end{equation}
as expected; the only current flowing through this loop is that flowing along the \(z\)-axis.
This was too easy! Yes, of course, one can solve this problem using rectangular coordinates, but more work is required, involving both algebra and trigonometry; the geometry is lost.
