Current in a wire

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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.