The Magnetic Field of a Straight Wire

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Section 17.4 The Magnetic Field of a Straight Wire

Consider the magnetic field of a finite segment of straight wire along the \(z\)-axis carrying a steady current \(\II=I\,\zhat\text{.}\)

Note 17.4.1. Finite wire segments.

Of course, a finite segment of wire cannot carry a steady current. But, because of the superposition principle for magnetic fields, if we want to find the magnetic field due to several individual segments of wire that together form a closed loop, we can simply add the contributions from each of the segments. Therefore, the result of this section is the building block for the result for a number of different interesting physical problems.

The Biot-Savart Law for a linear current density is:

\begin{gather} \BB(\vec{r}) = - {\mu_0 I\over 4\pi} \Int_{\textrm{Source}} {(\rr-\rrp)\times d\rrp\over|\rr-\rrp|^3} .\tag{17.4.1} \end{gather}

Choosing to put the wire on the axis of cylindrical coordinates, we have

\begin{align*} \rr \amp= r\,\rhat + z\,\zhat ,\\ \rrp \amp= z'\,\zhat , \end{align*}

so that the numerator of Equation (17.4.1) becomes

\begin{align*} - (\rr-\rrp)\times d\rr \amp= - \bigl(r\,\rhat + (z-z')\,\zhat\bigr) \times dz'\,\zhat\\ \amp= r\,dz'\,\phat \end{align*}

which gives the expected right-hand rule behavior for the direction of the magnetic field. We therefore have

\begin{align} \BB(r, \phi, z) \amp= \frac{\mu_0 I}{4\pi} \> r\,\phat \Int_{-L}^{L} \frac{dz'}{\bigl(r^2+(z-z')^2\bigr)^{3/2}}\notag\\ \amp= - \frac{\mu_0 I}{4\pi}\> \frac{\phat}{r} \frac{(z-z')}{\sqrt{r^2+(z-z')^2}} \Bigg|_{-L}^L\notag\\ \amp= - \frac{\mu_0 I}{4\pi}\> \frac{\phat}{r} \left(\frac{(z-L)}{\sqrt{r^2+(z-L)^2}} -\frac{(z+L)}{\sqrt{r^2+(z+L)^2}}\right)\tag{17.4.2} \end{align}

Activity 17.4.1. Limiting Case: The Infinite Wire.

An important special case is an infinite wire. Take the limit of Equation (17.4.2) as \(L\to\infty\text{,}\) to obtain an expression for the magentic field due to an infinite wire.

\begin{gather} \BB(r,\phi,z) = {\mu_0 I\over 2\pi} \> \frac{\phat}{r}\tag{17.4.3} \end{gather}

We will in Section 18.1 that the result Equation (17.4.3) for the infinite wire can be derived more simply using symmetry and Ampère's Law, an approach which fails however for the finite wire.