Question 5.5: Find the magnetic field a distance s from a long straight wi...

In the diagram,(d\acute{\pmb{I}}\times \hat{\pmb{\eta }})  points out of the page, and has the magnitude

d\acute{l}\sin\alpha =d\acute{l}\cos\theta.

Also, \acute{l} = s \tan \theta , so

and s = \eta \cos \theta , so

\frac{1}{\eta^{2}}=\frac{\cos^{2}\theta}{s^{2}}.

Thus

B=\frac{\mu _{0}I}{4\pi}\int_{\theta_{1}}^{\theta_{2}}{\left(\frac{\cos^{2}\theta}{s^{2}} \right) \left(\frac{s}{\cos^{2}\theta} \right)\cos\theta d\theta } \\= \frac{\mu _{0}I}{4\pi}\int_{\theta_{1}}^{\theta_{2}}{\cos\theta d\theta}=\frac{\mu _{0}I}{4\pi}(\sin\theta_{2}-\sin\theta_{1}).        (5.37)

Equation 5.37 gives the field of any straight segment of wire, in terms of the initial and final angles θ_1   and   θ_2 (Fig. 5.19). Of course, a finite segment by itself could never support a steady current (where would the charge go when it got to the end?), but it might be a piece of some closed circuit, and Eq. 5.37 would then represent its contribution to the total field. In the case of an infinite wire, θ_1 = −π/2     and    θ_2 = π/2, so we obtain

B=\frac{\mu _{0}I}{2\pi s}.        (5.38)

Notice that the field is inversely proportional to the distance from the wire— just like the electric field of an infinite line charge. In the region below the wire, B points into the page, and in general, it “circles around” the wire, in accordance with the right-hand rule (Fig. 5.3):

\pmb{B}=\frac{\mu _{0}I}{2\pi s}\hat{\pmb{\phi }}.            (5.39)

As an application, let’s find the force of attraction between two long, parallel wires a distance d apart, carrying currents I_1     and    I_2 (Fig. 5.20). The field at (2) due to (1) is

and it points into the page. The Lorentz force law (in the form appropriate to line currents, Eq. 5.17) predicts a force directed towards (1), of magnitude

F_{mag}=I\int{(d\pmb{I}\times \pmb{B})}.      (5.17)

The total force, not surprisingly, is infinite, but the force per unit length is

f=\frac{\mu _{0}}{2\pi} \frac{I_{1}I_{2}}{d}.                   (5.40)

If the currents are antiparallel (one up, one down), the force is repulsive— consistent again with the qualitative observations in Sect. 5.1.1.