Chapter 7: Problem 9
A current sheet \(\mathbf{K}=8 \mathbf{a}_{x} \mathrm{~A} / \mathrm{m}\) flows
in the region \(-2
Chapter 7: Problem 9
A current sheet \(\mathbf{K}=8 \mathbf{a}_{x} \mathrm{~A} / \mathrm{m}\) flows
in the region \(-2
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Get started for freeA square filamentary differential current loop, \(d L\) on a side, is centered at the origin in the \(z=0\) plane in free space. The current \(I\) flows generally in the \(\mathbf{a}_{\phi}\) direction. ( \(a\) ) Assuming that \(r>>d L\), and following a method similar to that in Section \(4.7\), show that $$d \mathbf{A}=\frac{\mu_{0} I(d L)^{2} \sin \theta}{4 \pi r^{2}} \mathbf{a}_{\phi}$$ (b) Show that $$d \mathbf{H}=\frac{I(d L)^{2}}{4 \pi r^{3}}\left(2 \cos \theta \mathbf{a}_{r}+\sin \theta \mathbf{a}_{\theta}\right)$$ The square loop is one form of a magnetic dipole.
Consider a sphere of radius \(r=4\) centered at \((0,0,3)\). Let \(S_{1}\) be that portion of the spherical surface that lies above the \(x y\) plane. Find \(\int_{S_{1}}(\nabla \times \mathbf{H}) \cdot d \mathbf{S}\) if \(\mathbf{H}=3 \rho \mathbf{a}_{\phi}\) in cylindrical coordinates.
Infinitely long filamentary conductors are located in the \(y=0\) plane at \(x=n\) meters where \(n=0, \pm 1, \pm 2, \ldots\) Each carries \(1 \mathrm{~A}\) in the \(\mathbf{a}_{z}\) direction. (a) Find \(\mathbf{H}\) on the \(y\) axis. As a help, $$\sum_{n=1}^{\infty} \frac{y}{y^{2}+n^{2}}=\frac{\pi}{2}-\frac{1}{2 y}+\frac{\pi}{e^{2 \pi y}-1}$$ (b) Compare your result of part \((a)\) to that obtained if the filaments are replaced by a current sheet in the \(y=0\) plane that carries surface current density \(\mathbf{K}=1 \mathbf{a}_{z} \mathrm{~A} / \mathrm{m}\).
Given the field \(\mathbf{H}=20 \rho^{2} \mathbf{a}_{\phi} \mathrm{A} / \mathrm{m}:(a)\) Determine the current density \(\mathbf{J}\). (b) Integrate \(\mathbf{J}\) over the circular surface \(\rho \leq 1,0<\phi<2 \pi, z=0\), to determine the total current passing through that surface in the \(\mathbf{a}_{z}\) direction. (c) Find the total current once more, this time by a line integral around the circular path \(\rho=1,0<\phi<2 \pi, z=0 .\)
A long, straight, nonmagnetic conductor of \(0.2 \mathrm{~mm}\) radius carries a uniformly distributed current of 2 A dc. \((a)\) Find \(J\) within the conductor. (b) Use Ampère's circuital law to find \(\mathbf{H}\) and \(\mathbf{B}\) within the conductor. (c) Show that \(\nabla \times \mathbf{H}=\mathbf{J}\) within the conductor. \((d)\) Find \(\mathbf{H}\) and \(\mathbf{B}\) outside the conductor. \((e)\) Show that \(\nabla \times \mathbf{H}=\mathbf{J}\) outside the conductor.
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