Chapter 7: Problem 3
Two semi-infinite filaments on the \(z\) axis lie in the regions \(-\infty
Chapter 7: Problem 3
Two semi-infinite filaments on the \(z\) axis lie in the regions \(-\infty
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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.
Let \(\mathbf{A}=(3 y-z) \mathbf{a}_{x}+2 x z \mathbf{a}_{y} \mathrm{~Wb} / \mathrm{m}\) in a certain region of free space. (a) Show that \(\nabla \cdot \mathbf{A}=0 .(b)\) At \(P(2,-1,3)\), find \(\mathbf{A}, \mathbf{B}, \mathbf{H}\), and \(\mathbf{J}\).
( \(a\) ) Find \(\mathbf{H}\) in rectangular components at \(P(2,3,4)\) if there is a current filament on the \(z\) axis carrying \(8 \mathrm{~mA}\) in the \(\mathbf{a}_{z}\) direction. ( \(b\) ) Repeat if the filament is located at \(x=-1, y=2\). ( \(c\) ) Find \(\mathbf{H}\) if both filaments are present.
A toroid having a cross section of rectangular shape is defined by the following surfaces: the cylinders \(\rho=2\) and \(\rho=3 \mathrm{~cm}\), and the planes \(z=1\) and \(z=2.5 \mathrm{~cm}\). The toroid carries a surface current density of \(-50 \mathbf{a}_{z} \mathrm{~A} / \mathrm{m}\) on the surface \(\rho=3 \mathrm{~cm}\). Find \(\mathbf{H}\) at the point \(P(\rho, \phi, z):(a) P_{A}(1.5 \mathrm{~cm}, 0\), \(2 \mathrm{~cm}) ;\left(\right.\) b) \(P_{B}(2.1 \mathrm{~cm}, 0,2 \mathrm{~cm}) ;\) (c) \(P_{C}(2.7 \mathrm{~cm}, \pi / 2,2 \mathrm{~cm}) ;\) (d) \(P_{D}(3.5 \mathrm{~cm},\), \(\pi / 2,2 \mathrm{~cm})\)
The magnetic field intensity is given in a certain region of space as \(\mathbf{H}=\) \(\left[(x+2 y) / z^{2}\right] \mathbf{a}_{y}+(2 / z) \mathbf{a}_{z} \mathrm{~A} / \mathrm{m} .(a)\) Find \(\nabla \times \mathbf{H} .(b)\) Find \(\mathbf{J} .(c)\) Use \(\mathbf{J}\) to find the total current passing through the surface \(z=4,1 \leq x \leq 2,3 \leq z \leq 5\), in the \(\mathbf{a}_{z}\) direction. ( \(d\) ) Show that the same result is obtained using the other side of Stokes' theorem.
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