Chapter 7: Problem 3
Two semi-infinite filaments on the \(z\) axis lie in the regions \(-\infty
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Chapter 7: Problem 3
Two semi-infinite filaments on the \(z\) axis lie in the regions \(-\infty
These are the key concepts you need to understand to accurately answer the question.
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Get started for freeShow that \(\nabla_{2}\left(1 / R_{12}\right)=-\nabla_{1}\left(1 / R_{12}\right)=\mathbf{R}_{21} / R_{12}^{3}\).
A filamentary conductor carrying current \(I\) in the \(\mathbf{a}_{z}\) direction extends along the entire negative \(z\) axis. At \(z=0\) it connects to a copper sheet that fills the \(x>0, y>0\) quadrant of the \(x y\) plane. \((a)\) Set up the Biot-Savart law and find \(\mathrm{H}\) everywhere on the \(z\) axis; \((b)\) repeat part \((a)\), but with the copper sheet occupying the entire \(x y\) plane (Hint: express \(\mathbf{a}_{\phi}\) in terms of \(\mathbf{a}_{x}\) and \(\mathbf{a}_{y}\) and angle \(\phi\) in the integral).
The free space region defined by \(1
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.
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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