Chapter 3: Problem 23
(a) A point charge \(Q\) lies at the origin. Show that div \(\mathbf{D}\) is zero
everywhere except at the origin. (b) Replace the point charge with a uniform
volume charge density \(\rho_{v 0}\) for \(0
Chapter 3: Problem 23
(a) A point charge \(Q\) lies at the origin. Show that div \(\mathbf{D}\) is zero
everywhere except at the origin. (b) Replace the point charge with a uniform
volume charge density \(\rho_{v 0}\) for \(0
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Get started for freeIn free space, a volume charge of constant density \(\rho_{v}=\rho_{0}\) exists
within the region \(-\infty
The cylindrical surface \(\rho=8 \mathrm{~cm}\) contains the surface charge
density, \(\rho_{S}=\) \(5 e^{-20|z|} \mathrm{nC} / \mathrm{m}^{2} .(a)\) What is
the total amount of charge present? \((b)\) How much electric flux leaves the
surface \(\rho=8 \mathrm{~cm}, 1 \mathrm{~cm}
An electric field in free space is \(\mathbf{E}=\left(5 z^{2} / \epsilon_{0}\right) \hat{\mathbf{a}}_{z} \mathrm{~V} / \mathrm{m}\). Find the total charge contained within a cube, centered at the origin, of \(4-\mathrm{m}\) side length, in which all sides are parallel to coordinate axes (and therefore each side intersects an axis at \(\pm 2\) ).
Calculate \(\nabla \cdot \mathbf{D}\) at the point specified if \((a) \mathbf{D}=\left(1 / z^{2}\right)\left[10 x y z \mathbf{a}_{x}+\right.\) \(\left.5 x^{2} z \mathbf{a}_{y}+\left(2 z^{3}-5 x^{2} y\right) \mathbf{a}_{z}\right]\) at \(P(-2,3,5) ;(b) \mathbf{D}=5 z^{2} \mathbf{a}_{\rho}+10 \rho z \mathbf{a}_{z}\) at \(P\left(3,-45^{\circ}, 5\right) ;(c) \mathbf{D}=2 r \sin \theta \sin \phi \mathbf{a}_{r}+r \cos \theta \sin \phi \mathbf{a}_{\theta}+r \cos \phi \mathbf{a}_{\phi}\) at \(P\left(3,45^{\circ},-45^{\circ}\right) .\)
Suppose that the Faraday concentric sphere experiment is performed in free space using a central charge at the origin, \(Q_{1}\), and with hemispheres of radius a. A second charge \(Q_{2}\) (this time a point charge) is located at distance \(R\) from \(Q_{1}\), where \(R>>a .(a)\) What is the force on the point charge before the hemispheres are assembled around \(Q_{1} ?\) (b) What is the force on the point charge after the hemispheres are assembled but before they are discharged? ( \(c\) ) What is the force on the point charge after the hemispheres are assembled and after they are discharged? ( \(d\) ) Qualitatively, describe what happens as \(Q_{2}\) is moved toward the sphere assembly to the extent that the condition \(R>>a\) is no longer valid.
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