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 freeThe 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}
A cube is defined by \(1
State whether the divergence of the following vector fields is positive, negative, or zero: ( \(a\) ) the thermal energy flow in \(\mathrm{J} /\left(\mathrm{m}^{2}-\mathrm{s}\right)\) at any point in a freezing ice cube; \((b)\) the current density in \(\mathrm{A} / \mathrm{m}^{2}\) in a bus bar carrying direct current; \((c)\) the mass flow rate in \(\mathrm{kg} /\left(\mathrm{m}^{2}-\mathrm{s}\right)\) below the surface of water in a basin, in which the water is circulating clockwise as viewed from above.
(a) Use Maxwell's first equation, \(\nabla \cdot \mathbf{D}=\rho_{v}\), to describe the variation of the electric field intensity with \(x\) in a region in which no charge density exists and in which a nonhomogeneous dielectric has a permittivity that increases exponentially with \(x\). The field has an \(x\) component only; \((b)\) repeat part \((a)\), but with a radially directed electric field (spherical coordinates), in which again \(\rho_{v}=0\), but in which the permittivity decreases exponentially with \(r\).
Spherical surfaces at \(r=2,4\), and \(6 \mathrm{~m}\) carry uniform surface charge densities of \(20 \mathrm{nC} / \mathrm{m}^{2},-4 \mathrm{n} \mathrm{C} / \mathrm{m}^{2}\), and \(\rho_{\mathrm{so}}\), respectively. \((a)\) Find \(\mathbf{D}\) at \(r=1\), 3 , and \(5 \mathrm{~m}\). (b) Determine \(\rho_{S 0}\) such that \(\mathbf{D}=0\) at \(r=7 \mathrm{~m}\).
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