91Ó°ÊÓ

If we have a perfect gas of mass density \(\rho_{m} \mathrm{~kg} / \mathrm{m}^{3}\), and we assign a velocity \(\mathbf{U} \mathrm{m} / \mathrm{s}\) to each differential element, then the mass flow rate is \(\rho_{m} \mathbf{U} \mathrm{kg} /\left(\mathrm{m}^{2}-\mathrm{s}\right)\). Physical reasoning then leads to the continuity equation, \(\nabla \cdot\left(\rho_{m} \mathbf{U}\right)=-\partial \rho_{m} / \partial t .(a)\) Explain in words the physical interpretation of this equation. (b) Show that \(\oint_{s} \rho_{m} \mathbf{U} \cdot d \mathbf{S}=-d M / d t\), where \(M\) is the total mass of the gas within the constant closed surface \(S\), and explain the physical significance of the equation.

Let \(\mathbf{D}=5.00 r^{2} \mathbf{a}_{r} \mathrm{mC} / \mathrm{m}^{2}\) for \(r \leq 0.08 \mathrm{~m}\) and \(\mathbf{D}=0.205 \mathrm{a}_{r} / r^{2} \mu \mathrm{C} / \mathrm{m}^{2}\) for \(r \geq 0.08 \mathrm{~m} .(a)\) Find \(\rho_{v}\) for \(r=0.06 \mathrm{~m} .(b)\) Find \(\rho_{v}\) for \(r=0.1 \mathrm{~m} .(c)\) What surface charge density could be located at \(r=0.08 \mathrm{~m}\) to cause \(\mathbf{D}=0\) for \(r>0.08 \mathrm{~m} ?\)

(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 \(r\).

Given the flux density \(\mathbf{D}=\frac{16}{r} \cos (2 \theta) \mathbf{a}_{\theta} \mathrm{C} / \mathrm{m}^{2}\), use two different methods to find the total charge within the region \(1