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(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 \(r\).

Short Answer

Expert verified
Answer: The expression for the variation of radially directed electric field intensity with radial distance (r) in a region with a nonhomogeneous dielectric is given by \(E_r(r) = \frac{C_2}{\epsilon(r) r^2}\), where \(C_2\) is a constant.

Step by step solution

01

Write Maxwell's first equation in terms of given variables

In this problem, we are given the equation \(\nabla \cdot \mathbf{D}=\rho_{v}\). Since no charge density exists, we have \(\rho_{v}=0\) and the equation becomes \(\nabla \cdot \mathbf{D} = 0\).
02

Express \(\mathbf{D}\) in terms of electric field intensity \(E_x\) and permittivity

For a nonhomogeneous dielectric with increasing permittivity, we have \(\mathbf{D} = \epsilon \mathbf{E}\). Since the electric field only has an \(x\) component, we can write: \(\mathbf{D} = \epsilon(x) E_x(x) \hat{x}\)
03

Calculate divergence of \(\mathbf{D}\)

Now, we need to find the divergence of \(\mathbf{D}\): \(\nabla \cdot \mathbf{D} = \frac{\partial (\epsilon(x) E_x(x))}{\partial x}\)
04

Set the divergence to zero and solve for \(E_x\)

Using the derived divergence from step 3 and the fact that \(\rho_v=0\), we now have: \(\frac{\partial (\epsilon(x) E_x(x))}{\partial x} = 0\) Integrating both sides with respect to \(x\) yields: \(\epsilon(x) E_x(x) = C_1\) We finally get the expression for the variation of electric field intensity with \(x\): \(E_x(x) = \frac{C_1}{\epsilon(x)}\) (b)
05

Repeat steps 1 and 2 for spherical coordinates

Using Maxwell's first equation in spherical coordinates, the divergence of \(\mathbf{D}\) for a radially varying electric field is given by: \(\nabla \cdot \mathbf{D} = \frac{1}{r^2} \frac{\partial (\epsilon(r) E_r(r) r^2)}{\partial r}\) Since \(\rho_{v} = 0\), the equation becomes: \(\frac{1}{r^2} \frac{\partial (\epsilon(r) E_r(r) r^2)}{\partial r} = 0\)
06

Solve the equation for \(E_r\)

Now, we need to solve this equation for \(E_r(r)\): \(\frac{\partial (\epsilon(r) E_r(r) r^2)}{\partial r} = 0\) Integrating both sides with respect to \(r\) yields: \(\epsilon(r) E_r(r) r^2 = C_2\) Finally, we get the expression for the variation of radially directed electric field intensity with \(r\): \(E_r(r) = \frac{C_2}{\epsilon(r) r^2}\)

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Most popular questions from this chapter

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 \(r>0.08 \mathrm{~m} ?\)

An infinitely long cylindrical dielectric of radius \(b\) contains charge within its volume of density \(\rho_{v}=a \rho^{2}\), where \(a\) is a constant. Find the electric field strength, \(\mathbf{E}\), both inside and outside the cylinder.

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) A flux density field is given as \(\mathbf{F}_{1}=5 \mathbf{a}_{z} .\) Evaluate the outward flux of \(\mathbf{F}_{1}\) through the hemispherical surface, \(r=a, 0<\theta<\pi / 2,0<\phi<2 \pi\) (b) What simple observation would have saved a lot of work in part \(a ?\) (c) Now suppose the field is given by \(\mathbf{F}_{2}=5 z \mathbf{a}_{z} .\) Using the appropriate surface integrals, evaluate the net outward flux of \(\mathbf{F}_{2}\) through the closed surface consisting of the hemisphere of part \(a\) and its circular base in the \(x y\) plane. ( \(d\) ) Repeat part \(c\) by using the divergence theorem and an appropriate volume integral.

A radial electric field distribution in free space is given in spherical coordinates as: $$ \begin{array}{l} \mathbf{E}_{1}=\frac{r \rho_{0}}{3 \epsilon_{0}} \mathbf{a}_{r} \quad(r \leq a) \\ \mathbf{E}_{2}=\frac{\left(2 a^{3}-r^{3}\right) \rho_{0}}{3 \epsilon_{0} r^{2}} \mathbf{a}_{r} \quad(a \leq r \leq b) \\ \mathbf{E}_{3}=\frac{\left(2 a^{3}-b^{3}\right) \rho_{0}}{3 \epsilon_{0} r^{2}} \mathbf{a}_{r} \quad(r \geq b) \end{array} $$ where \(\rho_{0}, a\), and \(b\) are constants. \((a)\) Determine the volume charge density in the entire region \((0 \leq r \leq \infty)\) by the appropriate use of \(\nabla \cdot \mathbf{D}=\rho_{v} \cdot(b) \mathrm{In}\) terms of given parameters, find the total charge, \(Q\), within a sphere of radius \(r\) where \(r>b\).

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