Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

(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 \(r\): \(E_r(r) = \frac{C_2}{\epsilon(r) r^2}\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The sun radiates a total power of about \(3.86 \times 10^{26}\) watts \((\mathrm{W})\). If we imagine the sun's surface to be marked off in latitude and longitude and assume uniform radiation, \((a)\) what power is radiated by the region lying between latitude \(50^{\circ} \mathrm{N}\) and \(60^{\circ} \mathrm{N}\) and longitude \(12^{\circ} \mathrm{W}\) and \(27^{\circ} \mathrm{W} ?(b)\) What is the power density on a spherical surface \(93,000,000\) miles from the sun in \(\mathrm{W} / \mathrm{m}^{2} ?\)

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 \(r>b\).

In a region in free space, electric flux density is found to be $$ \mathbf{D}=\left\\{\begin{array}{lr} \rho_{0}(z+2 d) \mathbf{a}_{z} \mathrm{C} / \mathrm{m}^{2} & (-2 d \leq z \leq 0) \\ -\rho_{0}(z-2 d) \mathbf{a}_{z} \mathrm{C} / \mathrm{m}^{2} & (0 \leq z \leq 2 d) \end{array}\right. $$ Everywhere else, \(\mathbf{D}=0 .\left(\right.\) a) Using \(\nabla \cdot \mathbf{D}=\rho_{v}\), find the volume charge density as a function of position everywhere. (b) Determine the electric flux that passes through the surface defined by \(z=0,-a \leq x \leq a,-b \leq y \leq b\). (c) Determine the total charge contained within the region \(-a \leq x \leq a\), \(-b \leq y \leq b,-d \leq z \leq d .(d)\) Determine the total charge contained within the region \(-a \leq x \leq a,-b \leq y \leq b, 0 \leq z \leq 2 d\).

An electric flux density is given by \(\mathbf{D}=D_{0} \mathbf{a}_{\rho}\), where \(D_{0}\) is a given constant. (a) What charge density generates this field? \((b)\) For the specified field, what total charge is contained within a cylinder of radius \(a\) and height \(b\), where the cylinder axis is the \(z\) axis?

(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

See all solutions

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free