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Volume charge density is located as follows: \(\rho_{v}=0\) for \(\rho<1 \mathrm{~mm}\) and for \(\rho>2 \mathrm{~mm}, \rho_{v}=4 \rho \mu \mathrm{C} / \mathrm{m}^{3}\) for \(1<\rho<2 \mathrm{~mm} .\) (a) Calculate the total charge in the region \(0<\rho<\rho_{1}, 0

Short Answer

Expert verified
To summarize the solutions of each part: a) The total charge in the region is given by: $$ Q = 2\pi L\cdot \frac{4\mu C/m^3(\rho_1^3 - 1)}{3}. $$ b) The electric displacement \(D_\rho\) at \(\rho = \rho_1\) is given by: $$ D_\rho = \frac{4\mu C/m^3(\rho_1^3 - 1)}{3\rho_1}. $$ c) The values of \(D_\rho\) at the specified values of \(\rho\) are: - \(\rho = 0.8\) mm: \(D_\rho = 0 \mu C/m^2\) - \(\rho = 1.6\) mm: \(D_\rho \approx 1.409 \mu C/m^2\) - \(\rho = 2.4\) mm: \(D_\rho \approx 1.667 \mu C/m^2\).

Step by step solution

01

Identify volume charge density function

The volume charge density \(\rho_v\) is given as follows: $$\rho_{v}=0 \text{ for } \rho<1 \mathrm{~mm} \text{ and for } \rho>2 \mathrm{~mm},$$ $$\rho_{v}=4 \rho \mu \mathrm{C} / \mathrm{m}^{3} \text{ for } 1<\rho<2 \mathrm{~mm}.$$
02

Calculate the total charge

The total charge in the region is given by integrating the volume charge density over the volume under consideration: $$Q = \int_{0}^{2\pi} \int_{0}^{L} \int_{0}^{\rho_1} \rho_v \cdot \rho d\rho dz d\phi. $$ Since \(\rho_v = 0\) for \(\rho < 1\) mm, we can consider only the region where \(\rho_v = 4\rho \mu C / m^3\). Thus, the integral will give: $$Q = \int_{0}^{2\pi} d\phi \int_{0}^{L} dz \int_{1}^{\rho_1} (4 \rho \mu C/m^3) \rho d\rho.$$ We can split the multi-variable integral to make calculations easier: $$Q= [\int_{0}^{2\pi} d\phi][\int_{0}^{L} dz][\int_{1}^{\rho_1} (4 \rho^2 \mu C/m^3) d\rho].$$
03

Evaluate the integrals

Compute the integrals separately: $$\int_{0}^{2\pi} d\phi = 2\pi,$$ $$\int_{0}^{L} dz = L,$$ $$\int_{1}^{\rho_1} (4\rho^2\mu C/m^3) d\rho = 4 \mu C/m^3 \int_{1}^{\rho_1} \rho^2 d\rho.$$ Evaluate the last integral: $$4\mu C/m^3 [(\rho_1^3 / 3) - (1^3 / 3)] = 4\mu C/m^3 (\rho_1^3 - 1)/3.$$
04

Multiply the results

Now, combine the results of the three integrals: $$Q = 2\pi L\cdot \frac{4\mu C/m^3(\rho_1^3 - 1)}{3}.$$ This gives the total charge in the region of interest. #b) Determine \(D_{\rho}\) at \(\rho=\rho_{1}\)#
05

Apply Gauss's Law

We can use Gauss's law to determine the electric displacement \(D_\rho\): $$\oint \vec{D} \cdot \mathrm{d}\vec{A} = Q_{enc},$$ where \(Q_{enc}\) is the charge enclosed within the Gaussian surface.
06

Choose appropriate Gaussian surface

Select a cylindrical Gaussian surface with height \(L\), radius \(\rho_1\), and a base parallel to the \(xy\) plane. Only \(D_\rho\) contributes to the integral, as \(D_z\) is perpendicular to the curved surface area and \(D_\phi\) is perpendicular to the top and bottom surfaces.
07

Integrating over the surface

The integral becomes: $$\oint \vec{D} \cdot \mathrm{d}\vec{A} = D_\rho \int_{0}^{2\pi} \int_{0}^{L} \rho_1 dL d\phi = 2\pi L \cdot \rho_1 D_\rho.$$ Since we have found the total charge in part (a), we can substitute it into Gauss's Law: $$2\pi L \cdot \rho_1 D_\rho = 2\pi L\cdot \frac{4\mu C/m^3(\rho_1^3 - 1)}{3}.$$
08

Solve for \(D_\rho\)

Divide both sides by \(2\pi L \cdot \rho_1\): $$D_\rho = \frac{4\mu C/m^3(\rho_1^3 - 1)}{3\rho_1}.$$ This expression provides the value of \(D_\rho\) at \(\rho = \rho_1\). #c) Evaluate \(D_{\rho}\) at specified values of \(\rho\)# Using the result from part (b), we can now evaluate \(D_{\rho}\) for the three values of \(\rho\) given in the problem:
09

\(\rho = 0.8\) mm

The value of \(\rho\) is less than 1 mm, so no charge is enclosed. Therefore, \(D_\rho = 0\).
10

\(\rho = 1.6\) mm

Using the obtained formula: $$D_\rho = \frac{4\mu C/m^3(1.6^3 - 1)}{3(1.6)} \approx 1.409\mu C/m^2.$$
11

\(\rho = 2.4\) mm

Since the volume charge density is zero for \(\rho > 2\) mm, the enclosed charge will be the same as when \(\rho = 2\) mm. Using the formula: $$D_\rho = \frac{4\mu C/m^3(2^3 - 1)}{3(2.4)} \approx 1.667\mu C/m^2.$$

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Most popular questions from this chapter

In free space, a volume charge of constant density \(\rho_{v}=\rho_{0}\) exists within the region \(-\infty

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.

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 \(\pm 2\) ).

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 \(r=7 \mathrm{~m}\).

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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