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(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

Short Answer

Expert verified
Answer: For a point charge Q at the origin, the divergence of the electric flux density field is zero everywhere except at the origin. For a uniform volume charge density ρ_v0 for 0<r<a, the divergence of the electric flux density field is \(\frac{3Q}{4\pi a^3}\) for 0<r<a, and zero for r≥a.

Step by step solution

01

(a) Given Information and Setting up the Problem

We are given a point charge Q at the origin. Given this setup, the electric flux density \(\mathbf{D}\) for a point charge can be given by: \(\mathbf{D} = k \cdot \frac{Q}{r^2} \hat{r}\), where \(\mathbf{D}\) is the electric flux density vector, \(k\) is the Coulomb's constant, \(Q\) is the charge, \(r\) is the distance from the origin, and \(\hat{r}\) is the radial unit vector. We use Gauss's Law (\(\nabla \cdot \mathbf{D} = \rho_v\)) to find the divergence of \(\mathbf{D}\), where \(\rho_v\) is the volume charge density.
02

(a) Find the Divergence of \(\mathbf{D}\)

Calculating the divergence of the electric flux density vector using spherical coordinates: \(\nabla \cdot \mathbf{D} = \frac{1}{r^2} \frac{\partial (r^2 D_r)}{\partial r} + \frac{1}{r\sin{\theta}}\frac{\partial(D_\theta \sin{\theta})}{\partial \theta} + \frac{1}{r\sin{\theta}}\frac{\partial D_\phi}{\partial \phi}\) Since \(\mathbf{D} = k \cdot \frac{Q}{r^2} \hat{r}\), \(D_\theta=0\) and \(D_\phi=0\). Substitute into our divergence equation: \(\nabla \cdot \mathbf{D} = \frac{1}{r^2} \frac{\partial (r^2 k \cdot \frac{Q}{r^2})}{\partial r}\)
03

(a) Evaluate the Derivative and Result

Evaluate the derivative with respect to r: \(\nabla \cdot \mathbf{D} = \frac{1}{r^2} \frac{\partial (kQ)}{\partial r} = 0\) Thus, the divergence of the electric flux density is zero everywhere except at the origin. At the origin, we cannot evaluate the divergence because the electric flux density vector is undefined.
04

(b) Replace Point Charge with Uniform Volume Charge Density

We are given a uniform volume charge density \(\rho_{v 0}\) for \(0 < r < a\), such that the total charge is the same as that of the point charge in part (a). To relate \(\rho_{v 0}\), Q, and a, use: total charge = volume charge density × volume \(Q = \rho_{v 0} \cdot \frac{4\pi}{3} a^3\) \(\rho_{v 0} = \frac{3Q}{4\pi a^3}\)
05

(b) Find the Divergence of \(\mathbf{D}\) for Uniform Volume Charge Density

Using Gauss's Law: \(\nabla \cdot \mathbf{D} = \rho_{v 0}\) Substitute the expression for \(\rho_{v 0}\): \(\nabla \cdot \mathbf{D} = \frac{3Q}{4\pi a^3}\) Therefore, for \(0 < r < a\), the divergence of the electric flux density is \(\frac{3Q}{4\pi a^3}\). Outside this region (i.e., \(r \geq a\)), the divergence of the electric flux density is zero.

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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

The 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}

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 \(\pm 2\) ).

Calculate \(\nabla \cdot \mathbf{D}\) at the point specified if \((a) \mathbf{D}=\left(1 / z^{2}\right)\left[10 x y z \mathbf{a}_{x}+\right.\) \(\left.5 x^{2} z \mathbf{a}_{y}+\left(2 z^{3}-5 x^{2} y\right) \mathbf{a}_{z}\right]\) at \(P(-2,3,5) ;(b) \mathbf{D}=5 z^{2} \mathbf{a}_{\rho}+10 \rho z \mathbf{a}_{z}\) at \(P\left(3,-45^{\circ}, 5\right) ;(c) \mathbf{D}=2 r \sin \theta \sin \phi \mathbf{a}_{r}+r \cos \theta \sin \phi \mathbf{a}_{\theta}+r \cos \phi \mathbf{a}_{\phi}\) at \(P\left(3,45^{\circ},-45^{\circ}\right) .\)

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.

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