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Chapter 9: Problem 16

Derive the continuity equation from Maxwell's equations.

Short Answer

Expert verified
Answer: The continuity equation derived from Maxwell's equations is given by \(\nabla \cdot \vec{J} + \frac{\partial \rho}{\partial t} = 0\). This equation describes the conservation of electric charge in a conducting medium, ensuring that the total charge within the medium remains constant over time.

Step by step solution

01

1. Review Maxwell's Equations

We will work with the four Maxwell's equations in their differential form: 1. Gauss's Law for Electricity: \(\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}\) 2. Gauss's Law for Magnetism: \(\nabla \cdot \vec{B} = 0\) 3. Faraday's Law: \(\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}\) 4. Ampere's Law with Maxwell's addition: \(\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}\)
02

2. Apply Vector Calculus

Take the divergence of Ampere's Law with Maxwell's addition: \(\nabla \cdot (\nabla \times \vec{B}) = \nabla \cdot (\mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t})\)
03

3. Simplify Equation

Using the property \(\nabla \cdot (\nabla \times A) = 0\) for any vector A, we can simplify the left side of the equation: 0 = \(\nabla \cdot (\mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t})\)
04

4. Distribute Divergence Operator

Distribute the divergence operator on the right side of the equation: 0 = \(\mu_0 \nabla \cdot \vec{J} + \mu_0 \epsilon_0 \nabla \cdot \frac{\partial \vec{E}}{\partial t}\)
05

5. Apply Gauss's Law for Electricity

Replace the divergence of the electric field by its expression from Gauss's Law for Electricity: 0 = \(\mu_0 \nabla \cdot \vec{J} + \mu_0 \epsilon_0 \frac{\partial}{\partial t} \left( \frac{\rho}{\epsilon_0} \right)\)
06

6. Rearrange terms to obtain the Continuity Equation

Rearrange the terms to obtain the Continuity Equation: \(\nabla \cdot \vec{J} + \frac{\partial \rho}{\partial t} = 0\) The continuity equation is derived, describing the conservation of electric charge in a conducting medium.

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Most popular questions from this chapter

Write Maxwell's equations in point form in terms of \(\mathbf{E}\) and \(\mathbf{H}\) as they apply to a sourceless medium, where \(\mathbf{J}\) and \(\rho_{v}\) are both zero. Replace \(\epsilon\) by \(\mu, \mu\) by \(\epsilon, \mathbf{E}\) by \(\mathbf{H}\), and \(\mathbf{H}\) by \(-\mathbf{E}\), and show that the equations are unchanged. This is a more general expression of the duality principle in circuit theory.

In a region where \(\mu_{r}=\epsilon_{r}=1\) and \(\sigma=0\), the retarded potentials are given by \(V=x(z-c t) \mathrm{V}\) and \(\mathbf{A}=x\left(\frac{z}{c}-t\right) \mathbf{a}_{z} \mathrm{~Wb} / \mathrm{m}\), where \(c=1 \sqrt{\mu_{0} \epsilon_{0}}\) (a) Show that \(\nabla \cdot \mathbf{A}=-\mu \epsilon \frac{\partial V}{\partial t} .(b)\) Find \(\mathbf{B}, \mathbf{H}, \mathbf{E}\), and \(\mathbf{D} .(c)\) Show that these results satisfy Maxwell's equations if \(\mathbf{J}\) and \(\rho_{v}\) are zero.

In a sourceless medium in which \(\mathbf{J}=0\) and \(\rho_{v}=0\), assume a rectangular coordinate system in which \(\mathbf{E}\) and \(\mathbf{H}\) are functions only of \(z\) and \(t .\) The medium has permittivity \(\epsilon\) and permeability \(\mu .(a)\) If \(\mathbf{E}=E_{x} \mathbf{a}_{x}\) and \(\mathbf{H}=H_{y} \mathbf{a}_{y}\), begin with Maxwell's equations and determine the second-order partial differential equation that \(E_{x}\) must satisfy. \((b)\) Show that \(E_{x}=E_{0} \cos (\omega t-\beta z)\) is a solution of that equation for a particular value of \(\beta .(c)\) Find \(\beta\) as a function of given parameters.

In region \(1, z<0, \epsilon_{1}=2 \times 10^{-11} \mathrm{~F} / \mathrm{m}, \mu_{1}=2 \times 10^{-6} \mathrm{H} / \mathrm{m}\), and \(\sigma_{1}=\) \(4 \times 10^{-3} \mathrm{~S} / \mathrm{m} ;\) in region \(2, z>0, \epsilon_{2}=\epsilon_{1} / 2, \mu_{2}=2 \mu_{1}\), and \(\sigma_{2}=\sigma_{1} / 4\). It is known that \(\mathbf{E}_{1}=\left(30 \mathbf{a}_{x}+20 \mathbf{a}_{y}+10 \mathbf{a}_{z}\right) \cos 10^{9} t \mathrm{~V} / \mathrm{m}\) at \(P\left(0,0,0^{-}\right) \cdot(a)\) Find \(\mathbf{E}_{N 1}, \mathbf{E}_{t 1}, \mathbf{D}_{N 1}\), and \(\mathbf{D}_{t 1}\) at \(P_{1} \cdot(b)\) Find \(\mathbf{J}_{N 1}\) and \(\mathbf{J}_{t 1}\) at \(P_{1} \cdot(c)\) Find \(\mathbf{E}_{t 2}\), \(\mathbf{D}_{t 2}\), and \(\mathbf{J}_{t 2}\) at \(P_{2}\left(0,0,0^{+}\right) .(d)\) (Harder) Use the continuity equation to help show that \(J_{N 1}-J_{N 2}=\partial D_{N 2} / \partial t-\partial D_{N 1} / \partial t\), and then determine \(\mathbf{D}_{N 2}\) \(\mathbf{J}_{N 2}\), and \(\mathbf{E}_{N 2}\).

A vector potential is given as \(\mathbf{A}=A_{0} \cos (\omega t-k z) \mathbf{a}_{y} .(a)\) Assuming as many components as possible are zero, find \(\mathbf{H}, \mathbf{E}\), and \(V .(b)\) Specify \(k\) in terms of \(A_{0}, \omega\), and the constants of the lossless medium, \(\epsilon\) and \(\mu\).
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