Question

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?

Short answer

The charge density generating the electric flux density is \(\rho_v = \frac{D_0}{\rho}\), and the total charge within the cylinder is \(Q = D_0 b \pi a^2\).

Step-by-step solution

Recall the relationship between electric flux density and charge density

According to Gauss's law, the divergence of the electric flux density \(\mathbf{D}\) is equal to the charge density \(\rho_v\): \begin{equation} \nabla \cdot \mathbf{D} = \rho_v \end{equation} In cylindrical coordinates \((\rho,\phi,z)\), the divergence of a vector field \(\mathbf{F} = F_\rho \mathbf{a}_\rho + F_\phi \mathbf{a}_\phi + F_z \mathbf{a}_z\) is given by: \begin{equation} \nabla \cdot \mathbf{F} = \frac{1}{\rho}\frac{\partial (\rho F_\rho)}{\partial \rho} + \frac{1}{\rho}\frac{\partial F_\phi}{\partial \phi} + \frac{\partial F_z}{\partial z} \end{equation}

Calculate the divergence of the electric flux density

Given the electric flux density is \(\mathbf{D} = D_{0} \mathbf{a}_{\rho}\), with only the radial component \(D_\rho = D_0\), we can find the divergence of \(\mathbf{D}\): \begin{align} \nabla \cdot \mathbf{D} &= \frac{1}{\rho}\frac{\partial (\rho D_\rho)}{\partial \rho} \\ &= \frac{1}{\rho}\frac{\partial (\rho D_0)}{\partial \rho} \end{align}

Determine the charge density

Now we can find the charge density \(\rho_v\) from the divergence of \(\mathbf{D}\): \begin{align} \rho_v &= \nabla \cdot \mathbf{D} \\ &= \frac{1}{\rho}\frac{\partial (\rho D_0)}{\partial \rho} \\ \rho_v &= \frac{1}{\rho}D_0 \frac{\partial \rho}{\partial \rho} \\ \rho_v &= \frac{D_0}{\rho} \end{align} Thus, the charge density generating the field is \(\rho_v = \frac{D_0}{\rho}\).

Set up the integral to find the total charge within the cylinder

To find the total charge \(Q\) within a cylinder of radius \(a\) and height \(b\), we will integrate the charge density over the volume of the cylinder: \begin{equation} Q = \int_{V} \rho_v \, dV \end{equation} where \(dV = \rho d\rho d\phi dz\) in cylindrical coordinates.

Calculate the total charge within the cylinder

Now, we can find the total charge \(Q\) within the cylinder: \begin{align} Q &= \int_{0}^{b}\int_{0}^{2\pi}\int_{0}^{a} \frac{D_0}{\rho} \rho \, d\rho d\phi dz \\ &= D_0 \int_{0}^{b} dz \int_{0}^{2\pi} d\phi \int_{0}^{a} d\rho \\ &= D_0 \left[ z \right]_{0}^{b} \left[ \phi \right]_{0}^{2\pi} \left[ \frac{1}{2}\rho^2 \right]_{0}^{a} \\ &= D_0 (b - 0) (2\pi - 0) \left(\frac{1}{2}a^2 - 0\right) \\ Q &= D_0 b \pi a^2 \end{align} Therefore, the total charge within the cylinder is \(Q = D_0 b \pi a^2\).

Key concepts

Gauss's Law

Gauss's Law is a fundamental principle that relates the distribution of electric charge to the resulting electric field. It states that the net electric flux through any closed surface, also known as a Gaussian surface, is equal to the charge enclosed by that surface divided by the permittivity of the medium. Mathematically, Gauss's Law is represented as:
\[\begin{equation}\oint \mathbf{D} \cdot d\mathbf{A} = Q_{\text{enc}}\end{equation}\]where \(\mathbf{D}\) is the electric flux density, \(d\mathbf{A}\) is the differential area vector on the Gaussian surface, and \(Q_{\text{enc}}\) is the enclosed charge.
This fundamental law is essential for calculating electric fields generated by symmetric charge distributions, such as the one described in the exercise. In the given problem, the exercise walks us through applying Gauss's Law by considering the divergence of the electric flux density to determine the charge density responsible for the field.

Charge Density

Charge density is a measure of electric charge per unit volume. It provides an understanding of how charge is distributed in space. There are various forms of charge density, with volume charge density \(\rho_v\) being particularly relevant to our exercise. It can be defined mathematically by the equation:
\[\begin{equation}\rho_v = \frac{dq}{dV}\end{equation}\]where \(dq\) is an infinitesimal amount of charge and \(dV\) is the differential volume element. In our example, the charge density \(\rho_v\) is derived from the divergence of the electric flux density \(\mathbf{D}\), which results in \(\rho_v = \frac{D_0}{\rho}\), indicating that the charge density varies inversely with the radial distance \(\rho\) in cylindrical coordinates. This is a unique situation indicating a non-uniform charge distribution.

Cylindrical Coordinates

Cylindrical coordinates are a three-dimensional coordinate system that extends polar coordinates into three dimensions by adding a height variable \(z\). It is particularly useful in problems with cylindrical symmetry. The coordinates \((\rho, \phi, z)\) correspond to the radial distance from the origin, the angular displacement from a reference axis, and the height from the reference plane, respectively.
In relation to vector calculus and electric flux density, the divergence in cylindrical coordinates is given by a more complex formula due to the curvilinear coordinate system. This system's divergence is represented by the given equation in Step 2 of the solution, which incorporates all three components of a vector field in cylindrical coordinates. Understanding how to compute operations in this coordinate system is crucial for correctly solving problems involving cylindrical geometries, like the cylinder in the exercise.

Divergence of a Vector Field

Divergence is a vector operator that measures a vector field's tendency to converge toward or diverge from a point. For an electric flux density vector field \(\mathbf{D}\), divergence precisely quantifies the rate at which the electric charge leaves a small volume around a point in space. Mathematically, the divergence of a vector field \(\mathbf{F}\) is a scalar function denoted as \(abla \cdot \mathbf{F}\).
In the context of electromagnetism, the divergence of the electric flux density is directly related to the charge density by Gauss's law, which is central to the problem addressed in the textbook solution. The solution walks through the calculation of the divergence of the given electric flux density in cylindrical coordinates, which is crucial for determining the charge density in the scenario.

Electromagnetic Field Calculations

Electromagnetic field calculations involve determining electric and magnetic field vectors from known charges and currents. These calculations frequently utilize Maxwell's equations, which include Gauss's Law for electricity as one of the four fundamental equations. These equations help us to calculate fields in various geometries and conditions.
For the exercise at hand, the calculation involved integrating the charge density over a specific volume to find the total charge enclosed within a cylindrical Gaussian surface. This integration process exemplifies a basic electromagnetic field calculation where symmetry and coordinate choice simplify the process. By integrating the density over the volume in cylindrical coordinates, as shown in Step 5, we obtain the total charge within the cylinder. This case showcases how understanding the relationship between electric flux density, charge density, and Gaussian surfaces can facilitate complex electromagnetic field calculations.