Question

(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

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

(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.

(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: \(\nabla \cdot \mathbf{D} = \frac{1}{r^2} \frac{\partial (r^2 k \cdot \frac{Q}{r^2})}{\partial r}\)

(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.

(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 \(Q = \rho_{v 0} \cdot \frac{4\pi}{3} a^3\) \(\rho_{v 0} = \frac{3Q}{4\pi a^3}\)

(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.

Key concepts

Electric Flux Density

Electric flux density, often represented as \( \mathbf{D} \), plays a critical role in electromagnetic theory. It is a vector field that represents how electric fields are distributed in a given space due to charged particles.

In the case of a point charge \(Q\) positioned at the origin, the electric flux density can be described using the formula:

• \( \mathbf{D} = k \cdot \frac{Q}{r^2} \hat{r} \)

This equation illustrates that \( \mathbf{D} \) is inversely proportional to the square of the distance \(r\) from the charge, emphasizing how the influence of the charge diminishes with distance. Here, \( \hat{r} \) is the radial unit vector, and \( k \) is Coulomb's constant.

The electric flux density simplifies the calculation of electric fields, especially in symmetrical configurations, making it indispensable in both theoretical and practical applications.

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetic theory, providing a connection between electric field lines emanating from charges and the electric flux density. Mathematically, it is expressed as:

• \( abla \cdot \mathbf{D} = \rho_v \)

This equation indicates that the divergence of the electric flux density \( \mathbf{D} \) results in the volume charge density \( \rho_v \).

Gauss's Law is particularly useful for determining electric fields for symmetric charge distributions. In the scenario of the problem, it helps evaluate the divergence of \( \mathbf{D} \) for both point charges and volume charge densities. The law reveals that for a point charge, the divergence is zero everywhere except at the origin, which supports the concept of the density field spreading uniformly from a point source.

Point Charge

A point charge is an idealized model of a charged particle, characterized by a charge \( Q \) located at a single point in space. In theory, point charges are used to simplify computations and provide insights into electrostatic principles.

In the provided problem, the point charge is assumed to be at the origin, facilitating a straightforward application of Gauss's Law. Despite being a theoretical construct, the concept of a point charge is crucial in deriving the behavior of electric fields and understanding electrostatic interactions at a fundamental level.

The divergence of the electric flux density for a point charge is zero everywhere except precisely at the location of the charge, due to the distribution of the field over space.

Divergence

Divergence is a mathematical operation applied to vector fields in electromagnetic theory, revealing how much a vector field spreads out from a given point. In the context of electric fields, it measures the sources and sinks of the field, like charge distributions.

The divergence of the electric flux density \( \mathbf{D} \) gives the volume charge density \( \rho_v \) according to Gauss’s Law. Using spherical coordinates simplifies calculating the divergence of radial fields, such as those created by point charges or uniformly distributed charges within a defined space.

For a point charge, the divergence is zero everywhere besides the origin. For a uniform volume charge density within a sphere of radius \( a \), the divergence yields a constant value, demonstrating how charge density can affect electric flux distribution.

Volume Charge Density

Volume charge density, denoted as \( \rho_v \), is a measure of how electric charge is distributed within a volume of space. It is calculated as the total charge \( Q \) divided by the volume \( V \) it occupies, making it a crucial concept for understanding distributed charges.

In cases where a point charge is replaced by a uniform volume charge density \( \rho_{v0} \), as in the problem, the total charge must remain equal. The formula:

• \( Q = \rho_{v0} \cdot \frac{4\pi}{3} a^3 \)

relates the volume charge density to the charge and sphere volume for radius \( a \). This relationship ensures that the same total charge is maintained within the spatial region.

The volume charge density is crucial in applications where charge is spread across a volume, allowing for real-world measurements and predictions of electric fields.