Problem 19 - Calculus: Early Transcendental Functions

Question

Assume that \(\iint_{S} \mathbf{D} \cdot \mathbf{n} d S=q,\) for any closed surface \(S,\) where \(\mathbf{D}=\epsilon_{0} \mathbf{E}\) is the electric flux density and \(q\) is the charge enclosed by \(S\). Show that \(\nabla \cdot \mathbf{D}=Q,\) where \(Q\) is the charge density satisfying \(q=\iiint_{R} Q d V\).

Step-by-Step Solution

Answer

The divergence of the electric flux density \(\mathbf{D}\) is equal to the charge density \(Q\), that is, \(\nabla \cdot \mathbf{D} = Q\). The crucial steps involve the use of Gauss's law and the divergence theorem.

Step 1: Applying Gauss's law

Gauss's law indicates that the total electric flux through a closed surface \(S\) equals the charge enclosed by that surface divided by the permittivity of free space. In our problem, it's already given that the total electric flux is \(q\). However, it's important to note that this flux is not the electric field \(E\) itself, but a function of \(E\), specifically \(\mathbf{D}=\epsilon_{0} \mathbf{E}\), which is the electric flux density.

Step 2: Applying divergence theorem

To convert the surface integral to a volume integral, we use the divergence theorem, which states that the surface integral of a vector field through a closed surface \(S\) is equal to the volume integral of the divergence of that vector field over the volume \(R\) enclosed by \(S\). Therefore, we can write \(\iint_{S} \mathbf{D} \cdot \mathbf{n} d S = \iiint_{R} \nabla \cdot \mathbf{D} d V\).

Step 3: Equate the two integral expressions for charge

From Gauss's law, we know that \(\iint_{S} \mathbf{D} \cdot \mathbf{n} d S=q\), and from divergence theorem, \(\iint_{S} \mathbf{D} \cdot \mathbf{n} d S = \iiint_{R} \nabla \cdot \mathbf{D} d V\). Hence, we can equate the two expressions and find \(\iiint_{R} \nabla \cdot \mathbf{D} d V = q\).

Step 4: Identify charge enclosed by volume R

We are also given that \(q=\iiint_{R} Q d V\). So, \(\iiint_{R} Q d V = \iiint_{R} \nabla \cdot \mathbf{D} d V\). Considering that this relation holds for every volume \(R\), the integrands must be equal.

Step 5: Final Identity

Thus, equating the integrands from the volume integrals, we have \(\nabla \cdot \mathbf{D} = Q\), where \(Q\) is the charge density.

Key Concepts

Electric Flux Density

Electric flux density, often denoted as \(\mathbf{D}\), is a crucial concept in electromagnetism. It represents the amount of electric flux passing through a specific area and is related to the electric field \(\mathbf{E}\) by the equation \(\mathbf{D} = \epsilon_{0} \mathbf{E}\), where \(\epsilon_{0}\) is the permittivity of free space. This relationship highlights that \(\mathbf{D}\) is directly proportional to \(\mathbf{E}\), scaling by a constant factor. In simple terms, while the electric field describes the force a charged object would experience, the electric flux density describes how the field spreads out in space. This becomes particularly relevant when calculating how charges affect surfaces they surround.
Understanding \(\mathbf{D}\) is vital when applying Gauss's Law in electromagnetism. Gauss's Law essentially connects the electric flux through a closed surface to the charge contained within that surface. It reveals how charges affect the surrounding space, captured by \(\mathbf{D}\). Thus, gaining a good understanding of electric flux density helps in visualizing and computing the action of electric fields in various contexts.

Divergence Theorem

The divergence theorem is a powerful mathematical tool used in vector calculus, bridging the gap between surface and volume integrals. Specifically, it allows the transformation of a surface integral into a volume integral. The theorem states that for any vector field \(\mathbf{F}\) and closed surface \(S\) with an enclosed volume \(R\), the surface integral of the normal component of \(\mathbf{F}\) equals the volume integral of the divergence of \(\mathbf{F}\). Mathematically, this can be expressed as \(\iint_{S} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{R} abla \cdot \mathbf{F} \, dV\).
This theorem is particularly useful in electromagnetism for converting problems that are easier to conceptualize on surfaces into formulations over volumes, where analytical techniques might be more straightforward. Applying this to \(\mathbf{D}\), the divergence theorem states that the total outflux of \(\mathbf{D}\) over a surface is equivalent to the integration of \(abla \cdot \mathbf{D}\) over the enclosed volume. This makes understanding and applying Gauss’s law more practical, as it relates electric fields to charges through mathematical expressions.

Charge Density

Charge density, denoted as \(Q\), quantifies how much electric charge is present in a specific volume of space. It is a measure of electric charge per unit volume, enabling us to precisely describe how charges are distributed in a given region. The fundamental relationship between charge density and charge \(q\) is summarized by the equation \(q = \iiint_{R} Q \, dV\), where \(q\) is the total charge, and the integral is taken over the volume \(R\).
In the context of Gauss's Law, charge density becomes important because it serves as the "source" term in the field equations. The divergence of the electric flux density, \(abla \cdot \mathbf{D}\), equals the local charge density \(Q\). This illustrates the concept that the presence of charge within a volume directly impacts the divergence of the electric flux, which can be visualized as how densely electric field lines emanate from or converge at points within the volume. Understanding \(Q\) and its relation to \(\mathbf{D}\) is critical for solving problems involving electrostatic fields.

Surface and Volume Integrals

Surface and volume integrals are integral tools used to calculate quantities over surfaces and volumes respectively. A surface integral is used to compute the flux of a vector field across a given surface. It is given analytically by \(\iint_{S} \mathbf{F} \cdot \mathbf{n} \, dS\), where \(\mathbf{n}\) is the unit normal to the surface \(S\).
On the other hand, a volume integral accumulates values over a three-dimensional region. For example, to find the total charge within a volume \(R\), you would use \(q = \iiint_{R} Q \, dV\). Volume integrals are employed extensively with divergence theorem applications, allowing the conversion from surface integrals to expressions over a volume, simplifying analysis and computation.
Understanding how to transition between these integrals aids in translating physical laws, like Gauss's Law, into mathematical expressions that can represent the distribution and behavior of electric and magnetic fields. This skill is indispensable for physicists and engineers working to design and analyze systems governed by electromagnetic interactions.