Problem 36 - Calculus: Early Transcendental Functions

Question

Use Gauss' Law \(\nabla \cdot \mathbf{E}=\frac{\rho}{\epsilon_{0}}\) for an electric field E, charge density \(\rho\) and permittivity \(\epsilon_{0}\) The integral form of Gauss' Law is \(\iint_{S} \mathbf{E} \cdot \mathbf{n} d S=\frac{q}{\epsilon_{0}},\) where \(\mathbf{E}\) is an electric field, \(q\) is the total charge enclosed by \(S\) and \(\epsilon_{0}\) is the permittivity constant. Use equation (7.1) to derive the differential form of Gauss' Law: \(\nabla \cdot \mathbf{E}=\frac{\rho}{\epsilon_{0}},\) where \(\rho\) is the charge density.

Step-by-Step Solution

Answer

By expressing total charge as a volume integral of charge density and using the divergence theorem to convert surface integral into volume integral, the differential form of Gauss's Law, \(\nabla \cdot \mathbf{E}=\frac{\rho}{\epsilon_{0}}\), is derived from its integral form.

Step 1: Understanding the integral form of Gauss's Law

Equation (7.1) is the integral form of Gauss's law, \(\iint_{S} \mathbf{E} \cdot \mathbf{n} dS=\frac{q}{\epsilon_{0}}\). This states that the flux of the electric field \(\mathbf{E}\) through a closed surface \(S\) is equal to the total charge \(q\) enclosed by that surface divided by the permittivity of free space \(\epsilon_{0}\). Here, \(\mathbf{n}\) is the outward unit normal at each point on the surface \(S\).

Step 2: Express the total charge enclosed

The total charge \(q\) enclosed by the surface \(S\) can be expressed as the volume integral of the charge density \(\rho\), i.e., \(q = \iiint_{V} \rho dV\). This is based on the understanding that charge density is defined as the amount of charge per unit volume.

Step 3: Substitution

Substitute \(q = \iiint_{V} \rho dV\) into the integral form of Gauss's law. We get, \(\iint_{S} \mathbf{E} \cdot \mathbf{n} dS=\iiint_{V} \frac{\rho}{\epsilon_{0}} dV\). This equation implies that the surface integral of the electric field over surface \(S\) equals the volume integral of the charge density divided by the permittivity of free space over volume \(V\) enclosed by surface \(S\).

Step 4: Use of divergence theorem

By using the divergence theorem, the left-hand side of the equation, a surface integral, can be converted into a volume integral. Hence the equation becomes, \(\iiint_{V} \nabla \cdot \mathbf{E} dV=\iiint_{V} \frac{\rho}{\epsilon_{0}} dV\). The divergence theorem states that the surface integral of a vector field through a closed surface is equal to volume integral of divergence of that vector field over the volume enclosed by the surface.

Step 5: Derivation of the differential form of Gauss's Law

Since the volume \(V\) in the above equation is arbitrary, which means that it holds true for any volume, the integrands of both sides should be equal. Hence, equating the integrands yields the differential form of Gauss's Law, \(\nabla \cdot \mathbf{E}=\frac{\rho}{\epsilon_{0}}\). This law states that the divergence of the electric field at a point is equal to the charge density at that point divided by the permittivity of free space.

Key Concepts

Electric Field

An electric field is a fundamental concept in physics that describes the force exerted by electric charges. Visualize it as an invisible force field that surrounds charged particles, influencing other charges in its vicinity. The electric field is represented as \(\mathbf{E}\) and is a vector field, meaning it has both magnitude and direction.

Key characteristics of an electric field include:

• The electric field vector points away from positive charges and towards negative charges.
• Its strength is proportional to the amount of charge and inversely proportional to the square of the distance from the charge.
• It is measured in volts per meter (V/m).

Understanding the behavior of electric fields allows us to predict how charges will interact in various environments, which is crucial for applications in electrical engineering, electronics, and more.

Charge Density

Charge density, denoted by \(\rho\), is a measure of the amount of electric charge within a given volume. It expresses how much charge is packed into a unit volume of space and is typically measured in coulombs per cubic meter (C/m³).

There are two primary types of charge densities:

• **Volume Charge Density**: The amount of charge per unit volume in three-dimensional space.
• **Surface Charge Density**: The amount of charge per unit area, used when charges are distributed over a two-dimensional surface.

Charge density is integral to understanding how charges are distributed in space and is an essential factor in using Gauss's Law effectively. It helps to calculate total charge \(q\) enclosed within a given area or volume, which can then be used in various applications such as designing capacitors or analyzing electric fields.

Permittivity of Free Space

The permittivity of free space, denoted by \(\epsilon_{0}\), is a constant that characterizes the ability of the vacuum to permit the electric field. It is a fundamental physical constant used in electromagnetism that helps quantify how electric fields interact with the vacuum of space.

The constant has a value approximately equal to \(8.85 \times 10^{-12} \underline{\phantom{xxx}} \text{C}^2/\text{N}\cdot\text{m}^2\). It plays a crucial role in the formulas related to electric forces, such as Coulomb's Law and Gauss's Law.

• In Coulomb's Law, \(\epsilon_{0}\) appears in the denominator of the equation describing the force between two point charges.
• In Gauss's Law, it scales the relationship between electric flux through a closed surface and the enclosed charge.

Understanding \(\epsilon_{0}\) is essential for calculating electric fields and forces in a vacuum, having impacts on everything from atomic scale interactions to designing electronic components.

Divergence Theorem

The divergence theorem is a powerful tool in vector calculus, relating the flux of a vector field through a closed surface to the divergence of that vector field inside the volume. It is a key mathematical concept used to bridge surface integrals to volume integrals.

In the context of electromagnetism, the divergence theorem allows us to rewrite Gauss's Law from its integral form to its differential form. Here’s a simple way to think about it:

• The theorem states that the total "outflow" of a vector field from a closed surface is equal to the sum of all source strengths within that surface.
• This is useful for converting surface integrals into volume integrals, simplifying calculations.

By using the divergence theorem, mathematicians and physicists can see how individual parts of a volume contribute to the behavior noted on its surface. This ability to convert and simplify integrals is essential in solving complex electrostatic problems.