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Chapter 1: Problem 2

(a) Show that Gauss's law follows from Maxwell's equation $$ \nabla \cdot \mathbf{E}=\frac{\rho}{\varepsilon_{0}} $$ Here, \(\rho\) is the usual charge density. (b) Assuming that the electric field of a point charge \(q\) is spherically symmetric, show that Gauss's law implies the Coulomb inverse square expression $$ \mathbf{E}=\frac{q \hat{\mathbf{r}}}{4 \pi \varepsilon_{0} r^{2}} $$

Short Answer

Expert verified
Gauss's law follows Maxwell's equation as it restates the divergence form; it implies Coulomb's law from spherical symmetry.

Step by step solution

01

Understanding Gauss's Law

Gauss's Law for electricity states that the electric flux through a closed surface is proportional to the charge enclosed. Mathematically, it is expressed as \( \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enclosed}}}{\varepsilon_{0}} \).
02

Relating Gauss's Law to the Divergence of Electric Field

Gauss's Law, when expressed in differential form, states that the divergence of the electric field is equal to the charge density divided by \(\varepsilon_{0}\), which is \(abla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_{0}}\). This is exactly Maxwell's equation for the divergence of the electric field, showing that Gauss's Law is a restatement of this Maxwell's equation.
03

Considering a Point Charge's Electric Field

For a point charge \(q\), the electric field is spherically symmetric. This means the electric field depends only on the radial distance \(r\) from the charge. Using Gauss's law for a spherical surface of radius \(r\), the symmetry implies the electric field is directed radially.
04

Applying Gauss's Law to the Spherical Surface

Using the spherical symmetry and Gauss's law, the electric flux \( \oint \mathbf{E} \cdot d\mathbf{A} = E \cdot 4\pi r^2 \) equates to \( \frac{q}{\varepsilon_{0}} \), because the integral of the electric field over a sphere results in the electric field magnitude \(E\) times the surface area \(4\pi r^2\).
05

Solving for the Electric Field

Based on the equation \( E \cdot 4\pi r^2 = \frac{q}{\varepsilon_{0}} \), solving for \(E\) gives \( E = \frac{q}{4\pi \varepsilon_{0} r^2} \), confirming that the electric field of a point charge follows an inverse square law and agreeing with Coulomb's Law expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gauss's Law
Gauss's Law helps us comprehend how electric fields originate from electric charges. In simple terms, it tells us that the total electric field passing through a closed surface depends on the charge inside that surface.
This is expressed mathematically as:
  • \( \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enclosed}}}{\varepsilon_{0}} \)
Here, \(\mathbf{E}\) represents the electric field, \(d\mathbf{A}\) is a small area element on the surface, \(Q_{\text{enclosed}}\) is the total charge within the surface, and \(\varepsilon_{0}\) is the permittivity of free space.
Gauss's Law gives us a simple way to calculate electric fields if symmetry can be exploited.
Maxwell's Equations
Maxwell's Equations are a set of four mathematical equations that form the foundation of electromagnetism. They describe how electric and magnetic fields interact with each other and with charges and currents.
One of Maxwell鈥檚 equations, specifically related to electricity, is:
  • \( abla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_{0}} \)
This mirrors Gauss's Law in a differential form, equating the divergence of the electric field \(abla \cdot \mathbf{E}\) with the charge density \(\rho\) divided by the permittivity of free space \(\varepsilon_{0}\). This tells us that charges create electric fields.
Coulomb's Law
Coulomb鈥檚 Law provides us with the basic principle about the force between two point charges. It states that the force is directly proportional to the product of the absolute charges and inversely proportional to the square of the distance between them.
Mathematically, it is formulated as:
  • \( \mathbf{F} = k \frac{|q_1 q_2|}{r^2} \hat{\mathbf{r}} \)
where \(k\) is Coulomb's constant, \(q_1\) and \(q_2\) are the magnitudes of the charges, \(r\) is the distance between the charges, and \(\hat{\mathbf{r}}\) is the unit vector pointing from one charge to the other.
Here, Gauss's Law helps derive the electric field of a single point charge, confirming the inverse square nature of Coulomb's Law.
Electric Field
The electric field is a concept in physics that describes how a charge affects the space around it. It is represented by the symbol \(\mathbf{E}\) and describes the force a charge would experience in the presence of other charges.
For a point charge \(q\), the electric field at a distance \(r\) is:
  • \( \mathbf{E} = \frac{q \hat{\mathbf{r}}}{4\pi \varepsilon_{0} r^2} \)
This means the field decreases with the square of the distance, echoing the inverse square law found in Coulomb's formulation.
The magnitude of the electric field helps determine how strongly a charged particle will be acted upon by another charge.
Charge Density
Charge density, represented by \(\rho\), is the measure of electric charge per unit volume. It essentially tells us how much charge is packed into a certain area or volume.
In the context of electric fields, charge density helps determine how the electric field diverges from a region of space. The divergence of the electric field, indicated by \(abla \cdot \mathbf{E}\), depends directly on \(\rho\) as per Maxwell's equation:
  • \( abla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_{0}} \)
A higher charge density means a stronger electric field around the region. Understanding charge density is crucial when calculating electric fields in a volume rather than just from point charges.

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Most popular questions from this chapter

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