Ch.3 Gauss's Law

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Overview

Gauss's Law

• first of Maxwell's Equations
• electrix flux through a closed surface is proportional to the amount of charge enclosed within the surface
• powerful for charge distributions with high symmetry
• calculate electric flux through Gaussian surface
• electric flux is the amount of electric field passing through a surface
• shows several properties of conductors in electrostatic equilibrium:
  • any excess charge is on the surface
  • interior electric field is zero
  • external field is perpendicular to the surface

Symmetry

Recall:

1. field points away from positive charges, towards negative charges
2. electric field exerts a force on a charged particle

Suppose we have an charged cylinder with:

• infinite length
• unknown diameter
• unknown charge density
• positive charge
• cylindrically symmetric charge distribution
  • translation parallel to the cylinder axis (infinite length)
  • rotation around cylinder axis
  • reflections in any plane containing/perpendicular to the cylinder axis

Importantly: symmetry of the field must match the symmetry of the charge distribution
This means the field of a cylindrically symmetric infinite rod must radiate out

Three Fundamental Symmetries

• Planar Symmetry
  • Infinite Plane
  • Infinite Parallel-Plate Capacitor
  • field is perpendicular to the plane
• Cylindrical Symmetry
  • Infinite Cylinder
  • Coaxial Cylinders
  • field is radial toward/away from axis
• Spherical Symmetry
  • Concentric Spheres
  • field is radial toward/away from axis

Flux

Gaussian surface: a closed surface through which an electric field passes through

• electric field "flows" out of closed surface surrounding a region of positive charge; in to closed surface surrounding a region of negative charge
• electric field may flow through closed surface surrounding no net charge, but thers no net flow
• field pattern is simple if closed surface matches symmetry of charge distribution inside

electric flux: amount of electric field passing through a surface

• outward flux through closed surface around net positive charge
• inward flux through closed surface around net negative charge
• no net flux through closed surface around no net charge
  $\vec{E}\cdot\vec{A}=\|\vec{E}\|\|\vec{A}\|\cos\theta=\Phi=\int\vec{E}\cdot d\vec{A}\propto q_\text{in}$

Flux depends on the field $\vec{E}$ and the normal vector of the surface $\vec{A}=A\vec{n}$ ($\vec{n}$ is the principal normal vector). The angle between them is $\theta$
In a nonuniform field, the sum of fluxes through smaller surfaces equals the total flux.
$\Phi_e=\int_S\vec{E}\cdot d\vec{A}$
...and we're only doing things that evaluate to $EA$ or $0$

In a closed surface, the notation becomes
$\oint_S\vec{E}\cdot d\vec{A}$
Note: $d\vec{A}$ points outside by convention
The differential form is
$\nabla\cdot\vec{E}=\frac{\rho}{\epsilon_0}$ where $\rho$ is the volume charge density at a point.

Gauss's Law

• easier to find fields for continuous charges than Coulomb's Law
• is valid for any velocity (though Coulomb's Law is a close enough approximation for $v\ll c$)

Consider a point charge with a spherical surface around it. The field $E=q/4\pi\epsilon_0r^2$ is perpendicular at every point, thus the flux is
$\Phi_e=\oint\vec{E}\cdot d\vec{A}=EA_\text{sphere}=\frac{q}{4\pi\epsilon_0r^2}(4\pi r^2)=\frac{q}{\epsilon_0}$
The flux is dependent only on charge; not shape or radius. If the surface is not spherical, spherical approximations can be made to make tangent and perpendicular surfaces, so the flux through any closed surface around a point with charge $q$ is $q/\epsilon_0$

If the charge is outside the surface, any inward flux is perfectly canceled by outward flux, so the net flux is $0$.

If there are multiple charges, the principle of superposition applies
$\Phi_e=\Phi_1+\Phi_2+\cdots$
where each $\Phi_i$ is the flux due to charge $i$
If we let $Q_\text{in}$ be the total charge inside, then
$\Phi_e=\frac{Q_\text{in}}{\epsilon_0}$
This is Gauss's Law

• only applies to closed surfaces, called Gaussian surfaces
  • is an imaginary, mathematical surface
• can be used in combination with shapes identified by symmetry to find field

Electrostatic Equilibrium

Consider a conductor in electrostatic equilibrium. The field inside $\vec{E}_\text{in}$ must be $0$ or the charge carriers would be moving and violate electrostatic equilibrium.
Choose a Gaussian surface barely inside the conductor. Since field is $0$, flux is $0$, thus there must be no net charge inside. Thus all excess charges reside on the surface.

For equilibrium to be met with these surface charges, the resulting field right at the surface must be perpendicular to the surface. By Gauss's Law, we have the net flux being $AE_\text{surface}$, and total charge being $\eta A$$\implies$$\vec{E}_\text{surface}=\frac{\eta}{\epsilon_0}$ perpendicular to the surface
$\eta$ ussually depends on the shape of the conductor

Similarly, because the interior flux is zero, any interior surfaces or holes have no net field, and no net charge. We can use this to create regions of no electric field in a parallel-plate capacitor by simply adding a neutral box surrounding the region we want to exclude, called screening.
Solid metal walls is ideal, but wire screen/mesh is sufficient, a Faraday cage- though the result is a complex exterior field.

An interior charge attracts the opposite charge to the inner surface to maintain the neutral field. Conservation of charge then requires the opposite of the attracted charge (equal to the interior charge) added, which goes to the exterior surface.