Gauss's Laws

Necessary conditions for practical application:

• High symmetry in charge distribution
• Ability to find a Gaussian surface where $\vec{E}$ is either constant in magnitude or zero
• Electric field direction must be known from symmetry arguments

Useful symmetries:

1. Spherical symmetry: For point charges, spherical shells, or uniform spheres

   • Use spherical Gaussian surface
   • Field is radial and constant in magnitude at fixed radius

2. Cylindrical symmetry: For line charges or cylindrical charge distributions

   • Use cylindrical Gaussian surface
   • Field is perpendicular to axis and constant at fixed radius

3. Planar symmetry: For infinite plane charge distributions

   • Use "pill box" Gaussian surface
   • Field is perpendicular to plane and independent of distance

When these symmetries aren't present, direct integration of Coulomb's Law is usually more practical.

Proof using Gauss's Law:

1. In electrostatic equilibrium, free charges in a conductor can redistribute themselves
2. Draw a Gaussian surface entirely within the conductor
3. If $\vec{E} \neq 0$ anywhere inside, charges would move (contradicting equilibrium)
4. Therefore $\vec{E} = 0$ everywhere inside the conductor
5. From Gauss's Law: $\oint \vec{E}\cdot d\vec{S} = \frac{q_{enclosed}}{\varepsilon_0} = 0$
6. Since $\vec{E} = 0$ everywhere on the Gaussian surface, $q_{enclosed} = 0$

Consequences:

• Any excess charge on a conductor must reside entirely on its surface
• The interior of a conductor is electrically neutral
• Cavities within conductors remain field-free unless charges are placed inside them
• The field just outside a conductor must be perpendicular to the surface

This property enables Faraday cages and electromagnetic shielding.

An area vector $\Delta\vec{S}$ is:

• A vector quantity with magnitude equal to the area of a surface element
• Direction perpendicular (normal) to the surface element
• Used to simplify flux calculations using dot product notation

Properties:

• For a closed surface, area vectors conventionally point outward
• The dot product $\vec{E} \cdot \Delta\vec{S}$ automatically accounts for the angle between field and surface
• Mathematically equivalent to $E \Delta S \cos\theta$

This vector formulation allows for elegant expression of flux through any surface as: $\Phi = \oint \vec{E} \cdot d\vec{S}$

The area vector concept applies to all field calculations in physics, including electric fields, magnetic fields, and fluid flow.

Moving the charge away from the center doesn't change the total flux through the closed surface, but it fundamentally alters the flux distribution:

• For a point charge q at any position inside a closed surface:
  • The total flux remains exactly Φ = q/ε₀ (Gauss's Law)
  • The electric field is no longer uniformly distributed across the surface
  • The solid angle subtended by each surface element remains critical

Mathematical analysis:

1. The flux through an infinitesimal area element becomes: dΦ = (q/(4πε₀r²))·dS·cosθ where r now varies for different surface points and θ is the angle between the field and surface normal

2. Although the field strength and angles vary across the surface, when integrated: Φ = ∮E·dS = ∮(q/(4πε₀r²))·cosθ·dS = (q/4πε₀)·∮(cosθ/r²)·dS = (q/4πε₀)·4π = q/ε₀

This represents a profound insight: the solid angle subtended by the entire closed surface from any internal point is always 4π steradians, regardless of the charge position within the surface.

In generalizing from plane angles to solid angles:

• For plane angles:

  • A line segment Δl subtends angle Δθ = (Δl·cosα)/r
  • Units: radians (dimensionless)
  • A complete circle subtends 2π radians

• For solid angles:

  • A surface area element ΔS subtends solid angle ΔΩ = (ΔS·cosα)/r²
  • Units: steradians (sr), also dimensionless
  • A complete sphere subtends 4π steradians

• The mathematical pattern is preserved:

  • In 2D: angle = (projected length)/(radius)
  • In 3D: solid angle = (projected area)/(radius²)

• This relationship is fundamental in electric flux calculations, where flux Δϕ through a small area ΔS equals E·ΔS·cosθ, which can be rewritten as E·r²·ΔΩ

Inside a charged conductor in electrostatic equilibrium:

• The electric field is exactly zero at all points
• This occurs because free electrons redistribute within the conductor
• If any field existed inside, it would cause further movement of charges until equilibrium is reached
• This principle is a direct consequence of Gauss's Law when applied to a closed surface inside the conductor

This principle means:

• Any excess charge must reside entirely on the conductor's surface
• The interior of a conductor remains electrically neutral regardless of external fields
• This property is the basis for electrostatic shielding (Faraday cages)

Note: This is true only in electrostatic equilibrium, not during charge movement.

To find the electric field around an infinite line charge using Gauss's Law:

1. Choose a cylindrical Gaussian surface of radius r and length l that is coaxial with the line charge
2. The electric field points radially outward (for positive charge) and has the same magnitude at all points at distance r
3. Only the curved surface contributes to the flux (E is parallel to end caps)
4. Apply Gauss's Law: ∮E⃗·dS⃗ = q_enclosed/ε₀ E(2πrl) = λl/ε₀
5. Solve for E: E = λ/(2πε₀r)

This result shows the electric field decreases as 1/r (not 1/r²), unlike point charges.

The electric field strength dependencies reveal fundamental differences in geometric charge distributions:

For a line charge (λ):

• E = λ/(2πε₀r)
• Field decreases as 1/r
• Electric field lines spread out in 2D (cylindrical symmetry)
• At distance r, field lines pass through cylindrical surface area 2πrl
• Flux is distributed over an area that grows linearly with r

For a point charge (q):

• E = q/(4πε₀r²)
• Field decreases as 1/r²
• Electric field lines spread out in 3D (spherical symmetry)
• At distance r, field lines pass through spherical surface area 4πr²
• Flux is distributed over an area that grows as r²

This difference explains why electric fields from power lines remain significant at larger distances than fields from point charges of similar magnitude.

A cylindrical Gaussian surface is optimal for analyzing an infinite line charge because:

1. Flux contribution analysis:

   • Curved cylindrical surface: E is perpendicular to surface and constant magnitude, so E⃗·dS⃗ = E·dS
   • End caps: E is perpendicular to the normal vector (parallel to caps), so E⃗·dS⃗ = 0

2. Mathematical advantages:

   • Symmetry matches the physical system (cylindrical symmetry)
   • At fixed radius r, E has constant magnitude everywhere
   • Simplifies the surface integral calculation

3. Enclosed charge calculation:

   • For length l of cylinder: q_enclosed = λl
   • Directly relates to linear charge density

The flux therefore comes entirely from the curved surface: Φ = E(2πrl), allowing direct solution for E using Gauss's Law.

Application of Gauss's Law:

1. Draw a Gaussian sphere of radius r < R concentric with the charged sphere
2. Calculate enclosed charge: $q_{enclosed} = Q\frac{r^3}{R^3}$ (since charge density is uniform, enclosed charge ∝ volume ratio)
3. Apply Gauss's Law: $\oint \vec{E}\cdot d\vec{S} = \frac{q_{enclosed}}{\epsilon_0}$
4. By symmetry, E has same magnitude at all points on Gaussian surface and is radial
5. Therefore: $E(4\pi r^2) = \frac{Q\frac{r^3}{R^3}}{\epsilon_0}$
6. Solving for E: $E = \frac{Qr}{4\pi\epsilon_0R^3}$

This gives the linear relationship between field strength and distance inside the sphere, with zero field at center and maximum field at the surface.