Gauss's Laws

Derivation outline:

1. For internal point charge $q$: flux through small area $\Delta S$ is $\Delta\Phi = \frac{q}{4\pi\varepsilon_0 r^2}\Delta S\cos\alpha = \frac{q}{4\pi\varepsilon_0}\Delta\Omega$
2. Total flux = $\frac{q}{4\pi\varepsilon_0}\sum\Delta\Omega = \frac{q}{4\pi\varepsilon_0}(4\pi) = \frac{q}{\varepsilon_0}$
3. For external charge: total solid angle subtended is zero, so flux = 0
4. For multiple charges: superposition principle gives $\Phi = \frac{q_{total}}{\varepsilon_0}$

Relationship:

• Gauss's Law is a consequence of Coulomb's Law and the inverse-square nature of electric fields
• Conceptually, Gauss's Law is a restatement of Coulomb's Law in terms of field flux
• Historically, Coulomb's Law was discovered first, but Gauss's Law is more fundamental
• Gauss's Law applies to continuous charge distributions where direct application of Coulomb's Law would require integration

Charge distribution:

On the inner surface: $-q$ (induced) On the outer surface: $+q$ (if conductor is neutral overall)

Reasoning from Gauss's Law:

1. The electric field inside the conducting material must be zero in electrostatics
2. Draw a Gaussian surface inside the conducting material surrounding the cavity
3. Since $\vec{E} = 0$ on this surface, $\oint \vec{E}\cdot d\vec{S} = 0$
4. Therefore $q_{enclosed} = 0$, meaning the total charge inside must be zero
5. Since there's already +q in the cavity, the inner surface must have -q
6. By charge conservation, if the conductor was initially neutral, the outer surface must have +q

This induced charge on the inner surface perfectly shields the rest of the conductor from the field of the cavity charge.

Field and charge behavior:

1. Inside conducting material: $E = 0$ everywhere

2. Charge distributions:

   • On inner surface: Induced charge of $-q$, non-uniformly distributed (higher density closer to point charge)
   • On outer surface: Charge $+q$ distributed uniformly (if conductor was initially neutral)

3. Electric field:

   • In cavity: Non-uniform field from both point charge and non-uniform induced charge
   • Outside shell: Uniform spherical field identical to that of a point charge $+q$ at the center of the shell

Analysis with Gauss's Law:

• Any Gaussian surface drawn inside conductor material shows $\oint \vec{E}\cdot d\vec{S} = 0$, confirming zero enclosed net charge
• A Gaussian sphere outside the shell gives $\oint \vec{E}\cdot d\vec{S} = \frac{q}{\varepsilon_0}$, proving the external field behaves as if the charge were at the center

This demonstrates the shielding property of conductors and the method of image charges in electrostatics.

The solid angle Ω is related to the area S subtended on a sphere of radius r by:

$\Omega = \frac{S}{r^2}$

Where:

• Ω is measured in steradians (sr)
• S is the area on the sphere intercepted by the cone
• r is the radius of the sphere

This is analogous to how a plane angle θ relates to arc length l by: θ = l/r

Properties:

• A complete sphere subtends a solid angle of 4π steradians
• A closed surface subtends a solid angle of 4π sr at any internal point
• A closed surface subtends zero solid angle at any external point

The distance becomes irrelevant because:

• For a point charge q at the center of a sphere of radius r:
  • The electric field magnitude at the surface is E = q/(4πε₀r²)
  • The surface area of the sphere is A = 4πr²
  • The total flux is: Φ = E·A = (q/(4πε₀r²))·(4πr²) = q/ε₀

The r² terms cancel out because:

1. The electric field strength decreases with the square of distance (r²)
2. But the surface area increases with the square of distance (r²)

This mathematical cancellation reveals a fundamental property of Gauss's Law - the total electric flux through any closed surface depends only on the enclosed charge, not on the size or shape of the surface. This is what makes Gauss's Law such a powerful tool for calculating electric fields in symmetric charge distributions.

The relationship between solid angle (ΔΩ) and plane angle (Δθ) demonstrates the dimensional extension from 2D to 3D:

For plane angle: $\Delta\theta = \frac{\Delta l \cos\alpha}{r}$

• Measured in radians
• Ratio of arc length (adjusted by cosα) to radius
• One-dimensional measure (length/radius)

For solid angle: $\Delta\Omega = \frac{\Delta S \cos\alpha}{r^2}$

• Measured in steradians
• Ratio of area (adjusted by cosα) to radius squared
• Two-dimensional measure (area/radius²)

This shows how solid angle generalizes the concept of angle from arcs in a plane to areas on a sphere, with the denominator increasing from r to r² to account for the additional dimension.

When a small line segment Δl is perpendicular to the radial line from point O:

• The angle subtended is given by: Δθ = Δl/r
• Where r is the distance from point O to the line segment
• This is the simplest case where the full length of the segment contributes to the subtended angle
• Example: A 1 cm segment at 10 cm distance perpendicular to the radial line subtends an angle of 0.1 radians (≈ 5.7°)

The Gaussian surface is constructed as a cylinder with:

• Flat circular ends (cross-sections A and A') of equal area ΔS
• Both ends positioned at equal distances from the charged sheet
• The cylinder passes through the sheet perpendicular to its surface
• The curved surface of the cylinder is parallel to the electric field

This construction works because:

• By symmetry, the electric field must be perpendicular to the sheet
• The field has equal magnitude at equal distances from both sides
• No flux passes through the curved cylindrical surface (E is parallel to it)
• The total enclosed charge is σ·ΔS (where σ is surface charge density)

When applying Gauss's law to this surface, we get E = σ/(2ε₀), showing the field is uniform and independent of distance from the sheet.

The difference occurs because:

• For an isolated sheet of charge (non-conducting), the electric field extends symmetrically on both sides with magnitude E = σ/2ε₀ in each direction
• For a conducting surface with charge density σ, the field must be zero inside the conductor
• To maintain zero field inside, the charges within the conductor redistribute, creating an effective "image charge" effect
• This redistribution produces twice the field outside (E = σ/ε₀) compared to the isolated sheet case
• The conducting material enforces boundary conditions that all the field lines must exit perpendicular to the surface
• The conductor's interior remains at equipotential despite the surface charge

This demonstrates how material properties fundamentally alter field distributions.

Electric Field of a Uniformly Charged Sphere:

1. Mathematical Expressions:

   • Outside the sphere (r > R): $E = \frac{Q}{4\pi\varepsilon_0 r^2}$
   • Inside the sphere (r < R): $E = \frac{Qr}{4\pi\varepsilon_0 R^3}$

2. Physical Significance:

   • Outside: Behaves exactly like a point charge at the center
   • Inside: Increases linearly with distance from center (proportional to r)
   • At r = 0: E = 0 due to symmetrical cancellation from all directions
   • At r = R (surface): Both expressions yield identical values, ensuring field continuity

3. Application of Gauss's Law:

   • For points outside, only the total charge Q matters, not its distribution
   • For points inside, only charge enclosed within radius r contributes to the field
   • Spherical shells of charge outside a given radius produce zero net field inside (shell theorem)

4. Conceptual Understanding:

   • The linear increase inside shows that outer shells of charge have no effect on inner points
   • The identical behavior to a point charge outside demonstrates the principle of effective charge location
   • This behavior is a direct consequence of the 1/r² nature of Coulomb's Law