Gauss's Laws

Electric field variation for a uniformly charged sphere of radius R and total charge Q:

OUTSIDE the sphere (r > R):

• $E = \frac{Q}{4\pi\epsilon_0 r^2}$
• Field decreases as $\frac{1}{r^2}$ (inverse square law)
• Identical to a point charge at the center

INSIDE the sphere (r < R):

• $E = \frac{Qr}{4\pi\epsilon_0 R^3}$
• Field increases linearly with r
• Zero at the center (r = 0)

AT the boundary (r = R):

• Both expressions give the same value: $E = \frac{Q}{4\pi\epsilon_0 R^2}$
• The field is continuous across the boundary
• The rate of change of the field (the gradient) is discontinuous

This behavior reflects how the enclosed charge increases with the cube of radius inside the sphere, while outside the sphere all charge is enclosed.

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.

This charge redistribution is necessary because:

1. Gauss's Law requires the electric field inside the conducting material to be zero
2. For the field to be zero, the net charge enclosed by any Gaussian surface drawn inside the conductor must be zero
3. When a positive charge is placed in the cavity, free electrons in the conductor are attracted to the inner surface, creating a negative charge layer (-q)
4. This leaves a deficiency of electrons (positive charge +q) on the outer surface
5. This arrangement ensures the total charge in any volume completely inside the conductor remains zero

This principle underlies Faraday cages and electromagnetic shielding technologies.

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 at the center equals zero due to perfect symmetry:

• Charge is distributed uniformly in all directions
• For any small element of charge, there exists another element on the opposite side
• The electric field contributions from these opposite charge elements exactly cancel
• This perfect cancellation occurs in all directions (spherical symmetry)
• Mathematically: the net solid angle subtended at the center is 4π, resulting in zero net field

This is analogous to gravitational fields, where at the center of a uniform spherical mass, the gravitational field is also zero.

The electric field of a uniform plane sheet remains constant with distance because:

• The sheet can be modeled as an infinite collection of point charges
• As you move away, you see more of the sheet (larger solid angle)
• This exactly compensates for the 1/r² decrease from each point charge
• Mathematically: E = σ/(2ε₀) in all space outside the sheet

Comparison with other charge distributions:

• Point charge: E ∝ 1/r² (decreases rapidly with distance)
• Line charge: E ∝ 1/r (decreases more slowly)
• Plane sheet: E = constant (no distance dependence)
• Spherical shell: E = 0 inside, E ∝ 1/r² outside

This constant field property makes parallel plate capacitors useful in creating uniform electric fields and applies only when the sheet dimensions are much larger than the distance from the sheet.

Note: This property breaks down near the edges of a finite sheet, where the field begins to curve and decrease in magnitude.

When two parallel conducting surfaces carry opposite charge densities:

• Between the surfaces: E = σ/ε₀ (field is strengthened) • This occurs because each surface contributes E = σ/(2ε₀) in the same direction • Both fields add constructively: σ/(2ε₀) + σ/(2ε₀) = σ/ε₀

• Outside both surfaces: E = 0 • The equal but opposite charges create fields that exactly cancel • This creates a field-free region outside the pair of oppositely charged plates

This principle is the foundation for parallel plate capacitors, where the electric field is confined primarily to the region between the plates.

The electric field is proportional to the distance from the center:

• E increases linearly as one moves from center toward surface
• E = 0 at the center
• E = (Q/4πε₀)(r/R³) for r < R
• This linear increase contrasts with the external field which follows an inverse square law

This occurs because at radius r, only the charge contained within a sphere of radius r contributes to the field at that point.

For a charged sphere, the electric potential energy:

1. Physical meaning: Work required to assemble the charge distribution from infinity
2. For uniformly charged solid sphere: U = 3Q²/(20πε₀R)
3. For thin spherical shell: U = Q²/(8πε₀R)

Scaling properties:

• U ∝ Q² (quadratic relationship with charge)
• U ∝ 1/R (inverse relationship with radius)
• Doubling charge quadruples energy
• Doubling radius halves energy
• Energy density scales as Q²/R⁴

Physical explanation:

• Higher charge means stronger repulsive forces to overcome during assembly
• Larger radius means charges are further apart, reducing repulsive forces
• Potential energy is stored in the electric field, with energy density proportional to E²

Purpose and mechanism of earthing (grounding):

1. Purpose:

   • Maintains the potential of a conductor at zero
   • Provides safety by preventing charge accumulation on appliance bodies
   • Creates a low-resistance path to earth for fault currents
   • Protects users from electric shock

2. Mechanism:

   • A thick metal plate is buried deep in the earth
   • Earth wire connects appliance metal bodies to this plate
   • Ensures appliance casings remain at zero potential
   • If a live wire accidentally touches the metal body, current flows to earth rather than through a person

3. Implementation:

   • Modern electrical systems use three wires: live, neutral, and earth
   • Earth wire connects to metallic bodies of appliances
   • Earth wire provides low-resistance path for fault currents

This safety system works because the earth can absorb or supply charges without appreciable change in potential due to its large size.