Untitled Quiz

Study Notes

A type of integration applied on a closed surface.
Its value equals the area of the Gaussian surface.
Example formulas for different shapes:
- Sphere: ∫ dA = 4πr²
- Side Area of Cylinder: ∫ dA = 2πrh
- Circle: ∫ dA = πr²

A vector quantity with magnitude equal to the area and direction normal to the surface.
Its direction is outward from the Gaussian surface.

The total number of electric field lines passing through a surface.
It is proportional to the magnitude of the enclosed charge.
𝚽 = 1/ε₀ * ∑ Q, where ε₀ is the permittivity of free space.
Measured in units of N.m²/C.
Formula: 𝚽 = E . A = E A cos θ, where:
- A is the area vector normal to the surface.
- θ is the angle between E and A.

𝚽 = (∑ Q) / ε₀ = ∫ E ⋅ dA
Gauss's law states that the electric flux through any closed surface is proportional to the enclosed electric charge.

Steps to solve problems:
1. Identify the shape of the charge distribution.
  - Point charge or charged sphere (Q)
  - Charged line or charged cylinder (Q = λl)
  - Charged surface (plate, plane, sheet) (Q = σA)
2. Choose a Gaussian surface that encloses the charge distribution.
3. Apply Gauss's law: (∑ Q) / ε₀ = ∫ E ⋅ dA.
4. Calculate the electric field (E).

To calculate the electric field at a distance (r) from a point charge (q):
1. Choose a spherical Gaussian surface of radius r.
2. Apply Gauss's Law: q / ε₀ = ∫ E ⋅ dA = E ⋅ 4πr²
3. Solve for E: E = q / (4πε₀r²) = kq/r², where k = 1/(4πε₀).

Cylindrical Symmetry:
- Electric field at a distance (r) from an infinite line of charge with a constant charge per unit length (λ).
- Use a cylindrical Gaussian surface of radius r and length L.
- Gauss's Law: λl / ε₀ = ∫ E ⋅ dA = E ⋅ 2πrl.
- Solve for E: E = λ / (2πε₀r) = 2kλ/r.
Planar Symmetry:
- Electric field at a distance (r) from an infinite sheet of charge with a constant charge per unit area (σ).
- Use a cylindrical Gaussian surface with its axis perpendicular to the sheet.
- Gauss's Law: σA / ε₀ = ∫ E ⋅ dA = 2EA.
- Solve for E: E = σ / (2ε₀).
Spherical Symmetry:
- Electric field at a distance (r) from a uniformly charged sphere of radius R.
- Two cases:
  - r < R: E = (kQr) / R³, where Q is the total charge of the sphere.
  - r > R: E = (kQ) / r², same as a point charge.