Gauss's Law - Comprehensive Notes

Introduction

• Gauss’s law relates the total electric flux through a closed surface to the total enclosed charge divided by \varepsilon_0.
• It is not new information beyond the known results for a point charge and the superposition principle; it simply repackages them in a useful form.
• Two main applications:
  • Use Gauss’s law plus symmetry to find electric fields for highly symmetric charge distributions.
  • Understand charges and electric fields inside conductors.
• Key terminology:
  • Gaussian surface: any closed surface chosen for applying Gauss’s law (not necessarily a physical surface).
  • dA⃗: infinitesimal vector area on the Gaussian surface, pointing outward.
  • Flux through a surface: measure of how much of the electric field passes through the surface.

2 Electric Flux

• Electric flux through a surface quantifies how much of the electric field passes through that surface.
• Built from the special case of a uniform field and a flat surface, extended to general surfaces via integration.

2.1 Definition for a Uniform Field and Flat Surface

• Consider a uniform electric field \mathbf{E} and a flat surface of area A.
• Requirements for the flux definition:
  • Flux is proportional to both the magnitude of the field and the surface area: doubling E or A doubles the flux.
  • Only the component of the field perpendicular to the surface contributes to flux (parallel component does not pass through).
• Simple definition for the flat surface: \Phi_E = A E cos(\theta). \quad (1)
• Vector form: \Phi_E = \mathbf{E} \cdot \mathbf{A}, \quad (2) where \mathbf{A} = A \hat{n}
• Interpretation of orientation:
  • Flux is maximal positive when E⃗ and A⃗ are parallel (E perpendicular to the surface).
  • Flux is zero when E⃗ is perpendicular to A⃗ (E lies in the surface).
  • Flux is maximal negative when E⃗ and A⃗ are antiparallel.

2.2 General Definition of Electric Flux

• For nonuniform fields and curved surfaces, E⃗ and n̂ can vary across the surface.
• Divide the surface into infinitesimal patches with area dA⃗; on each patch, the field is approximately constant, so the infinitesimal flux is:
• \dΦ_E = \mathbf{E} \cdot d\mathbf{A}.
• The total flux through the surface is the sum (integral):
• \PhiE = \int dΦE = \int \mathbf{E} \cdot d\mathbf{A}. \quad (3)
• This is the general definition of electric flux.

3 Gauss’s Law

• Gauss’s law states:
• The total electric flux through a closed surface equals the total enclosed charge divided by \varepsilon_0:
• \ΦE = Q{enc} / \varepsilon_0, \quad (4)
• with the total flux defined as \Φ_E = \oint \mathbf{E} \cdot d\mathbf{A}. \quad (5)
• Consequences and clarifications:
  • The surface in Gauss’s law is a Gaussian surface and can be chosen arbitrarily; it does not have to correspond to a physical object.
  • Gauss’s law is a mathematical theorem; it is always true. Its usefulness depends on the symmetry of the problem.
  • The notation ∮ indicates a closed-surface integral; the outward normal convention makes dA⃗ point outward. Regions where the field points outward contribute positively to the flux.
  • Charges outside the surface do not contribute to the total flux through the surface (they create fields, but the net flux due to those fields through the closed surface sums to zero).
  • The total flux Φ_E is determined entirely by the enclosed charge; the shape and size of the surface are irrelevant for a given enclosed charge distribution.

4 Application 1: Electric Field Calculations

• Gauss’s law, combined with symmetry, lets us find E for highly symmetric charge distributions.
• Three symmetry classes:
  • Spherical symmetry: the charge distribution has the same symmetry as a sphere (e.g., uniformly charged spheres, spherical shells, point charges, and combinations).
  • Cylindrical symmetry: symmetry of an infinitely long cylinder (e.g., uniformly charged solid or hollow cylinders, thin rods).
  • Planar symmetry: symmetry of an infinite plane (e.g., uniformly charged sheets and slabs).
• Core idea: symmetry constrains the possible variations of the field, making the field simple enough to relate to enclosed charge via a suitable Gaussian surface.
• Focus here: Spherically Symmetric Charge Distributions (SSCDs).

4.2 Spherically Symmetric Charge Distributions (SSCDs)

• Step 1: Symmetry implications for the field at a point ⃗r from the center:
  • Due to symmetry, the field must point radially (either along \hat{r} or -\hat{r}) and its magnitude depends only on r = |⃗r|.
  • Therefore, the field has the form \mathbf{E}(\mathbf{r}) = E_r(r) \hat{r}. \quad (6)
• Step 2: Introduce a spherical Gaussian surface of radius r concentric with the SSCD and relate flux to Er(r):
  • Each patch on the Gaussian surface has d\mathbf{A}⃗ parallel to \hat{r}, so d\mathbf{A}⃗ = dA \hat{r}.
  • Infinitesimal flux through a patch: dΦE = \mathbf{E} \cdot d\mathbf{A} = (Er(r) \hat{r}) \cdot (dA \hat{r}) = E_r(r) dA. \quad (7)
  • The flux through the sphere is independent of patch location, and the total flux is
  • \ΦE(r) = \oint \mathbf{E} \cdot d\mathbf{A} = \oint Er(r) dA = Er(r) \oint dA = Er(r) 4 \pi r^2. \quad (8)
• Step 3: Use Gauss’s law to relate Er(r) to the enclosed charge Q_enc(r):
  • \ΦE(r) = Q{enc}(r) / \varepsilon_0.
  • Therefore, \mathbf{E}r(r) = Er(r) = \frac{1}{4 \pi \varepsilon0} \frac{Q{enc}(r)}{r^2}. \quad (9)
• Result: The electric field for any SSCD is
  • \mathbf{E}(\mathbf{r}) = Er(r) \hat{r}, with \; Er(r) = \frac{1}{4 \pi \varepsilon0} \frac{Q{enc}(r)}{r^2}.
• Interpretation:
  • This resembles the field of a point charge, but the enclosed charge is a function of r, i.e., Q_enc(r) depends on how much charge lies within radius r.
  • The specific distribution is encoded in Q_enc(r) and determines the radial field via the above relation.

5 Application 2: Charges and Electric Fields in Conductors

• Conductors contain many mobile charge carriers (e.g., conduction electrons);

  • These charges rearrange in response to internal forces and external fields.

• Electrostatic equilibrium: all charges are stationary within the conductor.

• When disturbed from equilibrium, free charges rearrange quickly to restore equilibrium (time scale ~ 10^{-16} s).

• Four basic facts about conductors in electrostatic equilibrium:

  • Fact 1: Inside a conductor (in the conducting material) the electric field is zero: \mathbf{E} = 0. Proof: a nonzero field would exert forces on charges causing movement until the field vanishes.
  • This holds even when an external field is present, as charges rearrange on the surface to cancel the interior field (Fig. 7).
  • Fact 2: Any excess charge resides on the surface of the conductor.
  • Inside the material, E⃗ = 0, so the flux through any Gaussian surface entirely inside is zero, implying the enclosed charge is zero; hence excess charge must be on the surface (Fig. 8(a)).
  • Fact 3: There is no charge on the surface of an empty cavity inside a conductor.
  • Since E⃗ = 0 in the conducting material, the flux through a surface enclosing the cavity is zero, so the total charge on the surface of the cavity is zero (Fig. 8(b)); more advanced analyses show there can be no charge on the cavity surface at all.
  • Fact 4: If there is a charge q in a cavity inside the conductor, it will be screened by an opposite charge on the cavity surface.
  • E = 0 inside the conductor; the flux through any surface enclosing the cavity is zero, so a surface charge must be present on the cavity boundary to neutralize the field from the cavity charge (Fig. 8(c)).

• Practical takeaway: Conductors in electrostatic equilibrium have all excess charges on the outer surface (and possibly on cavity surfaces to shield interior regions), with zero field inside the conducting material.