EM fields, lecture 14 - Applications of Gauss' law and symmetrry /geometry

Overview

Tim Gorringe applies Gauss's law to uniformly charged spheres, infinitely long cylinders, and infinite slabs of finite thickness. Each solution combines charge-distribution symmetry with a matching Gaussian surface to reduce the flux integral, deriving the fields inside and outside and showing how geometry determines their distance dependence.

Key takeaways

A useful Gaussian surface matches the symmetry of the charge distribution: a sphere for spherical symmetry, a concentric cylinder for cylindrical symmetry, and a pillbox for planar symmetry.
For a uniformly charged sphere, the exterior field is Q/(4πε₀r²), while the interior field is Qr/(4πε₀a³); the interior field is zero at the center and peaks at the surface.
For an infinite charged cylinder with linear charge density λ and radius a, the exterior field decreases as 1/ρ, while the interior field is λρ/(2πε₀a²).
For an infinite slab of thickness d and volume charge density ρ, the exterior field magnitude is ρd/(2ε₀), while inside the slab the signed field varies as E_z = ρz/ε₀.
The different exterior distance laws follow from how flux spreads through space: spherical flux crosses an area proportional to r², cylindrical flux crosses a sidewall proportional to ρ, and planar flux crosses constant-area end caps.
In each geometry, the interior field grows linearly with distance from the symmetry center or plane because the amount of enclosed charge grows with the Gaussian surface's radius or enclosed thickness.

Chapters

0:00 Gauss's Law: Relating Enclosed Charge to Electric Flux

The lecture works through three classic charge geometries: a sphere, an infinite cylinder, and a finite-thickness infinite slab.
Each calculation has two steps: use Gauss's law to relate flux to enclosed charge, then use symmetry to relate flux to the electric field.

2:39 Spherical Symmetry Determines the Field Direction

A uniformly charged sphere has spherical symmetry, so its electric field must point radially and depend only on the distance r from its center.
A concentric spherical Gaussian surface matches that symmetry and is used separately for points inside and outside the sphere.

9:51 Uniformly Charged Sphere: The Exterior Field

On a spherical Gaussian surface of radius r greater than the sphere radius a, the field is parallel to each outward area element and has constant magnitude.
The flux is E times the sphere area, 4πr², and the enclosed charge is the total charge Q.
The resulting field is E = Q/(4πε₀r²), identical to the field of a point charge Q at the sphere's center.

13:22 Uniformly Charged Sphere: The Interior Field

For a Gaussian sphere of radius r less than a, uniform volume charge density means the enclosed charge is Q(r³/a³).
Substituting that charge and the flux E(4πr²) into Gauss's law gives E = Qr/(4πε₀a³), so the field increases linearly with r.
The field is zero at the center by spherical symmetry and reaches its maximum at the sphere's surface.

18:49 Infinite Cylinder: Symmetry and Gaussian Surfaces

The uniformly charged infinite cylinder has cylindrical symmetry, so its field points radially from the axis and depends only on radial distance ρ.
Gorringe characterizes the cylinder's charge by linear density λ, measured in charge per unit length.
Concentric cylindrical Gaussian surfaces of radius ρ and length L are selected to evaluate the field inside and outside the charged cylinder.

26:06 Infinite Cylinder: Exterior Field from Sidewall Flux

For a Gaussian cylinder outside the charged cylinder, the radial field crosses the curved sidewall but runs parallel to the two end caps.
The sidewall area is 2πρL, while the enclosed charge is λL; the length L cancels from Gauss's law.
The exterior field is E = λ/(2πε₀ρ), decreasing as 1/ρ rather than the sphere's 1/r².

31:46 Infinite Cylinder: Interior Field and Enclosed Charge

Inside a uniformly charged cylinder of radius a, a Gaussian cylinder encloses the fraction ρ²/a² of the charge within its length L.
Using the sidewall flux E(2πρL) gives E = λρ/(2πε₀a²), which grows linearly with distance from the axis.
The field is zero on the axis and reaches its maximum at the charged cylinder's surface.

36:47 Infinite Slab: Planar Symmetry and Field Direction

The slab extends infinitely in x and y, has finite thickness d in z, and carries uniform volume charge density ρ.
Planar symmetry rules out x and y field components; the field points along z and depends only on z.
A cylindrical Gaussian surface centered on the slab has end caps perpendicular to z and sidewalls parallel to the field.

44:12 Infinite Slab: Exterior Field from Two End Caps

For an end-cap position beyond the slab, |z| > d/2, flux passes through both end caps, giving total flux 2EA; the sidewalls contribute zero.
The enclosed slab volume is Ad, so the enclosed charge is ρAd; canceling A gives an exterior field magnitude E = ρd/(2ε₀).
The exterior field is independent of distance from the slab and increases with both charge density and slab thickness.

48:19 Infinite Slab: Interior Field and Geometry-Based Results

Inside the slab, |z| < d/2, the Gaussian surface encloses a charged thickness 2|z|, replacing d in the enclosed-charge calculation.
Gauss's law gives the signed field E_z = ρz/ε₀, which is zero at the slab's center and grows linearly toward either surface.
The sphere, cylinder, and slab all have linearly increasing interior fields because enclosed charge grows with volume, cross-sectional area, or thickness, respectively.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, tim gorringe.