EM fields lecture 19 - field, potential & charge densities of conductor in electric fields

Overview

Tim Gorringe derives the electrostatic equilibrium properties of conductors from charge-carrier motion, Gauss’s law, and the field–potential relationship: the interior field and volume charge density vanish, the conductor is equipotential, and surface charge produces a normal external field. He applies these results to oppositely charged infinite sheets with a conducting slab between them, then to two wire-connected spheres, showing that equal potential makes sphere charge proportional to radius while surface charge density and surface field vary inversely with radius.

Key takeaways

Mobile charges in a conductor rearrange until the net electrostatic field inside is zero; a nonzero field would continue to exert force on those charges.
A conductor in electrostatic equilibrium is equipotential, and its surface electric field has no tangential component.
The macroscopic volume charge density inside an equilibrium conductor is zero, while surface charge can remain nonzero.
A Gaussian pillbox across a conductor boundary gives σ = ε₀E⊥, linking surface charge density directly to the perpendicular field just outside.
Two infinite sheets with charge densities +σ and −σ produce a field σ/ε₀ between them and zero field outside; a conductor inserted between them develops induced face charges −σ and +σ.
Wire-connected conducting spheres share the same potential; since a sphere’s potential scales as Q/R, its total charge scales with radius while its surface charge density and surface field scale as 1/R.

Chapters

0:00 Conductors, Insulators, and the Lecture’s Field–Potential Framework

The lecture applies electric field, flux, potential, and charge concepts to conductors in electrostatic fields.
Conductors have free charge carriers that move through their volume; metals use electrons, while salt solutions can conduct through mobile ions.
Insulators such as glass and plastic lack freely moving charge carriers, though they can still polarize.

3:19 A Conductor’s Macroscopic Equilibrium Before an External Field

A metal is modeled microscopically as positive ions in a lattice surrounded by mobile electrons.
Without an applied field, microscopic positive and negative fields average to zero macroscopically inside the conductor.
The macroscopic interior and surface charge densities are zero, and the potential is constant throughout the conductor.

6:23 External Fields Redistribute a Metal’s Free Electrons

When an external electric field is switched on, mobile electrons experience force opposite the field direction.
Electron accumulation on one side exposes positive charge on the other, creating an induced internal field.
The induced field combines with the applied field to change the field pattern around the conductor.

9:18 Electrostatic Equilibrium Requires Zero Interior and Tangential Fields

Electrons continue rearranging until the net electric field inside the conductor is zero; otherwise they would keep accelerating.
The tangential component of the field at the conductor’s surface must also vanish because surface charges would move along the surface if it did not.
The external field can retain a perpendicular component, with its direction depending on the surface charge.

14:40 Zero Field Makes the Conductor an Equipotential Surface

Using the relation between field and potential, zero electric field throughout the conductor means its potential is constant through its volume.
Because the tangential surface field is zero, moving along the conductor’s surface produces no potential change.
The entire connected conductor—including its interior and surface—has one potential value.

17:43 Gauss’s Law Shows the Conductor Has No Interior Volume Charge

A Gaussian surface placed anywhere wholly inside the conductor has zero electric field across it and therefore zero flux.
Gauss’s law then requires zero net enclosed charge for each such interior region.
In electrostatic equilibrium, the macroscopic volume charge density inside the conductor is zero.

23:31 A Gaussian Pillbox Relates Surface Charge to Normal Field

A small cylindrical Gaussian pillbox straddling the conductor’s surface isolates the charge on the enclosed surface patch.
The interior end cap contributes no flux, and the side wall contributes none because the field is normal to the surface.
The outer end cap gives the boundary relation σ = ε₀E⊥: surface charge density is proportional to the perpendicular electric field just outside.

31:19 From Conductor Rules to the Charged-Sheet Example

The analysis so far establishes zero interior field, constant conductor potential, zero volume charge density, and potentially nonzero surface charge density.
For two infinite sheets with charge densities +σ and −σ, each sheet produces a field of magnitude σ/(2ε₀).
Between the oppositely charged sheets, the fields point the same way and add to σ/ε₀; outside them, they cancel.

39:19 An Inserted Conducting Slab Gains Opposite Surface Charges

Placing a conducting slab between the charged sheets causes its free charges to rearrange until the net interior field is zero.
The slab’s induced internal field opposes the external field of magnitude σ/ε₀ between the plates.
The slab develops surface charge densities −σ on the face toward the positive sheet and +σ on the face toward the negative sheet, while its volume charge density remains zero.

43:27 Wire-Connected Spheres Share One Potential and Charge by Radius

Two conducting spheres connected by a wire act as one conductor, so both spheres and the wire have the same potential.
For an isolated spherical conductor, surface potential scales as Q/R; equal potentials therefore imply Q₁/Q₂ = R₁/R₂.
The larger sphere carries more total charge, in proportion to its radius.

47:34 Smaller Connected Spheres Have Greater Surface Charge Density and Field

A sphere’s charge is Q = 4πR²σ; combining this with Q proportional to R gives surface charge density proportional to 1/R.
The smaller connected sphere has less total charge but greater surface charge density than the larger sphere.
Because the normal surface field is E⊥ = σ/ε₀, the smaller sphere also has the stronger field immediately outside its surface.

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.