A conductor settles to zero internal field
You will be able to: Use mobile charges to explain zero internal field, constant potential and normal surface fields.
Why do excess charges move to the surface?
Place extra electrons on a metal sphere. They repel one another and move until the metal has no electric field that can keep driving them. The final state is electrostatic equilibrium, not a current-carrying wire.
A useful starting point: Field and potential gradient →
Words and symbols before equations
- Conductor
- A material with mobile charge carriers; in a metal these are electrons.
- Electrostatic equilibrium
- A settled charge distribution with no continuing macroscopic charge motion.
- Surface density σ
- Excess charge per area, measured in C/m².
- Normal direction
- Perpendicular to a surface, here pointing outward.
What this picture assumes
Local flat vacuum-facing metal surface at electrostatic equilibrium. Arrow indicates direction, not magnitude scale. E_out,n = σ/ε₀; E in the metal is zero. Edge shape and curvature are not modeled.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- Signed outward field = 2.26 N/C; field inside metal = 0. Outside arrow is direction only.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
If an electric field persisted inside ideal metal, its mobile charges would feel forces and rearrange. Equilibrium therefore requires E = 0 in the metal. A Gaussian surface lying wholly in it has zero flux and encloses zero net excess charge.
Because ΔV = −∫E·dℓ, any two points connected through the metal have the same potential. That constant need not be zero. Excess charge sits on surfaces; positive metal has a deficit of electrons, not freely moving positive nuclei.
A tangential surface field would drive charge sideways, so the external field is normal to the surface. A tiny Gaussian pillbox gives E_out,n = σ/ε₀ in vacuum. Charge density can vary over an irregular conductor and is often greatest at sharp regions. Uniform density requires symmetry.
A worked example, step by step
A flat vacuum-facing patch has σ = +17.7 pC/m². Find the normal field just outside and the field in the metal. Use ε₀ = 8.85 × 10⁻¹² F/m.
- Convert σ = 17.7 × 10⁻¹² C/m².
- A pillbox has flux E_out,n A because the metal-side field is zero.
- Gauss’s law gives E_out,n = σ/ε₀ = +2.00 N/C.
- The outside field points outward; the metal interior has E = 0 even if its potential is nonzero.
E = 0 in metal does not imply V = 0, and a uniform surface density is not guaranteed.
If the outside field had a tangential component, would the charges stay settled?
Compare with an explanation
No. Mobile surface charges would rearrange until that component vanished.
Predict. Change one thing. Explain.
Change the signed surface density. Predict the outside direction before comparing it with the zero field inside the metal. The arrow encodes direction; the readout gives magnitude.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Signed outward field = 2.26 N/C; field inside metal = 0. Outside arrow is direction only.
Local flat vacuum-facing metal surface at electrostatic equilibrium. Arrow indicates direction, not magnitude scale. E_out,n = σ/ε₀; E in the metal is zero. Edge shape and curvature are not modeled.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant conductor equilibrium, charge conservation, capacitance or energy relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionExplain why an irregular charged conductor can have unequal surface charge density while all its metal remains at one potential. Describe the field at a sharp region.
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Compare with the answer and four-point rubric
- 1 point: Mobile charges rearrange until the field in the metal vanishes.
- 1 point: The integral of the internal field gives zero potential difference.
- 1 point: Equal potential does not require equal surface density; geometry determines the distribution.
- 1 point: Greater local density generally gives a stronger normal external field by E_out,n = σ/ε₀.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Why is E zero inside equilibrium metal?
A persistent field would move its mobile carriers.
RECALL 2What direction can the surface field have?
Normal to the surface; no tangential component.
RECALL 3Does positive metal contain mobile positive ions?
In this model it has an electron deficit; the ions remain in the lattice.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
A conductor settles to zero internal field
- In electrostatic metal: E = 0 and V is constant.
- E_out,n = σ/ε₀ for a vacuum-facing surface.
- Tangential E at an equilibrium surface is zero.
Remember: E = 0 in metal does not imply V = 0, and a uniform surface density is not guaranteed.
Conditions: Local flat vacuum-facing metal surface at electrostatic equilibrium. Arrow indicates direction, not magnitude scale. E_out,n = σ/ε₀; E in the metal is zero. Edge shape and curvature are not modeled.
Refresh Kid · AP Physics C: Electricity and Magnetism Unit 3 (official Unit 10) · Objectives 10.1.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 10.1, objectives 10.1.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. This is E&M Unit 3: Conductors and Capacitors, numbered Unit 10 in the official combined Physics C sequence. Topics 10.1–10.4 retain their official identifiers. Models state the electrostatic conditions, geometry approximations and whether charge or voltage stays fixed. Capacitor geometries include parallel plates, concentric spheres and long coaxial cylinders. Dielectric comparisons assume a fully filling ideal linear material. The optional 3D plate view uses explicitly different gap and lateral scales to show the small separation. Checked with the Fall 2026 clarifications. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
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