A metal sphere has flat potential inside
You will be able to: Construct piecewise field and potential functions for an isolated conducting sphere.
How can potential stay nonzero where the field vanishes?
A charged metal ball is one connected conductor. Walking from its surface toward its center changes neither your electric potential nor the work needed per test charge, although a probe outside sees a radial field.
A useful starting point: A conductor settles to zero internal field →
Words and symbols before equations
- Radius R
- Distance from the center to the metal surface, in metres.
- Radial field Eᵣ
- Signed outward component of electric field, in N/C.
- Piecewise function
- Different expressions for different spatial regions.
- Infinity reference
- V approaches zero far from an isolated finite charge distribution.
What this picture assumes
Isolated conducting sphere in vacuum, no external fields, V = 0 at infinity. Radial graph uses physical metres. At the surface the electric field has distinct inside/outside limits; the readout reports them explicitly.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- V = 44.96 V. Eᵣ = 112.4 N/C. The orange graph line marks R.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
Spherical symmetry makes the external field identical to that of a point charge Q at the center: Eᵣ = kQ/r² for r > R. There is no such field in the metal: E = 0 for r < R.
Integrate inward from infinity to get V = kQ/r outside. Potential remains continuous at the surface and constant throughout the metal, so V = kQ/R inside. The field jumps at the charged surface; its inside and outside limits differ.
The negative potential slope equals the radial field. A flat potential is therefore a zero field, not necessarily a zero potential. This spherical model assumes no nearby objects or applied external field.
A worked example, step by step
An isolated sphere has R = 0.20 m and Q = +2 nC. Find V at the center and E at r = 0.40 m using k ≈ 9 × 10⁹.
- At the center use the inside expression V = kQ/R.
- V_center = 9×10⁹ × 2×10⁻⁹ / 0.20 = 90 V.
- The external point r = 0.40 m uses Eᵣ = kQ/r² = 112.5 N/C.
- E_center = 0. The nonzero 90 V and zero center field describe different quantities.
Do not extend the point-charge expression through the conducting interior or assign a single field value to the ideal charged surface.
Can a negative sphere have a flat negative potential inside?
Compare with an explanation
Yes. Its reference-dependent value is negative; its slope and internal field are zero.
Predict. Change one thing. Explain.
Move the probe from the center to outside the sphere. Compare the flat inside potential with the inverse-distance outside curve. Reverse Q and explain the sign changes.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
V = 44.96 V. Eᵣ = 112.4 N/C. The orange graph line marks R.
Isolated conducting sphere in vacuum, no external fields, V = 0 at infinity. Radial graph uses physical metres. At the surface the electric field has distinct inside/outside limits; the readout reports them explicitly.
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 questionSketch Eᵣ(r) and V(r) for a positively charged isolated conducting sphere. Mark the center, surface, inside/outside limits and infinity reference.
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Compare with the answer and four-point rubric
- 1 point: E = 0 for r < R.
- 1 point: Outside E = kQ/r², starting at kQ/R² immediately outside.
- 1 point: V is constant kQ/R inside and joins continuously to kQ/r outside.
- 1 point: V tends to zero at infinity; its negative slope matches E in each region.
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 1What is V at the sphere center?
kQ/R relative to infinity.
RECALL 2Which quantity jumps at the charged surface?
The normal electric field; potential remains continuous.
RECALL 3Why does the outside point-charge model work?
Spherical symmetry and Gauss’s law.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
A metal sphere has flat potential inside
- r < R: E = 0, V = kQ/R.
- r > R: Eᵣ = kQ/r², V = kQ/r.
- V is continuous at R; E has distinct one-sided limits.
Remember: Do not extend the point-charge expression through the conducting interior or assign a single field value to the ideal charged surface.
Conditions: Isolated conducting sphere in vacuum, no external fields, V = 0 at infinity. Radial graph uses physical metres. At the surface the electric field has distinct inside/outside limits; the readout reports them explicitly.
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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