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Electric charges, fields and Gauss's law practice problems AP Physics C: E&M, Unit 8

Written by STEM-Tutor's tutors · Updated October 2026 · 8 questions and 1 free-response question

Coulomb's law, electric fields of point charges and continuous distributions, electric flux and Gauss's law. Try each question before you look: use the hint if you are stuck, then reveal the worked solution one step at a time.

On this page
  1. Key equations
  2. Warm-up questions
  3. Exam-style questions
  4. Free-response practice
  5. Common mistakes
  6. Keep going

Key equations for electric charges, fields and gauss's law

F = k q1q2/r2Coulomb's law
E = kq/r2field of a point charge
ΦE = ∮ E·dA = qenc/ε0Gauss's law
E = λ/(2πε0r)infinite line of charge
E = σ/(2ε0)infinite sheet of charge

Constants as on the AP reference sheet: k = 9.0 × 109 N·m²/C², ε0 = 8.85 × 10−12 C²/(N·m²), μ0 = 4π × 10−7 T·m/A, e = 1.6 × 10−19 C. You can type answers like 3.6e5 or 3.6x10^5. The full sheet is in our AP Physics C equation sheet.

Warm-up questions

Quick checks on the core ideas. If these feel easy, go straight to the exam-style questions.

Q1Warm-up
Two charges of 2.0 μC and 3.0 μC are 0.30 m apart. What is the size of the force between them?
N

Hint

Use Coulomb's law, with the charges in coulombs.

Worked solution
  1. F = kq1q2/r2 = 9.0 × 109 × (2.0 × 10−6)(3.0 × 10−6)/0.302.
  2. = 9.0 × 109 × 6.0 × 10−12/0.09.
  3. F = 0.60 N.
Q2Warm-up
Electric field lines near a single positive point charge:

Hint

Field lines show the direction of the force on a positive test charge.

Worked solution
  1. A positive test charge is repelled by a positive charge.
  2. So the lines point radially outwards: B.
Q3Warm-up
What is the electric field 0.50 m from a 4.0 nC point charge?
N/C

Hint

E = kq/r2, with 1 nC = 10−9 C.

Worked solution
  1. E = 9.0 × 109 × 4.0 × 10−9/0.502.
  2. = 36/0.25.
  3. E = 144 N/C.

Exam-style questions

Questions at the level of the AP Physics C: E&M exam. The ones marked Challenge combine more than one idea.

Q4Exam-style
A closed surface surrounds a +5.0 nC charge. What is the electric flux through the surface?
N·m²/C

Hint

Gauss's law: the flux depends only on the charge enclosed.

Worked solution
  1. Φ = qenc/ε0.
  2. = 5.0 × 10−9/8.85 × 10−12.
  3. Φ = 565 N·m²/C, whatever the shape of the surface.
Q5Exam-style
A long straight wire carries a uniform charge of 2.0 μC per metre. What is the field 0.10 m from the wire?
N/C

Hint

Use a cylindrical Gaussian surface around the wire.

Worked solution
  1. Gauss's law with a cylinder of radius r and length L: E(2πrL) = λL/ε0.
  2. E = λ/(2πε0r) = 2.0 × 10−6/(2π × 8.85 × 10−12 × 0.10).
  3. E = 3.60 × 105 N/C.
Q6Exam-style
A thin spherical shell carries charge spread uniformly over its surface. The electric field inside the shell is:

Hint

Draw a Gaussian sphere inside the shell. How much charge does it enclose?

Worked solution
  1. A Gaussian sphere inside the shell encloses no charge.
  2. By symmetry E is the same all over that sphere, so E × 4πr2 = 0.
  3. So E = 0 everywhere inside: A.
Q7Challenge
A solid sphere of radius 0.20 m has a charge of 8.0 nC spread uniformly through its volume. What is the field 0.10 m from the centre?
N/C

Hint

Inside a uniform sphere, only the charge within radius r counts, and it grows as r3.

Worked solution
  1. Enclosed charge: Q(r/R)3. So E = kQr/R3.
  2. E = 9.0 × 109 × 8.0 × 10−9 × 0.10/0.203 = 7.2/0.008.
  3. E = 900 N/C.
Q8Exam-style
A large flat sheet carries 1.77 × 10−9 C/m². What is the field near it, away from the edges?
N/C

Hint

Use a Gaussian “pillbox” through the sheet.

Worked solution
  1. The flux leaves through both faces: E × 2A = σA/ε0.
  2. E = σ/(2ε0) = 1.77 × 10−9/(1.77 × 10−11).
  3. E = 100 N/C, independent of distance.

Free-response practice

One original question in the style of the exam's free-response section. Answer it in full, with working, before you open the scoring guide.

Mathematical routines style · 6 points · about 20 minutes

A solid insulating sphere of radius R has total charge Q spread uniformly through its volume.

  1. (a) Use Gauss's law to derive the electric field at a distance r < R from the centre.
  2. (b) Derive the field for r > R.
  3. (c) Sketch or describe the graph of E against r from r = 0 to r = 3R.

Write your full answer on paper first, then open the scoring guide and tick each point you earned.

Scoring guide and model answer

Part (a)

Enclosed charge Q(r/R)3. E(4πr2) = Qr3/(ε0R3), so E = Qr/(4πε0R3).

Part (b)

All the charge is enclosed: E = Q/(4πε0r2).

Part (c)

E rises linearly from 0 at the centre to a maximum at r = R, then falls as 1/r2.

Your mark: 0 of 6

Common mistakes in electric charges, fields and gauss's law

Keep going

Revise the ideas behind these questions:

Then try real exam questions: AP Physics C: E&M past papers, with official scoring guidelines. · All AP Physics C: E&M units