Quick answer: Electric flux is Φ = ∮E · dA, the field passing through a surface. This AP Physics C worksheet on gauss's law gives 20 problems from easy to hard: flux of a uniform field, charge from flux and flux from a point charge. The worked answers are on a separate page, and the worksheet is a free printable PDF.
These problems train the calculus-based treatment of electric fields: building E from continuous charge by integration, computing flux, and applying Gauss’s law with spherical, cylindrical and planar symmetry. Contexts include wires, plates, shells and cubes.
Level 1 problems use a single relation, level 2 problems combine superposition with symmetry, and level 3 problems need integration, non-uniform charge or conductors, as in AP Physics C: Electricity and Magnetism unit 8.
Level: AP Physics C (Mechanics and E&M, calculus-based) · Curriculum: AP PC:E&M Unit 8
Problem 1 Flux of a uniform field ★★★
1. the surface is face-on to the field (normal parallel to E)
2. the normal makes 60° with E
3. the surface is edge-on.
Problem 2 Charge from flux ★★★
The net outward flux through a closed surface is 3.0 × 10³ N m²/C. Calculate the charge enclosed.Problem 3 Flux from a point charge ★★★
A point charge of +4.0 nC sits at the centre of a sphere of radius 0.10 m. Calculate the flux through the sphere and say what happens if the sphere radius is doubled.Problem 4 Field of a point charge ★★★
Calculate the electric field 0.30 m from a point charge of +5.0 nC and state its direction.Problem 5 Net enclosed charge ★★★
A closed surface encloses charges of +3.0 nC and −1.0 nC. A further +8.0 nC charge lies outside it.1. Calculate the net flux through the surface.
2. State the effect of the outside charge.
Problem 6 Charge densities ★★★
1. A rod 0.60 m long carries 3.0 nC uniformly. Calculate λ.2. A plate of area 0.30 m² carries 12 nC uniformly on one face. Calculate σ.
Problem 7 Charged ring on its axis ★★★
A thin ring of radius 0.12 m carries a uniform charge of +5.0 nC. Calculate the field on its axis at 0.16 m from the centre, and state the field at the centre.Problem 8 Long wire ★★★
A long straight wire carries a uniform charge of 3.0 nC/m. Calculate the field 0.40 m from it.Problem 9 Large sheet ★★★
A large flat insulating sheet carries a uniform surface charge of +4.0 nC/m².1. Calculate the field 0.10 m from it.
2. Calculate the field 1.0 m from it.
Problem 10 Thin spherical shell ★★★
A thin spherical shell of radius 0.20 m carries a charge of −12 nC. Calculate the field at (a) r = 0.10 m and (b) r = 0.50 m.Problem 11 Solid insulating sphere ★★★
A solid insulating sphere of radius 0.15 m carries a uniform +10 nC. Calculate the field at r = 0.10 m and at r = 0.30 m.Problem 12 Reading an E–r graph ★★★
1. State the radius of the sphere.
2. Calculate the charge on it.
3. Calculate the field at r = 0.20 m.
Problem 13 Charge at the centre of a cube ★★★
Problem 14 Spot the error ★★★
A student writes: ‘In Gauss’s law the field E on the surface is the field of the enclosed charge only, so a charge outside the surface has no effect.’1. Identify the error.
2. State the correct statement.
Problem 15 Field between charged plates ★★★
Two large parallel plates carry equal and opposite surface charge densities of ±2.0 × 10⁻⁷ C/m².1. Calculate the field between them.
2. Calculate the acceleration of an electron placed there.
Problem 16 Maximum field on a ring’s axis ★★★
A ring of radius R = 0.12 m carries +5.0 nC, and E(x) = kQx / (x² + R²)^(3/2).1. Show that the field is greatest at x = R/√2.
2. Calculate the maximum field.
Problem 17 Field of a finite rod ★★★
A rod of length 0.40 m carries +8.0 nC uniformly. A point P lies on the line of the rod, 0.10 m beyond one end.1. Set up and evaluate the integral for E at P.
2. Calculate E.
Problem 18 Non-uniform sphere ★★★
A sphere of radius R = 0.20 m has volume charge density ρ = A r with A = 3.0 × 10⁻⁶ C/m⁴.1. Calculate the total charge.
2. Use Gauss’s law to find E at r = 0.10 m.
3. Calculate E at r = 0.50 m.
Problem 19 Charge inside a conducting shell ★★★
1. State the charge on the inner and outer surfaces.
2. Calculate E at r = 0.05 m and at r = 0.30 m.
3. State E inside the metal.
Problem 20 Long charged cylinder ★★★
A very long solid insulating cylinder of radius 0.050 m has uniform volume charge density ρ = +2.0 μC/m³.1. Use Gauss’s law to find E inside at r = 0.030 m.
2. Calculate E outside at r = 0.10 m.
