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The Timpson Lyceum · University Physics II (2026) – Electricity & Magnetism

Chapter 22 — Gauss's Law

Problem sheet · 12 points · 3 extra credit

Chapter 22 — Gauss's Law

Core: 12 points · Extra credit: 3 points. Show your reasoning; partial credit is given. Use ε0=8.85×10−12 C2/(N⋅m2)\varepsilon_0=8.85\times10^{-12}\,\text{C}^2/(\text{N}\cdot\text{m}^2) and k=8.99×109 N⋅m2/C2k=8.99\times10^{9}\,\text{N}\cdot\text{m}^2/\text{C}^2.

  1. 1. (2 pts ••) A uniform electric field E=6.0×103 N/CE=6.0\times10^{3}\,\text{N/C} points in the +x+x direction.

    (a) Find the electric flux through a flat 0.20 m×0.40 m0.20\,\text{m}\times0.40\,\text{m} rectangle whose normal makes a 30∘30^\circ angle with the field.

    (b) A closed box encloses a net charge of +8.0 nC+8.0\,\text{nC}. What is the net electric flux through the box?

  2. 2. (2 pts ••) A point charge q=+5.0 nCq=+5.0\,\text{nC} sits at the center of a cube.

    (a) Find the total electric flux through the cube.

    (b) Find the flux through one face, and explain how symmetry gives this in one more step.

  3. 3. (2 pts ••) A solid insulating sphere of radius R=0.10 mR=0.10\,\text{m} carries charge Q=+4.0 nCQ=+4.0\,\text{nC} spread uniformly through its volume.

    (a) Find the electric field at r=0.20 mr=0.20\,\text{m}.

    (b) Find the electric field at r=0.050 mr=0.050\,\text{m}.

  4. 4. (2 pts ••) An infinitely long line carries uniform linear charge density λ=3.0 nC/m\lambda=3.0\,\text{nC/m}. Using a Gaussian cylinder, find the magnitude and direction of the electric field at a distance r=5.0 cmr=5.0\,\text{cm} from the line.

  5. 5. (3 pts •••) A point charge q=+2.0 nCq=+2.0\,\text{nC} sits at the center of a neutral conducting spherical shell of inner radius aa and outer radius bb.

    (a) Find the charge induced on the inner and outer surfaces of the shell.

    (b) Find the electric field in the three regions r<ar<a, a<r<ba<r<b, and r>br>b.

    (c) How do your answers to (a) and (b) change if the shell instead carries a net charge of −5.0 nC-5.0\,\text{nC}?

  6. 6. (1 pt •) A closed surface encloses a +3.0 nC+3.0\,\text{nC} charge and a −5.0 nC-5.0\,\text{nC} charge and no others. What is the net electric flux through the surface?

  7. Extra Credit (3 pts •••)

    A solid insulating sphere of radius RR has a radially varying charge density ρ(r)=ρ0 ⁣(1−rR)\rho(r)=\rho_0\!\left(1-\dfrac{r}{R}\right) for r≤Rr\le R (and zero outside).

    (a) Find the total charge QQ of the sphere.

    (b) Use Gauss's law to find the electric field E(r)E(r) both inside (r<Rr<R) and outside (r>Rr>R).

    (c) Find the radius at which the field inside the sphere is maximum.