Chapter 22 — Gauss's Law

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 $\varepsilon_0=8.85\times10^{-12}\,\text{C}^2/(\text{N}\cdot\text{m}^2)$ and $k=8.99\times10^{9}\,\text{N}\cdot\text{m}^2/\text{C}^2$.

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

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

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

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2. 2. (2 pts ••) A point charge $q=+5.0\,\text{nC}$ 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.

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

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

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

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

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

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

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

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

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

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7. Extra Credit (3 pts •••)

   A solid insulating sphere of radius $R$ has a radially varying charge density $\rho(r)=\rho_0\!\left(1-\dfrac{r}{R}\right)$ for $r\le R$ (and zero outside).

   (a) Find the total charge $Q$ of the sphere.

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

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

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