Exercise 9a

Stewart, Section 14.9, Exercises 3, 12, 17, 18, 21, 22, 23.

Let $ \mathbf{E}= \frac{Q}{4 \pi \epsilon_0} \frac{\mathbf{r}}{r^3}$, where $ r = \sqrt{x^2 + y^2 + z^2}$, $ \mathbf{r}= (x\mathbf{i}+ y \mathbf{j}+ z\mathbf{k})$, and $ Q,
\epsilon_0$ are positive constants. Find the surface integral of $ \mathbf{E}$ over the surface shown, with the given unit normal.

[Drawing: omitted. The drawing depicts a surface that encloses the origin, and has a ``tail'' that curves back over towards the origin, intersects the surface and then encloses the origin again for a second time.]

Let $ \mathbf{E}$ be the same as in Exercise 2, and let $ C$ be the curve $ x^2
+ y^2 =1, z=0$ oriented in the anti-clockwise direction when viewed from above.
Calculate directly the flux of $ \mathbf{E}$ through the bottom half of the unit sphere, with unit normal pointing towards the origin.
Use the divergence theorem to show that if if $ S$ is any oriented surface with oriented boundary $ C$, and $ S$ does not intersect the non-negative part of the $ z$-axis, then the flux of $ E$ through $ S$ is $ \frac{Q}{2 \epsilon_0}$.

Math 13 Winter 1999