Divergence Practice including Curvilinear Coordinates

• assignment Curl Practice including Curvilinear Coordinates

assignment Homework

Curl Practice including Curvilinear Coordinates

Calculate the curl of each of the following vector fields. You may look up the formulas for curl in curvilinear coordinates.

1. $$\vec{F}=z^2\,\hat{x} + x^2 \,\hat{y} -y^2 \,\hat{z}$$
2. $$\vec{G} = e^{-x} \,\hat{x} + e^{-y} \,\hat{y} +e^{-z} \,\hat{z}$$
3. $$\vec{H} = yz\,\hat{x} + zx\,\hat{y} + xy\,\hat{z}$$
4. $$\vec{I} = x^2\,\hat{x} + z^2\,\hat{y} + y^2\,\hat{z}$$
5. $$\vec{J} = xy\,\hat{x} + xz\,\hat{y} + yz\,\hat{z}$$
6. $$\vec{K} = s^2\,\hat{s}$$
7. $$\vec{L} = r^3\,\hat{\phi}$$

assignment Homework

Static Fields 2023 (4 years)

Find the gradient of each of the following functions:

1. $$f(x,y,z)=e^{(x+y)}+x^2 y^3 \ln \frac{x}{z}$$
2. $$\sigma(\theta,\phi)=\cos\theta \sin^2\phi$$
3. $$\rho(s,\phi,z)=(s+3z)^2\cos\phi$$

assignment Homework

Consider the fields at a point $\vec{r}$ due to a point charge located at $\vec{r}'$.

1. Write down an expression for the electrostatic potential $V(\vec{r})$ at a point $\vec{r}$ due to a point charge located at $\vec{r}'$. (There is nothing to calculate here.)
2. Write down an expression for the electric field $\vec{E}(\vec{r})$ at a point $\vec{r}$ due to a point charge located at $\vec{r}'$. (There is nothing to calculate here.)
3. Working in rectangular coordinates, compute the gradient of $V$.
4. Write several sentences comparing your answers to the last two questions.

• assignment The Gradient for a Point Charge

assignment Homework

The Gradient for a Point Charge

Static Fields 2023 (6 years)

The electrostatic potential due to a point charge at the origin is given by: $$V=\frac{1}{4\pi\epsilon_0} \frac{q}{r}$$

1. Find the electric field due to a point charge at the origin as a gradient in rectangular coordinates.
2. Find the electric field due to a point charge at the origin as a gradient in spherical coordinates.
3. Find the electric field due to a point charge at the origin as a gradient in cylindrical coordinates.

• assignment Gravitational Field and Mass

assignment Homework

Gravitational Field and Mass
Static Fields 2023 (5 years)

The gravitational field due to a spherical shell of matter (or equivalently, the electric field due to a spherical shell of charge) is given by: $$\vec g = \begin{cases} 0&\textrm{for } r<a\\ -G \,\frac{M}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ -G\,\frac{M}{r^2}\, \hat r & \textrm{for } r>b \\ \end{cases}$$

This problem explores the consequences of the divergence theorem for this shell.

1. Using the given description of the gravitational field, find the divergence of the gravitational field everywhere in space. You will need to divide this question up into three parts: $r<a$, $a<r<b$, and $r>b$.
2. Briefly discuss the physical meaning of the divergence in this particular example.
3. For this gravitational field, verify the divergence theorem on a sphere, concentric with the shell, with radius $Q$, where $a<Q<b$. ("Verify" the divergence theorem means calculate the integrals from both sides of the divergence theorem and show that they give the same answer.)
4. Briefly discuss how this example would change if you were discussing the electric field of a uniformly charged spherical shell.

• assignment Divergence through a Prism

assignment Homework

Divergence through a Prism
Static Fields 2023 (6 years)

Consider the vector field $\vec F=(x+2)\hat{x} +(z+2)\hat{z}$.

1. Calculate the divergence of $\vec F$.
2. In which direction does the vector field $\vec F$ point on the plane $z=x$? What is the value of $\vec F\cdot \hat n$ on this plane where $\hat n$ is the unit normal to the plane?
3. Verify the divergence theorem for this vector field where the volume involved is drawn below. (“Verify” means calculate both sides of the divergence theorem, separately, for this example and show that they are the same.)

• assignment Line Sources Using the Gradient

assignment Homework

Static Fields 2023 (6 years)
1. Find the electric field around an infinite, uniformly charged, straight wire, starting from the following expression for the electrostatic potential: $$V(\vec r)=\frac{2\lambda}{4\pi\epsilon_0}\, \ln\left( \frac{ s_0}{s} \right)$$

• assignment Electric Field of a Finite Line

assignment Homework

Electric Field of a Finite Line

Consider the finite line with a uniform charge density from class.

1. Write an integral expression for the electric field at any point in space due to the finite line. In addition to your usual physics sense-making, you must include a clearly labeled figure and discuss what happens to the direction of the unit vectors as you integrate.Consider the finite line with a uniform charge density from class.
2. Perform the integral to find the $z$-component of the electric field. In addition to your usual physics sense-making, you must compare your result to the gradient of the electric potential we found in class. (If you want to challenge yourself, do the $s$-component as well!)

• assignment Total Current, Square Cross-Section

assignment Homework

Total Current, Square Cross-Section

Integration Sequence

Static Fields 2023 (6 years)
1. Current $I$ flows down a wire with square cross-section. The length of the square side is $L$. If the current is uniformly distributed over the entire area, find the current density .
2. If the current is uniformly distributed over the outer surface only, find the current density .
• assignment Directional Derivative

assignment Homework

Directional Derivative

Static Fields 2023 (6 years)

You are on a hike. The altitude nearby is described by the function $f(x, y)= k x^{2}y$, where $k=20 \mathrm{\frac{m}{km^3}}$ is a constant, $x$ and $y$ are east and north coordinates, respectively, with units of kilometers. You're standing at the spot $(3~\mathrm{km},2~\mathrm{km})$ and there is a cottage located at $(1~\mathrm{km}, 2~\mathrm{km})$. You drop your water bottle and the water spills out.

1. Plot the function $f(x, y)$ and also its level curves in your favorite plotting software. Include images of these graphs. Special note: If you use a computer program written by someone else, you must reference that appropriately.
2. In which direction in space does the water flow?
3. At the spot you're standing, what is the slope of the ground in the direction of the cottage?
4. Does your result to part (c) make sense from the graph?

• Calculate the divergence of each of the following vector fields. You may look up the formulas for divergence in curvilinear coordinates.

1. $$\hat{F}=z^2\,\hat{x} + x^2 \,\hat{y} -y^2 \,\hat{z}$$
2. $$\hat{G} = e^{-x} \,\hat{x} + e^{-y} \,\hat{y} +e^{-z} \,\hat{z}$$
3. $$\hat{H} = yz\,\hat{x} + zx\,\hat{y} + xy\,\hat{z}$$
4. $$\hat{I} = x^2\,\hat{x} + z^2\,\hat{y} + y^2\,\hat{z}$$
5. $$\hat{J} = xy\,\hat{x} + xz\,\hat{y} + yz\,\hat{z}$$
6. $$\hat{K} = s^2\,\hat{s}$$
7. $$\hat{L} = r^3\,\hat{\phi}$$