Curl Practice including Curvilinear Coordinates

    • assignment Divergence Practice including Curvilinear Coordinates

      assignment Homework

      Divergence Practice including Curvilinear Coordinates

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

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

    • assignment Gradient Point Charge

      assignment Homework

      Gradient Point Charge

      Gradient Sequence

      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 Gradient Practice

      assignment Homework

      Gradient Practice

      Gradient Sequence

      Static Fields 2022 (4 years)

      Find the gradient of each of the following functions:

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

    • assignment The Gradient for a Point Charge

      assignment Homework

      The Gradient for a Point Charge

      Gradient Sequence

      Static Fields 2022 (6 years)

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

      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 2022 (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: \begin{equation} \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} \end{equation}

      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 2022 (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

      Line Sources Using the Gradient

      Gradient Sequence

      Static Fields 2022 (6 years)
      1. Find the electric field around an infinite, uniformly charged, straight wire, starting from the following expression for the electrostatic potential: \begin{equation} V(\vec r)=\frac{2\lambda}{4\pi\epsilon_0}\, \ln\left( \frac{ s_0}{s} \right) \end{equation}

    • 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 Directional Derivative

      assignment Homework

      Directional Derivative

      Gradient Sequence

      Static Fields 2022 (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?

    • assignment Differential Form of Gauss's Law

      assignment Homework

      Differential Form of Gauss's Law
      Static Fields 2022 (6 years)

      For an infinitesimally thin cylindrical shell of radius \(b\) with uniform surface charge density \(\sigma\), the electric field is zero for \(s<b\) and \(\vec{E}= \frac{\sigma b}{\epsilon_0 s}\, \hat s\) for \(s > b\). Use the differential form of Gauss' Law to find the charge density everywhere in space.

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

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