Gradient Practice

    • 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 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. \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}

    • assignment Mass of a Slab

      assignment Homework

      Mass of a Slab
      Static Fields 2023 (6 years)

      Determine the total mass of each of the slabs below.

      1. A square slab of side length \(L\) with thickness \(h\), resting on a table top at \(z=0\), whose mass density is given by \begin{equation} \rho=A\pi\sin(\pi z/h). \end{equation}
      2. A square slab of side length \(L\) with thickness \(h\), resting on a table top at \(z=0\), whose mass density is given by \begin{equation} \rho = 2A \Big( \Theta(z)-\Theta(z-h) \Big) \end{equation}
      3. An infinitesimally thin square sheet of side length \(L\), resting on a table top at \(z=0\), whose surface density is given by \(\sigma=2Ah\).
      4. An infinitesimally thin square sheet of side length \(L\), resting on a table top at \(z=0\), whose mass density is given by \(\rho=2Ah\,\delta(z)\).
      5. What are the dimensions of \(A\)?
      6. Write several sentences comparing your answers to the different cases above.

    • 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 Volume Charge Density, Version 2

      assignment Homework

      Volume Charge Density, Version 2
      charge density delta function Static Fields 2023 (6 years)

      You have a charge distribution on the \(x\)-axis composed of two point charges: one with charge \(+3q\) located at \(x=-d\) and the other with charge \(-q\) located at \(x=+d\).

      1. Sketch the charge distribution.
      2. Write an expression for the volume charge density \(\rho (\vec{r})\) everywhere in space.

    • assignment Line Sources Using the Gradient

      assignment Homework

      Line Sources Using the Gradient

      Gradient Sequence

      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: \begin{equation} V(\vec r)=\frac{2\lambda}{4\pi\epsilon_0}\, \ln\left( \frac{ s_0}{s} \right) \end{equation}

    • assignment Total Current, Circular Cross Section

      assignment Homework

      Total Current, Circular Cross Section

      Integration Sequence

      Static Fields 2023 (5 years)

      A current \(I\) flows down a cylindrical wire of radius \(R\).

      1. If it is uniformly distributed over the surface, give a formula for the surface current density \(\vec K\).
      2. If it is distributed in such a way that the volume current density, \(|\vec J|\), is inversely proportional to the distance from the axis, give a formula for \(\vec J\).

    • 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 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 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: \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.

  • Static Fields 2023 (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}