Current in a Wire

    • 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 Magnetic Field and Current

      assignment Homework

      Magnetic Field and Current
      Static Fields 2023 (4 years) Consider the magnetic field \[ \vec{B}(s,\phi,z)= \begin{cases} 0&0\le s<a\\ \alpha \frac{1}{s}(s^4-a^4)\, \hat{\phi}&a<s<b\\ 0&s>b \end{cases} \]
      1. Use step and/or delta functions to write this magnetic field as a single expression valid everywhere in space.
      2. Find a formula for the current density that creates this magnetic field.
      3. Interpret your formula for the current density, i.e. explain briefly in words where the current is.
    • accessibility_new Acting Out Current Density

      accessibility_new Kinesthetic

      10 min.

      Acting Out Current Density
      Static Fields 2023 (6 years)

      Steady current current density magnetic field idealization

      Integration Sequence

      Ring Cycle Sequence

      Students, pretending they are point charges, move around the room so as to make an imaginary magnetic field meter register a constant magnetic field, introducing the concept of steady current. Students act out linear \(\vec{I}\), surface \(\vec{K}\), and volume \(\vec{J}\) current densities. The instructor demonstrates what it means to measure these quantities by counting how many students pass through a gate.
    • assignment Line Sources Using Coulomb's Law

      assignment Homework

      Line Sources Using Coulomb's Law
      Static Fields 2023 (6 years)
      1. Find the electric field around a finite, uniformly charged, straight rod, at a point a distance \(s\) straight out from the midpoint, starting from Coulomb's Law.
      2. Find the electric field around an infinite, uniformly charged, straight rod, starting from the result for a finite rod.
    • 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 Symmetry Arguments for Gauss's Law

      assignment Homework

      Symmetry Arguments for Gauss's Law
      Static Fields 2023 (5 years)

      Instructions for 2022: You will need to complete this assignment in a 15 minute appointment on Zoom or in person with one of the members of the teaching team between 1/21 and 10 pm on 1/26. Here is a link to a sign-up page.

      You are required to watch a sample video for how to make symmetry arguments here. As demonstrated in the video you should bring with you to the meeting a cylinder, an observer, and a vector.

      Use good symmetry arguments to find the possible direction for the electric field due to a charged wire. Also, use good symmetry arguments to find the possible functional dependence of the electric field due to a charged wire. Rather than writing this up to turn in, you should find a member of the teaching team and make the arguments to them verbally.

    • assignment The puddle

      assignment Homework

      The puddle
      differentials Static Fields 2023 (5 years) The depth of a puddle in millimeters is given by \[h=\frac{1}{10} \bigl(1+\sin(\pi xy)\bigr)\] Your path through the puddle is given by \[x=3t \qquad y=4t\] and your current position is \(x=3\), \(y=4\), with \(x\) and \(y\) also in millimeters, and \(t\) in seconds.
      1. At your current position, how fast is the depth of water through which you are walking changing per unit time?
      2. At your current position, how fast is the depth of water through which you are walking changing per unit distance?
      3. FOOD FOR THOUGHT (optional)
        There is a walkway over the puddle at \(x=10\). At your current position, how fast is the depth of water through which you are walking changing per unit distance towards the walkway.
    • assignment Icecream Mass

      assignment Homework

      Icecream Mass
      Static Fields 2023 (6 years)

      Use integration to find the total mass of the icecream in a packed cone (both the cone and the hemisphere of icecream on top).

    • assignment Flux through a Plane

      assignment Homework

      Flux through a Plane
      Static Fields 2023 (4 years) Find the upward pointing flux of the vector field \(\boldsymbol{\vec{H}}=2z\,\boldsymbol{\hat{x}} +\frac{1}{x^2+1}\boldsymbol{\hat{y}}+(3+2z)\boldsymbol{\hat{z}}\) through the rectangle \(R\) with one edge along the \(y\) axis and the other in the \(xz\)-plane along the line \(z=x\), with \(0\le y\le2\) and \(0\le x\le3\).
  • Static Fields 2023 (4 years) The current density in a cylindrical wire of radius \(R\) is given by \(\vec{J}(\vec{r})=\alpha s^3\cos^2\phi\,\hat{z}\). Find the total current in the wire.