Total Current, Square Cross-Section

    • 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 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.
    • assignment Current in a Wire

      assignment Homework

      Current in a Wire
      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.
    • 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 Current from a Spinning Cylinder

      assignment Homework

      Current from a Spinning Cylinder
      A solid cylinder with radius \(R\) and height \(H\) has its base on the \(x,y\)-plane and is symmetric around the \(z\)-axis. There is a fixed volume charge density on the cylinder \(\rho=\alpha z\). If the cylinder is spinning with period \(T\):
      1. Find the volume current density.
      2. Find the total current.
    • format_list_numbered Ring Cycle Sequence

      format_list_numbered Sequence

      Ring Cycle Sequence
      Students calculate electrostatic fields (\(V\), \(\vec{E}\)) and magnetostatic fields (\(\vec{A}\), \(\vec{B}\)) from charge and current sources with a common geometry. The sequence of activities is arranged so that the mathematical complexity of the formulas students encounter increases with each activity. Several auxiliary activities allow students to focus on the geometric/physical meaning of the distance formula, charge densities, and steady currents. A meta goal of the entire sequence is that students gain confidence in their ability to parse and manipulate complicated equations.
    • group Magnetic Vector Potential Due to a Spinning Charged Ring

      group Small Group Activity

      30 min.

      Magnetic Vector Potential Due to a Spinning Charged Ring
      Static Fields 2023 (6 years)

      compare and contrast mathematica magnetic vector potential magnetic fields vector field symmetry

      Power Series Sequence (E&M)

      Ring Cycle Sequence

      Students work in small groups to use the superposition principle \[\vec{A}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert}\, d\tau^{\prime}\] to find an integral expression for the magnetic vector potential, \(\vec{A}(\vec{r})\), due to a spinning ring of charge.

      In an optional extension, students find a series expansion for \(\vec{A}(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

    • assignment Directional Derivative

      assignment Homework

      Directional Derivative

      Gradient Sequence

      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?

    • 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.
    • face Warm-Up Powerpoint

      face Lecture

      10 min.

      Warm-Up Powerpoint

      Warm-Up

      The attached powerpoint articulates the possible paths through the curriculum for new graduate students at OSU. Make sure to update this powerpoint yearly to reflect current course offerings and sequencing. It was partially, but not completely edited in fall 2022.
  • 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 .