Magnetic Field and Current

• 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 2022 (4 years)

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

  Power Series Sequence (E&M)

  Ring Cycle Sequence

  Students work in groups of three 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.

• group Magnetic Field Due to a Spinning Ring of Charge

  group Small Group Activity

  30 min.

  Magnetic Field Due to a Spinning Ring of Charge

  Static Fields 2022 (5 years)

  magnetic fields current Biot-Savart law vector field symmetry

  Power Series Sequence (E&M)

  Ring Cycle Sequence

  Students work in groups of three to use the Biot-Savart law \[\vec{B}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})\times \left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the magnetic field, \(\vec{B}(\vec{r})\), due to a spinning ring of charge.

  In an optional extension, students find a series expansion for \(\vec{B}(\vec{r})\) 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 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.
• assignment Total Current, Circular Cross Section

  assignment Homework

  Total Current, Circular Cross Section

  Integration Sequence

  Static Fields 2022 (3 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 Current in a Wire

  assignment Homework

  Current in a Wire

  Static Fields 2022 (3 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 2022 (4 years)

  Steady current current density magnetic field idealization

  Ring Cycle Sequence

  Integration 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.
• group Static Fields Equation Sheet

  group Small Group Activity

  5 min.

  Static Fields Equation Sheet

  Static Fields 2022 (3 years)
• assignment Total Current, Square Cross-Section

  assignment Homework

  Total Current, Square Cross-Section

  Integration Sequence

  Static Fields 2022 (4 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 Electric Field and Charge

  assignment Homework

  Electric Field and Charge

  divergence charge density Maxwell's equations electric field Static Fields 2022 (3 years) Consider the electric field \begin{equation} \vec E(r,\theta,\phi) = \begin{cases} 0&\textrm{for } r<a\\ \frac{1}{4\pi\epsilon_0} \,\frac{Q}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ 0 & \textrm{for } r>b \\ \end{cases} \end{equation}

  1. Use step and/or delta functions to write this electric field as a single expression valid everywhere in space.
  2. Find a formula for the charge density that creates this electric field.
  3. Interpret your formula for the charge density, i.e. explain briefly in words where the charge is.
• 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.
• Static Fields 2022 (3 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.