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accessibility_new Kinesthetic

10 min.

Acting Out Charge Densities

density charge density mass density linear density uniform idealization

Ring Cycle Sequence

Integration Sequence

Students, pretending they are point charges, move around the room acting out various prompts from the instructor regarding charge densities, including linear \(\lambda\), surface \(\sigma\), and volume \(\rho\) charge densities, both uniform and non-uniform. The instructor demonstrates what it means to measure these quantities. In a remote setting, we have students manipulate 10 coins to model the prompts in this activity and the we demonstrate the answers with coins under a doc cam.

assignment Homework

Total Current, Square Cross-Section
  1. Current \(I\) flows down a wire (length \(L\)) with square cross-section (side \(a\)). If it is uniformly distributed over the entire area, what is the magnitude of the volume current density \(\vec{J}\)?
  2. If the current is uniformly distributed over the outer surface only, what is the magnitude of the surface current density \(\vec{K}\)?

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 Homework

Divergence through a Prism

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.

assignment Homework

Gravitational Field and Mass

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 value 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. 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. Discuss how this example would change if you were discussing the electric field of a uniformly charged spherical shell.

accessibility_new Kinesthetic

10 min.

Acting Out Current Density

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.
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