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Results: charge density

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.

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.

assignment Homework

Total Charge

For each case below, find the total charge.

  1. A positively charged (dielectric) spherical shell of inner radius \(a\) and outer radius \(b\) with a spherically symmetric internal charge density \begin{equation} \rho(\vec{r})=3\alpha\, e^{(kr)^3} \end{equation}
  2. A positively charged (dielectric) cylindrical shell of inner radius \(a\) and outer radius \(b\) with a cylindrically symmetric internal charge density \begin{equation} \rho(\vec{r})=\alpha\, \frac{1}{s}\, e^{ks} \end{equation}

group Small Group Activity

30 min.

Total Charge

charge volume charge density multiple integral scalar field coordinate systems differential elements curvilinear coordinates

Integration Sequence

In this small group activity, students integrate over non-uniform charge densities in cylindrical and spherical coordinates to calculate total charge.

assignment Homework

Cube Charge
  1. Charge is distributed throughout the volume of a dielectric cube with charge density \(\rho=\beta z^2\), where \(z\) is the height from the bottom of the cube, and where each side of the cube has length \(L\). What is the total charge inside the cube? Do this problem in two ways as both a single integral and as a triple integral.
  2. In a new physical situation: Charge is distributed on the surface of a cube with charge density \(\sigma=\alpha z\) where \(z\) is the height from the bottom of the cube, and where each side of the cube has length \(L\). What is the total charge on the cube? Don't forget about the top and bottom of the cube.

assignment Homework

Volume Charge Density Practice

You have a charge distribution 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 Homework

Spherical Shell Step Functions
Step Function

One way to write volume charge densities without using piecewise functions is to use step \((\Theta)\) or \(\delta\) functions. If you need to review this, see the following link in the math-physics book: https://books.physics.oregonstate.edu/GMM/step.html

Consider a spherical shell with charge density \(\rho (\vec{r})=\alpha3e^{(k r)^3}\) between the inner radius \(a\) and the outer radius \(b\). The charge density is zero everywhere else. Use step functions to write this charge density as a single function valid everywhere in space.

assignment Homework

Electric Field and Charge
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.
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