## Activity: Total Charge

Static Fields 2023 (6 years)
In this small group activity, students integrate over non-uniform charge densities in cylindrical and spherical coordinates to calculate total charge.
• This activity is used in the following sequences
• Media

Calculating Total Charge

Each group will be given one of the charge distributions given below: ($\alpha$ and $k$ are constants with dimensions appropriate for the specific example.)

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

For your group's case, answer the following questions:

1. Find the total charge. (If the total charge is infinite, decide what you should calculate instead to provide a meaningful answer.)
2. Find the dimensions of the constants $\alpha$ and $k$.

## Instructor's Guide

### Introduction

We usually start with a mini-lecture reminder that total charge is calculated by integrating over the charge density by chopping up the charge density, multiplying by the appropriate geometric differential (length, area, or volume element), and adding up the contribution from each of the pieces. Chop, Multiply, Add is a mantra that we want students to use whenever they are doing integration in a physical context.

The students should already know formulas for the volume elements in cylindrical and spherical coordinates. We recommend Scalar Surface and Volume Elements as a prerequisite.

We start the activity with the formulas $Q=\int\rho(\vec{r}')d\tau'$, $Q=\int\sigma(\vec{r}')dA'$, and $Q=\int\lambda(\vec{r}')ds'$ written on the board. We emphasize that choosing the appropriate formula by looking at the geometry of the problem they are doing, is part of the task.

Each student group is assigned a particular charge density that varies in space and asked to calculate the total charge. This activity is an example of https://paradigms.oregonstate.edu/whitepaper/compare-and-contrast-activity.

### Student Conversations

This activity helps students practice the mechanics of making total charge calculations.

• Order of Integration When doing multiple integrals, students rarely think about the geometric interpretation of the order of integration. If they do the $r$ integral first, then they are integrating along a radial line. What about $\theta$ and $\phi$. If this topic does not come up in the small groups, it makes a rich discussion in the wrap-up.
• Limits of Integration some students need some practice determining the limits of the integrals. This issue becomes especially important for the groups working with a cylinder - the handout does not give the students a height of the cylinder. There are two acceptable resolutions to this situation. Students can “name the thing they don't know” and leave the height as a parameter of the problem. Students can also give the answer as the total charge per unit length. We usually talk the groups through both of these options.
• Dimensions Students have some trouble determining the dimensions of constants. Making students talk through their reasoning is an excellent exercise. In particular, they should know that the argument of the exponential function (indeed, the argument of any special fuction other than the logarithm) must be dimensionless.
• Integration Some students need a refresher in integrating exponentials and making $u$-substitutions.

### Wrap-up

You might ask two groups to present their solutions, one spherical and one cylindrical so that everyone can see an example of both. Examples (b) and (f) are nice illustrative examples.

• assignment Total Charge

assignment Homework

##### Total Charge
charge density curvilinear coordinates

Integration Sequence

Static Fields 2023 (6 years)

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

• assignment Spherical Shell Step Functions

assignment Homework

##### Spherical Shell Step Functions
step function charge density Static Fields 2023 (6 years)

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://paradigms.oregonstate.eduhttps://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 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 Cube Charge

assignment Homework

##### Cube Charge
charge density

Integration Sequence

Static Fields 2023 (6 years)
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. On a different cube: 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 Gravitational Field and Mass

assignment Homework

##### Gravitational Field and Mass
Static Fields 2023 (5 years)

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: $$\vec g = \begin{cases} 0&\textrm{for } r<a\\ -G \,\frac{M}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ -G\,\frac{M}{r^2}\, \hat r & \textrm{for } r>b \\ \end{cases}$$

This problem explores the consequences of the divergence theorem for this shell.

1. Using the given description 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. Briefly 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. Briefly discuss how this example would change if you were discussing the electric field of a uniformly charged spherical shell.

• assignment Electric Field and Charge

assignment Homework

##### Electric Field and Charge
divergence charge density Maxwell's equations electric field Static Fields 2023 (4 years) Consider the electric field $$\vec E(r,\theta,\phi) = \begin{cases} 0&\textrm{for } r<a\\ \frac{1}{4\pi\epsilon_0} \,\frac{Q}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ 0 & \textrm{for } r>b \\ \end{cases}$$
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.
• assignment Charge on a Spiral

assignment Homework

##### Charge on a Spiral
Static Fields 2023 (2 years) A charged spiral in the $x,y$-plane has 6 turns from the origin out to a maximum radius $R$ , with $\phi$ increasing proportionally to the distance from the center of the spiral. Charge is distributed on the spiral so that the charge density increases linearly as the radial distance from the center increases. At the center of the spiral the linear charge density is $0~\frac{\textrm{C}}{\textrm{m}}$. At the end of the spiral, the linear charge density is $13~\frac{\textrm{C}}{\textrm{m}}$. What is the total charge on the spiral?
• assignment Potential vs. Potential Energy

assignment Homework

##### Potential vs. Potential Energy
Static Fields 2023 (6 years)

In this course, two of the primary examples we will be using are the potential due to gravity and the potential due to an electric charge. Both of these forces vary like $\frac{1}{r}$, so they will have many, many similarities. Most of the calculations we do for the one case will be true for the other. But there are some extremely important differences:

1. Find the value of the electrostatic potential energy of a system consisting of a hydrogen nucleus and an electron separated by the Bohr radius. Find the value of the gravitational potential energy of the same two particles at the same radius. Use the same system of units in both cases. Compare and the contrast the two answers.
2. Find the value of the electrostatic potential due to the nucleus of a hydrogen atom at the Bohr radius. Find the gravitational potential due to the nucleus at the same radius. Use the same system of units in both cases. Compare and contrast the two answers.
3. Briefly discuss at least one other fundamental difference between electromagnetic and gravitational systems. Hint: Why are we bound to the earth gravitationally, but not electromagnetically?

• assignment Central Force Definition

assignment Homework

##### Central Force Definition
Central Forces 2023 (3 years)

Which of the following forces can be central forces? which cannot? If the force CAN be a central force, explain the circumstances that would allow it to be a central force.

1. The force on a test mass $m$ in a gravitational field $\vec{g~}$, i.e. $m\vec g$
2. The force on a test charge $q$ in an electric field $\vec E$, i.e. $q\vec E$
3. The force on a test charge $q$ moving at velocity $\vec{v~}$ in a magnetic field $\vec B$, i.e. $q\vec v \times \vec B$

• group Electrostatic Potential Due to a Pair of Charges (without Series)

group Small Group Activity

30 min.

##### Electrostatic Potential Due to a Pair of Charges (without Series)
Static Fields 2023 (4 years) Students work in small groups to use the superposition principle $V(\vec{r}) = \frac{1}{4\pi\epsilon_0}\sum_i \frac{q_i}{\vert\vec{r}-\vec{r}_i\vert}$ to find the electrostatic potential $V$ everywhere in space due to a pair of charges (either identical charges or a dipole). This activity can be paired with activity 29 to find the limiting cases of the potential on the axes of symmetry.

Author Information
Corinne Manogue, Tevian Dray
Learning Outcomes