Learning Scenario - Fields and Sheets (Excel)

Basic Model:

Description

Scientists will often use a scalar model to demonstrate the interactions between two charged particles. This model uses a similar method with varying colors to depict the resulting vectors from the electromagnetic interactions between two particles. Users can change the charge and placements of each of the particles in order to see the effects of size and separation have on the particles' surrounding charge. Particle charges are broken up into x and y components, both of which are displayed on their own worksheet. When these two are superimposed, the resulting vector is displayed on the main screen.

Background Information

Electromagnetic fields are broken up into two components, the electrical and the magnetic waves. These waves work at right angles from each other. If the electric field were a vertical wave, the magnetic wave would be horizontal. When the electromagnetic field from one particle interacts with another, the resulting vector field is the sum (or difference) between the two waves. If the two particles attract each other, the charges will be summed, and if they repel each other, they will be subtracted from each other. The model assigns different colors corresponding to the magnitude of charges. With this guide, it is easy to visualize the summation or subtraction of electromagnetically charged particles.

Science/Math

The fundamental concept behind this model is HAVE = HAD + CHANGE. For each time the simulation is run, the following things are calculated and recorded:

1. The zoom of the graph is set according to the user-input "gridspace"
2. The particles are plotted on the graph according to the user-input coordinates
3. The magnitude of the charges is calculated based on the user-input charge for each particle
4. The x and y components of the field are calculated and graphed on their own sheets
5. The two components are combined to find the resulting total magnitude

The two particles in this situation are completely dependent upon the preset variables. CHANGE, therefore, does not actually happen in real life situations the way it happens in the model, but the CHANGE does allow for a study between multiple particles of different magnitude and positions.

Teaching Strategies

An effective way of introducing this model is to study vectors in relation to things other than charged particles. A review on electromagnetism could also be helpful, especially on the two components of an electromagnetic field, electricity and magnetism. Ask the following questions:

1. What are the two components to an electromagnetic field? How are the two related to each other?
2. What do you think would happen if two electromagnetic fields were to interact with each other?
3. How would the magnitude of the waves be changed because of the interactions? What would this depend on?
4. Why do you think a visual depiction of the interactions would be more helpful than a scalar representation, which is traditionally done? Explain.

Students should write down their answers and compare them to their results after using the model.

Implementation:

How to use the model

This advanced model has a few variables that may be changed to study how the different sizes and positions of charged particles can affect each other and how they are viewed:

1. The charge1 variable determines the charge of the first particle, while charge2 does the same for the second
2. The x_1, y_1, and x_2, y_2 variables determine the positions of the two particles, respectively
3. The "gridspace" variable may be changed to view the model at different zoom levels

All of these parameters may be changed by simply typing the value into the corresponding cells. The charge1 variable may be changed by moving the slider next to it as well. Any changes will be instantly calculated and applied to the model.

The calculations are run and the data points are recorded in the table on each page. The graph itself is shaded based on the resulting magnitude between the two charges. Both x and y components may be viewed on their respective worksheets with their graphs.

Learning Objectives

1. Understand the x and y components of two charged fields and the variables' effect on their shape
2. Understand the interactions between charged particles and resulting vector fields

Objective 1

This objective will be best accomplished by starting out with separated fields. The second particle's charge should be set at 0 and the position of the first's put in a place where it can be easily studied. Ask students to manipulate the variables and compare the first field to the x and y components on the second and third worksheets. The following questions will guide the students in their discovery:

1. What is the overall shape of the field magnitude? Compare this to the shape of the x and y components. How and why do these differ?
2. Change the position of the particle. Does this have any effect on the shape and magnitude of the field? Why or why not?
3. Add back in the second particle and make sure that the two are decently close together. Why does the shape change?
4. View the colors around the particles. How do they change as the two grow closer together? Why? What does this imply?
5. Move the particle around the other. How do the colors and charges change this time? What is the relationship between magnitude and the distance between the two particles?

Objective 2

Magnitudes are calculated by using the formula M=(E_x^2+E_y^2). This is how the main worksheet is calculated, which is based off of the x and y sheets. The components are subtracted from each other to find the vector charge, which is represented in a scalar graph. Students should understand the calculations that work behind the model and the real life application of them. Ask the following questions:

1. How do the colors change as the two particles grow closer together? Study the numbers that correspond to each color. Does it seem like the numbers are increasing or decreasing? Why do you think this is?
2. Switch to the E_x sheet. Do the two particles show different magnitudes, even if the particles have the same charge? Explain.
3. Switch to the E_y sheet. Are there any similarities between this graph and the E_x graph in terms of charges? How can you see both of the charges represented in the E-field graph?
4. Move one particle so that it is close but not touching the second particle. Is the field a perfect sphere, or is there some deformity? How could this happen if the two fields do not seem to be close enough to interact?

Extensions:

1. Understand how electricity and magnetism are related to each other
2. Research the application of electromagnetic waves to cancer research
3. Understand the 3D nature of electromagnetic fields and how this model fails to accurately portray certain aspects of electromagnetic geometry

Extension 1

As mentioned above, electricity and magnetism are located at a 90-degree angle from each other in an electromagnetic wave. The two are more related than that, though. In fact, the movement of electrons along the electricity plane creates the magnetic waves. Apply this concept to the model and ask the following questions:

1. 1. How are electricity and magnetism related? What does the movement of electrons create?
2. 2. With this knowledge in mind, how are the graphs of E_x and E_y related? Does their similar shape make sense? Is it possible for them not to be the same shape? Explain.

Extension 2

Electromagnetic waves are used for many different purposes. Students may already know about electromagnets, but electromagnetism will soon be harnessed for medical purposes. Some researchers have been using electromagnetism in an attempt to treat malaria. Have students research this application and tie it to the knowledge of electromagnetism that they already know. See the source link below for more information. Ask the following questions:

1. How is electromagnetism being used to treat and eradicate malaria? Why would doctors want to move towards electromagnetism and away from traditional medication?
2. What makes this technology so effective against the parasites?

Additional Materials: A pdf

Extension 3

While the graph for the Fields and Sheets model depicted the electromagnetic field as being a two-dimensional, in reality it is three-dimensional in a sphere-like shape. Electricity is found on one axis, magnetism on the other, and as they move across space they follow the z-axis. Students should understand the shortcomings of the model being only two-dimensional. Ask the following questions:

1. What is the actual shape of an electromagnetic field? How does the model display the field?
2. How are electricity and magnetism connected in a three-dimensional field?
3. What are some things that the model cannot depict, since it is only two-dimensional? How would you improve the model?

Supplemental Materials:

1. Random Number Algorithm
2. Fourmilab Hotbits
3. Random Number Generator

Related Models

• Bunny Hopping Model
  

  The Fields and Sheets model used a method of displaying different colors to show varying charges. This same method is applied to other fields than physics as well. For example, geographers use the same scheme of applying different colors to label a certain elevation. Color-coding was important in the Fields and Sheets model to understand the interactions between the two particles. In the Bunny Hopping model, it is used to understand trends in bunny dispersion and clumping. The model connects mathematics and geography to teach concepts relating to functions, such as relative and absolute extrema.

• Malaria Epidemiology
  

  Electromagnetic waves may be used one day to treat malaria. In order to use this technology, doctors must have an idea of how the disease spreads. This model will allow students to study a hypothetical village that is stricken by malaria. The interactions between sick and healthy people and mosquitoes are graphed and recorded. Extension 2 would be a good complement and introduction to this model.