# Faraday Loop Model

The EJS Faraday Loop Model shows a conducting loop in a time-dependent constant magnetic field. The model is initially incomplete and students must complete the model (by adding the appropriate equations). Ejs must be installed to complete the model. Information about EJS is available at: <http://www.um.es/fem/Ejs/> and in the OSP ComPADRE collection <http://www.compadre.org/OSP/>.

## Exercises:

1. Run the simulation. Notice that you get a plot of the external magnetic field as a function of time, the plot of the emf (red line) is constant. This is because Faraday's Law has not been programmed into the model. What do each of the buttons and sliders do? Measure the period of oscillation. Which button or slider is related to this? What is the relation?
2. In order to calculate the induced emf in the loop due to the changing magnetic field, what do you want the simulation to calculate? How should it depend on the parameters "r" and "angle" controlled by the sliders below the picture of the loop and the field?
3. Right-click on the simulation and Open EJS Model. Look at the Evolution page. Right now, the evolution page is simply:*

The Evolution page is labeled Derivative because the emf is found by calculating the time derivative of the magnetic flux:

emf = d(B•A)/dt

You are going to have EJS calculate the derivative by calculating:

emf = A•(ΔB/Δt)

for small time steps. Explain (in words, not equations) what ΔB is and how you could find it if you were given the following table of values of B as a function of time:

Your calculation of ΔB/Δt at various times:

t | ΔB/Δt
.01 |
.02 |
.03 |

4. One method for computing ΔB/Δt is to take the difference between the current value of B and the previous value of B and divide by the time step. Think back to Calculus I, what is the definition of a derivative? How is this similar to what you learned as the definition of a derivative in Calculus I?
5. You are going to use the method described in the previous step for the Evolution of the model. Notice that in the Evolution page, because BxOld is defined as Bx and then the time is increased using the expression t = t + dt, BxOld is indeed the previous value of Bx. Therefore, using emf = -(Bx-BxOld)/dt sets the emf equal to ΔB/Δt. Why?
6. Try inputing this code into the Evolution page and verify that it works as expected. Does it calculate the derivative? Check that it does the correct thing not only for sine waves, but square waves and triangular waves.
7. Once the model is working correctly, remove the message Incomplete Model from the plot: from the Evolution page set ModelComplete = true (or simply remove the line ModelComplete=false).
8. The model, however, is not finished. You have not taken into account the size of the loop (to calculate the area) or the angle between the loop and the field. Include this as well in your equation for the emf. (Note: To get a calculation of sin x, you will need to write the code as Math.sin(x).) Your final equation for the emf:

emf = ________________________.

7. Sketch the output for a triangle wave input and verify that the derivative is correct (check the slope of the triangular wave and the height of the square wave).
8. When the input is a square wave, why does the output have "spikes"?

*For those of you who have used the built-in ODE solvers in EJS before and are wondering why you can not simply use that, an ODE solver does a numerical integration, not differentiation. Why?

### References:

- Giancoli, Physics for Scientists and Engineers, 4th edition, Chapter 29 (2008).

### Credits:

The Faraday Loop Model and Exercises were created by Anne J Cox using the Easy Java Simulations (EJS) authoring and modeling tool.

You can examine and modify a compiled EJS model if you run the program by double clicking on the model's jar file. Right-click within the running program and select "Open EJS Model" from the pop-up menu to copy the model's XML description into EJS. You must, of course, have EJS installed on your computer.

Information about EJS is available at: <http://www.um.es/fem/Ejs/> and in the OSP ComPADRE collection <http://www.compadre.org/OSP/>.