Part 1) Faraday s Law OBJECTIVES In this part of the lab you will Use Faraday s law to predict the emf produced in a coil from a time-varying magnetic field Measure the emf produced in a coil for a time-varying magnetic field Faraday s law states that a changing magnetic flux will produce an electric field: d E dl=- B nda ˆ dt NC If you place a coil with appropriate orientation in the region of this changing magnetic field, the electric field will cause a voltage difference between the ends of the coil. In this lab you create a changing magnetic field by moving a bar magnet away from a coil, and measure the resulting potential difference (emf). Before you make the measurement, you will attempt to calculate what the emf will be. 1) Warm Up Problem Problem 1) The north pole of a bar magnet points toward a thin circular coil of wire containing 40 turns. The magnet is moved away from the coil, so that the flux through one turn inside the coil decreases by 0.3 T m 2 in 0.2 s. What is the average emf induced in the coil during this time interval? Viewed from the right side (opposite the bar magnet), does the induced current run clockwise or counterclockwise? Explain briefly. CHECKPOINT 1: Ask an instructor to check your work for credit. You may proceed while you wait to be checked off
2) Predicting the emf a) Take out the following items for this part of the lab: 1 compass 1 bar magnet 1 ruler or meter stick 1 coil of 1600 turns 1 PASport voltage sensor b) Quickly determine and record the magnetic dipole moment of the bar magnet in your kit. If you do not remember how to do this see Lab 9, Section 3 (you can do this with a compass and a ruler). If you examine your 1600 turn coil you will notice that the inside and outside diameters of the coil are rather different. Hold the bar magnet so the end of the magnet is just outside the coil, as shown below. This will be the initial position. You need to calculate the approximate magnetic flux through one turn of the coil, due to the magnet at this location, and then find the total flux through all the turns of the coil. Magnet Coil c) Determine the average area of one turn of the coil (read on for details). If you take the coil to be a square with internal width and height w and outside width and height W, then the average area is:
3 3 W -w A= 3 W-w Which is obtained by using the average value of a function, on an interval defined as: to 1 f = a-b a b 1 A = W-w 1 A = W-w f x dx W w W 2 x dx w 1 x A = W-w 3 3 3 W -w A = 3 W-w A x dx d) Determine the average magnetic field inside the coil from the bar magnet, by deriving a formula in the same way we derived the formula for average area. 3 W Recall that the magnitude of the magnetic field of a bar magnet on axis is: w The lower limit of the integral is l, the distance from the center of the magnet to the front of the coil. The upper limit of the integral is L, the distance from the center of the magnet to the far end of the coil. e) Determine the average magnetic flux through one turn of the coil in this initial situation. f) Determine the average magnetic flux through all turns of the coil. Now hold your magnet 30 cm away from the coil. This is your final position g) Determine the average magnetic field inside the coil from the bar magnet. h) Determine the average magnetic flux through one turn of the coil in this final situation. i) Determine the average magnetic flux through all turns of the coil.
You want to move the magnet rapidly away from the coil so that the change in flux is large. A typical time for rapidly moving your hand 30 cm is 0.05 s. There is a motion sensor attached to the TA s computer where you can check how fast you can move your hand if you wish. j) Use either the typical t = 0.05 s or the time you measured for t along with the change in flux from initial to final position, to predict the emf you should observe when you quickly move the magnet from near the coil to 30 cm away. CHECKPOINT 2: Ask an instructor to check your work for credit. You may proceed while you wait to be checked off 3) Measuring the emf a) Log on to the lab computer and double click on Faraday.ds located on laptop desktop. It should automatically start software known as Datastudio. b) Connect the PASport voltage sensor cables to the coil. c) Click START to start recording, and move the magnet back and forth a few times to see what happens. d) Move the magnet slowly. e) Describe qualitatively what you observe. f) Move the magnet rapidly. g) Describe qualitatively what is different in your observations. h) Now measure the peak emf in the coil when you move the magnet away rapidly several times (you do not need to stop your hand when it is 30 cm away). Start at the initial location you used in your prediction calculation. i) Draw a sketch of a typical peak with scale and units in your work space j) Compare your experimental value to your predicted value. k) Given the approximations you made in estimating the emf, are your measured and predicted values in good agreement? CHECKPOINT 3: Ask an instructor to check your work for credit. You may proceed while you wait to be checked off
Part 2) RLC Oscillating Circuits OBJECTIVES In this part of the lab you will: Write a VPython program to simulate a circuit consisting of a capacitor and inductor (LC circuit) Add resistance to your LC circuit program to make a RLC circuit Adjust the resistance in your RLC circuit to observe different circuit behaviors An inductor is a device which stores energy in a magnetic field. A capacitor stores energy in an electric field. If you wire the two devices together the potential energy in the circuit will oscillate between that in the electric field and that in the magnetic field. You will use VPython to simulate the behavior of one of these circuits (an LC circuit), then you will add in resistance (an RLC), due to a resistor or the intrinsic resistance in the circuit itself, to make your model into a more realistic circuit. 1) Warm Up Problems Problem 2) A circuit consists of a capacitor and in inductor. Given the that voltage change across a capacitor is Q di and the voltage change across and inductor is -L, C dt solve the energy conservation equation for di. Problem 3) To the above circuit, add a resistor in series, across which the potential change is -IR. Solve the new energy conservation equation for di. CHECKPOINT 4: Ask an instructor to check your work for credit. You may proceed while you wait to be checked off 2) Modeling a RLC Circuit with VPython a) Open a New IDLE window and enter the basic skeleton of a program, like that below. from visual import * from visual.graph import * from future import division # Define Constants # Initial Values
# Create Objects # Calculations Do the following in you constants section. You need to input the characteristics of the circuit which are the resistance, the inductance, and the capacitance. b) Set the inductance ( L ) of the inductor in your simulated circuit to be 0.052 H. c) Se the capacitance ( C ) of the capacitor in your circuit to be 1 F. d) Set the resistance ( R ) of your circuit to be 0 Ohms. You will run this circuit for a set amount of time so you need to define a time interval. e) Set your time interval ( deltat ) to be 0.005 s. Do the following in your initial values section. The initial state of your circuit will be the capacitor charged to 3 V and no current in the circuit. f) Set the initial charge ( Q ) to be the charge on the capacitor if it were charged to 3 V. g) Set the initial current ( I ) to be 0. h) Set the time ( t ) to 0. Do the following in your create objects section. You want to graph both current, which relates to the potential energy stored in the inductor, and charge, which relates to the potential energy stored in the capacitor. i) Create a graph called Qgraph with the following line of code. Qgraph = gcurve(color=color.cyan) j) Create a graph called Igraph. Igraph = gcurve(color=color.red) Do the following in your calculations section. The period of a RLC circuit is given by
T=2π LC k) Make VPython calculate and print out the period of your RLC circuit (in VPython the word pi is equal to the the value of pi). You want to watch the behavior of your circuit as time passes so you will need to use a loop containing update equations. l) Construct a while loop that will stop after 10 s (in VPython time). Do the following in the loop The current will change as time passes, so you need to enter a current update equation. m) Input the current update equation. This will be given by I=I+dI n) Put in the expression you found in Problem 3 of the Warm Up for di. As time passes, the charge on the capacitor will also change. o) Input the charge update equation given by p) Update the time Q=Q-Idt q) Add data to your graph with the following lines of code. Qgraph.plot( pos=(t,q)) Igraph.plot( pos=(t,i)) r) Run your code. i. Does the period on your graph match the period VPython printed out? If the periods do not match there is a problem with your code. You should see a graph of the amount of charge on the capacitor and the amount of current versus time. A capacitor stores potential energy in an electric field while an inductor stores potential energy in a magnetic field. What you are simulating is a circuit which transfers potential energy back and forth between the two circuit elements. This is like a weight on a vertical spring. That system transfers potential energy between two forms as well, gravity and spring potential. If each of these systems has no energy losses they will oscillate forever,
however, each one gives up energy as heat. The weight gives up energy as heat when the metal of the spring bends and also loses energy due to air resistance. The circuit gives up energy to heat through the resistance of the metal from which the circuit is made. ii. In your graph what is a noticeable relation between current and charge on the capacitor There are three types of behavior you should observe depending on your resistance. You have already seen the first one involving no resistance (undamped oscillations). s) Slowly increase the resistance of your circuit until you see the other two behaviors. If your resistance becomes too high the program will crash. t) Sketch a graph of all three behaviors in your work space. CHECKPOINT 5: Ask an instructor to check your work for credit.