LAB 5: Induction: A Linear Generator

Transcription

1 1 Name Date Partner(s) OBJECTIVES LAB 5: Induction: A Linear Generator To understand how a changing magnetic field induces an electric field. To observe the effect of induction by measuring the generated current. To experimentally determine the induction of a coil. OVERVIEW Faraday s Law of Induction describes the induced voltage in a loop of conductive wire. This voltage is induced when there is a change in the magnetic field passing through the loop of wire, a change in flux. Flux is magnetic field per unit of area represented as: B represents the magnitude of the magnetic field, A represents the area of the loop, and cos(θ) represents the cosine of the angle the loop makes with the magnetic field. Thus the maximum flux is when the plane of the loop is perpendicular to the magnetic field, and the minimum flux is when the plane of the loop is parallel to the magnetic field. Voltage is induced when there is a change in the flux through the loop. This produces a current that opposes the change in flux through the loop. Using the right hand rule, one can determine which direction the current needs to flow to produce an opposing change in flux. It is important to note that the current does not necessarily oppose the direction of the magnetic field, but rather the change in flux of the magnetic field. For example, if a loop is oriented in the x-y plane and the flux is decreasing in the -z direction, then a current will be produced that increases the flux in the -z direction. The induced voltage, sometimes referred to as the electromotive force or emf is equal to the time derivative of magnetic flux: For a coil of wire, the voltage is simply the time derivative of the magnetic flux multiplied by the number of turns in the coil: This is because each turn of the coil is experiencing its own induced voltage, which add in series. Thus the more turns there are the greater the induced voltage. Coils of wire also have a property called inductance, represented by L, which describes a coil s response to a change in the current flowing through it. The voltage induced by an inductor when there is a change in current, sometimes referred to as the back emf, is the product of the inductance and the time derivative of the current through it:

2 2 Interestingly enough, the inductance of a coil can be determined entirely by the geometry of the coil. It follows from Faraday s Laws we can derive inductance in terms of its geometry. Using the definition of magnetic flux, we find (1) For a solenoidal coil, we know from Ampère s law that the magnetic field inside the coil is given by where is the length of the coil. Plugging this expression for the magnetic field back into the induction equation (1) yields So, we can write the induction of the coil in purely geometric quantities as Theoretical Calculation of Inductance You will first be determining the inductance of a coil based on its geometry. You will later compare this result to your experimentally calculated inductance. You will need a field coil and a ruler to measure the area and length of the coil. Indicate the values you measure below.. Coil Area: Coil Length: Number of turns in the Coil: Calculated inductance: (Pay attention to units!) Experimental Determination of Inductance You will be constructing a linear generator. Essentially you will be constructing an apparatus to drop a magnet through a field coil. As the magnet falls, it produces a changing magnetic field, and therefore a change in flux, through the field coil, which will induce a voltage and a current in the coil. You will be able to calculate the inductance by analyzing this voltage and current.

3 3 Equipment: Two vertical metal poles, placed in equipment holes on table surface Field coil with post Three right angle clamps Two pulleys and posts String Mass Hanger and Masses Neodymium Magnet with hook PASCO 850 data acquisition interface PASCO VI sensor Computer running Capstone and Graphical Analysis or some other graphical software Setup: Please refer to the diagram below as you set up the equipment. 1. Set up the upright poles and fasten the field coil to one horizontally using a right angle clamp. The plane of the coil should be horizontal so that the magnet will pass along the central axis of the coil. 2. Attach the pulleys near the top of the poles so string can move freely across the two. 3. Next connect the neodymium magnet to a string and fasten the other end to the mass hanger. 4. Drape the string over the pulleys with the mass hanger on one side and the magnet hanging over the center of the field coil. 5. Insert the VI sensor into the Pasco 850 and connect the Voltmeter to the positive and negative terminals of the field coil. Connect the Ammeter to the negative and positive terminals of the field coil as well. 6. On Capstone, set up the hardware to measure voltage and current versus time.

4 4 Activity 1-1: Induced Voltage and Current 1. As an initial test, take the magnet on the string and position it over the center of the coil. Start data recording, and rapidly lower the magnet at constant speed through the coil, trying to keep the magnet moving along the central axis of the coil. You can try lowering the magnet at different speeds to get the best results. In the graphs below, sketch the resulting voltage and current that you measure for one trial of lowering the magnet (or print them out and attach them to the lab): Voltage vs. Time Current vs. Time Question 1-1: On the graphs, indicate at what point the vertical position of the magnet was the same as the plane of the coil. In terms of changing flux, explain the shape and the relative signs of the different portions of the voltage and current graphs. For example, why is the voltage positive and then negative, or vice versa?

5 5 Activity 1-2: Measurement of the Coil Inductance If you haven t already, attach the string to the mass hanger so that the motion of the magnet and mass hanger are coupled. The smooth motion of the system should give better data. You may vary the mass on the mass hanger to change the rate at which the magnet falls. 1. Pull the mass hanger down until the magnet is suspended several centimeters above the field coil. 2. Start Pasco data collection run. 3. Release the mass hanger. 4. Once the magnet has stopped falling, end the data collection run. Activity 1-3: Analysis Capstone-based Analysis: 1. Go to Capstone s calculate tab and create a new calculation for the derivative of the measured current (This can also be done by hand, but that would take longer.). 2. Create another calculation: (Voltage V)/(Derivative of I); this will yield the induction as you can see from the formulae in the Overview. 3. Graph or tabulate the second calculation and average the values ignoring any extreme peaks. Attach your data or graphs to the lab write-up. Question 1-2: What numerical value do you get for the inductance using this method? How does it compare to what you calculated using the geometric quantities? Question 1-3: Do you have any explanations for what might have caused strange peaks in your data? Graphical Analysis Regression Analysis: 1. Copy your voltage data into Graphical Analysis 2. Select one full sine wave from the data 3. Use best fit line to fit a sine function to the dataset; record the resulting equation in the table on the next page. Attach a copy of the plot and fit to this lab write-up. 4. Copy your current data into Graphical Analysis 5. Select one full sine wave from the data

6 6 6. Use best fit line to fit a sine function to the dataset; record the resulting equation in the table, below. Attach a copy of the plot and fit to this lab write-up. 7. Find the derivative of the best fit line for the current 8. Divide the amplitude of the best fit voltage equation the amplitude of the best fit derivative of the current equation Best fit Voltage Equation Best fit Current Equation Derivative of Current Equation Amplitude of Voltage Equation Amplitude of Current Equation Inductance (Amplitude of V / Amplitude of di/dt) Question 1-4: How do your two values of inductance compare to one another? Question 1-5: How do they compare to the inductance you calculated based on geometry? Question 1-6: What possible sources of error or uncertainty could account for these differences? [There is no additional write-up required for this lab. You can add extra graphical material if you like.]