Lab #18 Induction page 1 Faraday s and Lenz s Law: Induction Reading: Giambatista, Richardson, and Richardson Chapter 20 (20.1-20.9). Summary: In order for power stations to provide electrical current with minimal resistive losses, high voltage transmission lines are used. Since the resistive power drop across the length of the transmission line is proportional to I 2 R (where I is current in the line and R is the transmission line resistance), reducing the current and increasing the voltage produces the smallest resistive power losses. Typical power station voltages are hundreds of thousands of volts. However the appliances in your home can only take a few hundred volts, so the high voltage needs to both be decreased and converted into current. Step-down transformers are used to perform this task. In this lab, you will simulate the high voltage from the power station with a timechanging voltage (called an alternating current voltage or ac voltage) across a pair of Helmholtz coils. You will make a step-down transformer by inserting a solenoid inside the Helmholtz coils. The ac voltage will cause the magnetic field in the Helmholtz coil to change with time, which will induce both a current and voltage in the solenoid. In this lab, you will experimentally measure the magnitude and direction of the induced voltage. You will also determine by what factor the high voltage of the Helmholtz coils is steppeddown to produce the current and voltage in the solenoid. Note 1: Answers to all questions are to be given in full sentences, such that by looking at the answer, the reader knows what question is being answered without having to read the question. Any answers not written this way, even if they are correct, will receive 0 credit. Note 2: All voltages in this lab refer to root mean square voltages (or rms voltages). Please read the Alternating Current sections in your textbooks that describe what is meant by this term. Note 3: Please keep track of significant digits in all calculations. You will lose points for not having the correct number of significant digits on all answers. Pre-Lab Analysis In this lab you will have two coils, a solenoid and a set of Helmholtz coils. Throughout the lab, the Helmholtz coils will be called the primary coil, because of the attached power supply, while the solenoid will be called the secondary coil and will be unpowered. The solenoid will be at the center of the Helmholtz coils with both coil axes parallel to each other, as in Figure 1. Applying a voltage across the Helmholtz coils will cause a current to flow in the coils, producing a magnetic field inside the coils. This magnetic field passes through the center of the solenoid. As the voltage across the Helmholtz coils changes (AC voltage), the resulting magnetic field will also change with time. Since this changing magnetic field passes through the solenoid, the magnetic flux will vary with time, which will induce a voltage in the solenoid according to Faraday s Law. Lenz s Law determines the polarity of this induced voltage. Figure 1: Helmholtz coils and solenoid with axes aligned parallel.
Lab #18 Induction page 2 1.) A voltage is induced in the solenoid (secondary coil) as described above. a.) Relate (using Faraday s Law) the voltage induced in the secondary coils (Vs) to the DF number of turns in the secondary coils (Ns) and s Ë Á D t the rate at which magnetic flux passing through the secondary solenoid changes with time. (Look this up in your textbook.) [4 pts] 1 b.) Explain the meaning of the minus sign in Faraday s Law. [2 pts] 2 c.) Write an expression for the flux through one loop of the secondary solenoid (Fs) due to the magnetic field produced by the primary solenoid (Bp). Assume this magnetic flux is constant throughout the secondary solenoid and everywhere perpendicular to its cross-sectional area (As). (Again look this up in your textbook.) [4 pts] 3 It turns out through some complicated calculus, that the magnetic field produced by the primary (Helmholtz) coil (Bp) is related to the current through this coil (Ip which produces Bp). The units on Ip are Amps per turn and, if we assume the number of turns in the primary coil is Np with a coil radius of Rp, then B p = 8m 0 Á N p I Ë R p p If Ip is an AC current, that is Ip=Ip0sin(wt), then Bp will change with time similarly. The flux through the secondary coil (solenoid), which depends on the magnetic field produced by the primary (Helmholtz) coil, therefore, will change with time. 2.) The cross-sectional area of the secondary coil (As) does not change with time. a.) Show that the flux of Bp through one secondary coil loop can be written as: [4 pts] 4 F s = 8m 0 Á N p Ë R s I p = 8m 0 Á N p p Ë R s I p0 sin w t p b.) Next substitute F s = 8m 0 Á N p Ë R s I p into Faraday s Law and show that the induced p voltage across all Ns loops of the secondary coil (solenoid) can be written as: [4 pts] 5 V s = - 8m 0 5 3 N s N p / 2 R s DI p p Dt So what we have at this point is that, when the current in the primary coil changes with time, a voltage is induced in the secondary solenoid. However, to tell whether more or less voltage was induced in the secondary coil (Vs) in comparison to the primary applied voltage (Vp), we really need a formula that relates these two voltages to each other. To do this, we need to correlate the current in the Helmholtz coils with the voltage across them. The Helmholtz coils are inductors. To measure any voltage across an inductor, a time-changing voltage must be applied across the inductor. This time-changing voltage, in ( )
Lab #18 Induction page 3 turn, induces what is called a back-emf in the inductor. By experiment, the back-emf is proportional to the rate of change of the current caused by the applied voltage. By definition, the constant of proportionality is called the inductance (L) of the inductor. That is: V back = - { L D I p Dt in Henries where the minus sign indicates that the back-emf opposes the change in the applied current. If no resistors are present in the circuit, then the magnitude of the induced EMF will be the same as the magnitude of the applied voltage. Thus V applied = V p = L D I p Dt Substituting this equation into the one given in Question 2b relates the voltage applied to the Helmholtz coils to the voltage induced in the solenoid coils: V s = 8m 0 5 3 N s N p / 2 R p L Á s Ë V p In this lab, the quantities Vs, Vp, Ns, Np, µ0, As, and Rp are all directly measurable. The one quantity that is not is the inductance of the primary solenoid, L. 3.) How would you graph the measurable quantities so that the slope of the resultant line R p would be Á L? [8 pts] Ë 8m 0 N s N p A 6 s 4.) Table 1 contains data of voltage applied to the Helmholtz coils and the induced voltage measured in the solenoid. a.) Plot this data as you determined in Question 3 and fit a straight line to it. [5 pts] 7 b.) From the slope of this line, determine the inductance of the primary (Helmholtz) coils, if: the primary coils have 120 turns, the secondary (solenoid) coils have 300 turns, the cross-sectional area of the secondary coils is 40 cm 2, and the radius of the Helmholtz coils is 10 cm. Be sure to convert the units on your answer into Henries (the units of inductance in SI units). [8 pts] 8 Vp (V) Vs (mv) 0.9 179 2.2 441 3.1 622 4.0 799 4.9 950 Table 1: Voltage data for finding the Helmholtz Coil inductance.
Lab #18 Induction page 4 5.) Now suppose (as you will do in the lab) the voltage across the primary solenoid is increased to 10 V. a.) What will be the magnitude of the induced voltage in the secondary solenoid (given the same values as in Question 4)? [5 pts)] 9 b.) Would this be considered a step-up or step-down transformer and why? [6 pts] 10 6.) Outline the lab following the format of Outline Format posted on the Electronic Reserves web page. (20 pts) 11 Equipment to be used in this lab: 2 digital multimeters 1 power supply 1 solenoid 1 pair of Helmholtz coils A. Apparatus Dimensions needed for Calculations r In your lab notebook sketch the arrangement of the Helmholtz coils and Solenoid as you find them on your bench (should look something like in Figure 1). (3 pts) 12 Please do not move the solenoid it has been carefully placed to give the best results for this experiment. r On your drawing record the mean radius of the Helmholtz coils and the purple number marked on the base of the Helmholtz coils. (2 pts) 13 r Also on your sketch record all dimensions of the solenoid and the green number marked on the base of the solenoid. (3 pts: ) 14 r Table 2 contains the average number of turns in each Helmholtz coil and Solenoid. In your notebook record next to your Helmholtz and solenoid purple and green numbers, the actual number of turns in each coil. (2 pts: ) 15 Coil # Helmholtz Coils # of turns Solenoid # of turns 1 130 175 2 130 312 3 127 302 4 114 261 5 119 180 6 115 195 7 120 270 8 122 240 9 115 207 Table 2: Voltage data for finding the Helmholtz coil inductance. B. Equipment Set-up: Voltage Connections r The circuit will be pre-wired for you. Under no circumstances should you make connections with the power supply on. These voltages can be dangerous. r One multimeter will be attached to the Helmholtz coils and the other multimeter to the solenoid. Never exceed the listed voltages. Turn off the power immediately when not actually needed for the measurement. The power supply should be connected only to the Helmholtz coils. r Check that the alternating voltage Variac and step-down transformer are connected to the Helmholtz coil digital multimeter.
Lab #18 Induction page 5 r Check that this multimeter is connected to the Helmhotz coils as shown in Figure 2. Alternating current provides the time-changing current that causes the time-changing magnetic field, that induces the EMF in the solenoid. The voltage changes sinusoidally in time with a 60 Hz frequency. r Check that the second multimeter is connected across the solenoid in a parallel connection like the multimeter for the Helmholtz coils. r Check that the little black switch in the upper left corner of both multimeters is set to read AC. r Turn the large knob in the middle of the Helmholtz multimeter to the 20 V scale. r Set the main knob on the solenoid multimeter to the 2 V scale. Helmholtz Coils Voltmeter AC supply = Variac + step-down tansformer Figure 2. Helmholtz coil wiring diagram. r Do not exceed 10 V applied to the Helmholtz coils to avoid damaging the Helmholtz coils. r Do not touch either set of coils when current is flowing to the Helmholtz coils to avoid electrocuting yourself or your partner. C. Taking the Data r Make sure the big dial on the Variac is turned completely counterclockwise. r Turn on the Variac by flipping the toggle switch up. r Slowly increase the voltage (big dial) until the Helmholtz voltmeter begins to register 0.5 V. Record the precise value in your notebook. r Record the voltage induced in the solenoid. r Increase the Helmholtz voltage in 0.5 V steps up to 5 V and record the Helmholtz and solenoid voltages each time. [25 pts] 16 D. Analyzing Your Data to find the Helmholtz Coil Inductance r Using Excel, plot your data as you did in the Prelab. (5 pts) 17 r Fit a linear Trendline to your data and print your fitted plot. (4 pts) 18 r From the slope of the fitted line, find the Helmholtz coil inductance. In your notebook, on the same page as your attached plot, show your work for determining this inductance and circle your answer. (Don t forget units.) (11 pts) 19 E. Predicting the Induced Voltage for a Different Applied Voltage r Given this inductance, what should be the induced voltage in the solenoid if the rms voltage applied to the Helmholtz coils was made 8 V? Show all your work and keep track of units in your calculations. (7 pts) 20
Lab #18 Induction page 6 F. Checking your Prediction r Apply 8 V to the Helmholtz coils and record the solenoid s induced voltage. (6 pts) 21 r Compare your measured and predicted values. If you end up with a 0% or greater than 30% difference, figure out what s wrong and redo the measurements. (5 pts) 22 7.) Cleaning Up r When you are finished with your experiments and calculations, turn both multimeters OFF. Make sure all the power connections are correct as you used them in the lab. r Call your TA over to check that everything is wired correctly. r Once your TA has checked everything, you may leave. [5 pts] 23