Magnetic Fields, Hall effect and Electromagnetic induction (Electricity and Magnetism) Umer Hassan, Wasif Zia and Sabieh Anwar LUMS School of Science and Engineering May 18, 2009 Why does a magnet rotate a current carrying loop placed close to it? Why does the secondary winding of a transformer carry a current when it is not connected to a voltage source? How does a bicycle dynamo work? How does the Mangla Power House generate electricity? Let’s find out the answers to some of these questions with a simple experiment. KEYWORDS Faraday’s Law · Magnetic Field · Magnetic Flux · Induced EMF · Magnetic Dipole Moment · Hall Sensor · Solenoid APPROXIMATE PERFORMANCE TIME 4 hours 1 Conceptual Objectives In this experiment, we will, 1. understand one of the fundamental laws of electromagnetism, 2. understand the meaning of magnetic fields, flux, solenoids, magnets and electromagnetic induction, 3. appreciate the working of magnetic data storage, such as in hard disks, and 4. interpret the physical meaning of differentiation and integration. 2 Experimental Objectives The experimental objective is to use a Hall sensor and to find the field and magnetization of a magnet. We will also gain practical knowledge of, 1. magnetic field transducers, 2. hard disk operation and data storage, 3. visually and analytically determining the relationship between induced EMF and magnetic flux, and 4. indirect measurement of the speed of a motor. 1 3 The Magnetic Field B and Flux Φ The magnetic field exists when we have moving electric charges. About 150 years ago, physicists found that, unlike the electric field, which is present even when the charge is not moving, the magnetic field is produced only when the charge moves. This discovery allowed physicists to learn interesting ideas about materials. In the twentieth century, scientists determined the configuration of elementary particles in atoms and they realized that electrons inside atoms also produce tiny magnetic fields. This field is found in all materials. The magnetic field is mapped out by magnetic field lines. Magnetic field lines are like stretched rubber bands, closely packed near the poles .This is why the closer we get to the poles of a magnet, the higher the magnetic field. The number of magnetic field lines passing through an area is known as magnetic flux φ. Mathematically, we divide an area through which we want to find the flux into identical area elements ~ perpendicular and away from the surface as shown in the margin figure. A scalar product between ∆A ~ and ∆A ~ is, the magnetic field vector B ∆A φ = B~1 · ∆A~1 + B~2 · ∆A~2 . . . , ∆A B ∆A Subsequently, we may also write φ= X (1) ~i · ∆A ~i B (2) i 4 Electromagnetic Induction Extensive work was done on current carrying conductors in the nineteenth century, major ground work was set by Faraday (1831) and following him Lenz (1834) [1]. Faraday discovered that a changing magnetic field across a conductor generates electric field. When a charge moves around a closed circuit this electric field does work on the charge. Like the electromotive force (EMF) of a battery this induced EMF is capable of driving a current around the circuit. V EMF registered in millivolts N North pole of bar magnet approaching a single loop S Figure 1: Principle of electromagnetic induction. Faraday’s law asserts that the EMF produced is directly proportional to the rate at which the magnetic field lines per unit area or magnetic flux ‘cuts’ the conducting loop. Lenz’s law is incorporated into Faraday’s Law with a negative sign which shows that the EMF produced opposes the relative motion between the conductor and magnet, it tries to resist the change in flux. 2 Mathematically both of these laws are expressed together as, ε=− dφ , dt (3) for a single loop of conductor, where ε is the electromotive force induced, φ is the magnetic flux. dφ dt is time rate of change of magnetic flux. The rate depends on the speed at which the magnet moves relative to the conductor loop, as well as the strength of the field. Electric power plants or more commonly; generators, are a physical manifestation of laws of induction. The principle is to change the magnetic flux over large stationary coils. The ‘change’ of flux is brought about mechanically, either by falling water or by running a turbine. The changing flux induces an EMF in the coils. 5 Q 1. What are the units of ε and φ? Q 2. Rewrite Equation 3 for N number of loops. How does the EMF depend on N? Solenoids Shown in Figure 2 is a coil of wire wound around a core. Magnetically it behaves like a bar magnet, producing a magnetic field when the current flows. It remains a magnet till the time current is flowing through the conductor. The mathematical expression for magnetic field generated inside an ideal solenoid is, B = µ0 nI, (4) where µ0 is the permeability in free space, value; 1.26 × 10−6 H/m, n is the number of turns of the conductor per unit length and I is the current through the conductor. The magnetic field B, is measured in Tesla. In our experiment we will use a changing magnetic field near a solenoid to N S Figure 2: Solenoid made from an enameled copper wire wound on a plastic pipe induce an EMF in it. This is the the Faraday effect! 6 The Hall effect Imagine a sea. There is a sea of electrons in a conductor. When we apply a potential this ‘sea’ flows from the higher to the lower potential. Further, if we place this conductor, in which current is flowing in a magnetic field the moving charges tend to interact with the applied magnetic field and also deflect. This deflection results in a potential difference across or perpendicular to the conduction path, know as the Hall voltage. 3 Thickness (T) (a) Current flowing due to applied EMF width (w) VH Hall voltage registered across the conductor when magnetic field is applied (b) Charges accumulating on the surface Direction of electric field due to accumulation of charges Direction of applied field Figure 3: (a) Shows electrons flowing through the conductor. (b) Shows some charges accumulating on the front and back surfaces generating Hall voltage across the width w . Figure 3 illustrates how moving charges are deflected due to the applied magnetic field. The magnitude of this force (FB ) is given by, FB = Bqυ, (5) where q is the charge and υ is the velocity. The build-up of charges on one side generates an electric field (E⊥ ) perpendicular to the current as shown in Figure 3. These charges continue to accumulate till the time force (FE ) due to electric field, FE = qE⊥ , (6) is equal to the force due to the magnetic field (FB ). Mathematically this equilibrium means that, FE = FB , (7) Bqυ = qE⊥ . (8) VH = E⊥ w , (9) or The voltage developed due to E⊥ is, where (VH ) is the Hall voltage and w is the width of the conductor. Combining Equation 8 and Equation 9 we get, VH = υw B. (10) We know that the average velocity of electrons in terms of current (I) is given by, υ= I , neA (11) where n is the volume density of electrons and A is the cross-sectional area, a product of width (w ) and thickness (T ). Combining Equation 11 and Equation 10 we obtain the Hall voltage in terms of applied magnetic field, BI . (12) VH = neT 4 The Hall effect is important in the study of materials, for example it helps us to find the number of conducting particles in a wire and their charge. In our experiment, this effect holds a central importance as we will use sensors developed using this principle to probe the magnetic fields generated by magnets. Read heads in tape recorders and magnetic disk drives utilize this principle too. Q 3. A strip of copper 150 µ m thick is placed inside a magnetic field B = 0.65 T perpendicular to the plane of the strip, and a current I = 23 A is setup in the strip. What Hall potential difference would appear across the width of the strip if there were 8.49 × 1028 electrons/m3 ? 6.1 Comparison between the solenoid and the Hall probe The Hall probe and the solenoid are both transducers, they convert one form of energy to another. Figure 4 shows that Hall probes generate a measurable potential which varies with the direction and magnitude of the field and flux. This potential is then converted to magnetic field using a simple relation provided by the manufacturer of the Hall chip. On the other hand, a solenoid, directly measures the EMF. The value of EMF, of course depends on the rate of change of flux being measured. However a major role is also played by the number of turns of the solenoid. Magnetic field Area Hall probe Flux Solenoid Potential drop No. of turns and area Calibration Rate of change of flux linkage Magnetic field B EMF Figure 4: A comparison between the operation of the solenoid and Hall probe. 7 Data Storage on a hard disk Computers are digital. So, every letter of every language must be stored or processed in computers in “digital form”; i.e. as a sequence of 0’s and 1’s. Computers use ASCII (American Standard Code for Information Interchange). It is a 7 bit code for all English alphabets, Roman letters and many other symbols. We will be using a similar scheme for a 5 bit code in our experiment. Consider the following table, Table 1, which shows a possible binary conversion of English alphabets into bits. 5 Letter a b c d e f g h i Binary Code 00000 00001 00010 00011 00100 00101 00110 00111 01000 Letter j k l m n o p q r Binary Code 01001 01010 01011 01100 01101 01110 01111 10000 10001 Letter s t u v w x y z Binary code 10001 10010 10011 10100 10101 10110 10111 11000 Table 1: Binary Representation of English alphabets 8 8.1 The Experiment Building a Hall probe Hall probes are used to measure magnetic fields. The output voltage of a Hall sensor is proportional to the magnetic field being measured. The measured voltage is then converted to magnetic field using a calibration scheme provided by the manufacturer of the Hall sensing chip. This calibration curve will be given to you in the lab. Hall probe sensor is shown in Figure 5. Vcc is provided to the Hall sensor using the Universal Serial Bus (USB) port. All USB ports have a 5 V regulated output, so we will be using USB port as power supply to the Hall chip. Vcc 1 GND 2 Vout 3 Figure 5: Hall Probe sensor 8.2 Magnetic Field of a Disk Magnet Now let’s map the field of a disk magnet using the probe you just built. The lab has provided you small disk magnets based on iron. F Q 4. Following the scheme in Table 2 below to find the output voltage on the probe as you move along the magnetic axis as shown in Figure 6. Make sure that the flat face of the probe is perpendicular to the magnetic axis. F Q 5. Plot a graph between magnetic field strength Bmeasur ed and distance. F Q 6. Using the above graph, write your observations regarding the change in magnetic field with respect to distance. 6 Hall probe Vary the position of the Hall probe along x x a l Disk magnet Magnetic Axis Figure 6: Schematic shows disk magnet field mapping using a Hall probe. Distance (mm) 0 2 . . Output Volatge (volts) Measured magnetic field Bmeasur ed (Gauss) Table 2: Mapping the field of a disk magnet. For voltage to field conversion use the provided calibration sheet. 8.3 Magnetization of a disk magnet Magnetic materials are made up of atoms which have magnetic dipole moment µ ~ B . These randomly aligned dipoles have a net magnetic dipole µ ~ if we sum over a volume V , mathematically, X µ ~= µ ~B. (13) ~ as We can now define magnetization M P µ ~B V ~ = M (14) For a disk magnet the expression for the magnetic field strength as a function of distance is µ ¶ µ0 M x + l/2 x − l/2 p p B(x) = , (15) − 2 (x + l/2)2 + a2 (x − l/2)2 + a2 where M is the magnetization of a disk magnet, x is the distance along the magnetic axis from the disk magnet, l is the thickness of the disk magnet and a is the radius [4]. The term in brackets needs some mathematical detail in which we will not delve. However it is important to tell that it is a geometrical term which is the result of an integral depending upon the dimensions of the magnet and solved over the distance at which we are measuring the field. For the sake of simplicity lets replace µ ¶ µ0 x + l/2 x − l/2 p −p 2 (x + l/2)2 + a2 (x − l/2)2 + a2 (16) with the geometrical function f (x), obtaining B(x) = Mf (x). 7 (17) The goal is to find the magnetization of the disk magnet using Equation 17 F Q 7. For this perform the following procedure. 1. Find the thickness of the disk magnet with vernier callipers. 2. Find the radius of the disk magnet. 3. Run Matlab, type, >> magneticfield; 4. The programme prompts to enter radius, thickness and distance from the magnet at which magnetic field was measured. 5. The Matlab code returns the value of f (x) that should be equal to B(x) M . 6. Using the values of Bmeasur ed and f (x), evaluate the value of M for each distance using Equation 17. F Q 8. What is the mean value of the magnetization and the standard error of the mean. F Q 9. What are the units of M? F Q 10. Curve fit f (x) and Bmeasur ed to a suitable fitting function using least square curve fitting technique and evaluate the value of M. 9 Hard disk operation Now we are going to simulate the operation of a hard disk. The experimental setup consists of a AC motor which rotates a disk. There is also a switch and a regulator, to control the speed, as shown in the schematic Figure 8. Magnets are placed over the disk at fixed positions. There are two coils in an AC motor; a main winding and an auxiliary winding. The capacitor inside the box is connected in series with the stator winding and in parallel with the rotor winding, there is a difference in phase provided when a AC current passes through the capacitor, this difference in phase allows a couple of force to act on the axel which makes it rotate. Bushing assembly Soft Iron core Coil Axle Figure 7: Shows dismantled motor of a fan. Main and auxiliary windings are not visible as the are taped together. There are two possible ways of placing a magnet, i.e., either place its north or south facing upward. When the north is upward we call it as 0 and when south is upward we call it as 1. As the disk 8 rotates the magnetic flux linking the solenoid with the Hall probe is changed with time. In order to see the induced EMF and magnetic flux we use the solenoid and the Hall probe respectively in different experiments. Probe or solenoid Probe post Magnets Side view Capacitor Top view Regulator Switch Figure 8: Setup of the hard disk experiment. Observing induced EMF and changing magnetic flux using a solenoid EMF 9.1 Time Figure 9: EMF induced in the solenoid as the magnet passes below the solenoid. F Q 11. Carry out the following procedure. 1. Make a solenoid approximately 8 cm in length. Note the number of turns. You have been provided with enameled copper wire. 9 Magnetic Flux Time Figure 10: Changing magnetic flux as measured by the Hall probe. 2. Align the solenoid with the magnetic field of the magnets. 3. Turn ON the power supply, the motor starts and disk starts rotating. 4. Now run the “solenoid.vi” file. 5. Click on the Run button, the data starts acquiring. 6. Now, observe the waveform graphs. 7. Observe the EMF on the graph labeled as EMF. 8. Observe the magnetic flux on the graph labeled as Magnetic flux. F Q 12. You will observe something similar to Figure 9. Explain the graphs you observed. Is there a mathematical relation between the graphs you see? (HINT: Use the concepts of differentiation and integration.) F Q 13. Now change the number of turns of solenoid and observe the induced EMF. Draw the figures and note down your observations in your note books. F Q 14. 9.2 What are your major conclusions? Observing magnetic field using Hall Probe Now instead of the solenoid we’ll use the Hall probe as a magnetic field transducer. The voltage across the Hall chip is proportional to the magnetic flux. Depending on the front(flat) or the back(round) surface or the direction of the field, voltage will either drop below or jump above the voltage when there is no field present. Figure 10 shows the corresponding flux from a typical experiment. F Q 15. Now carry out the following procedure to observe the behavior of the Hall Probe. 1. Place the Hall probe sensor close enough the magnets. 2. Turn ON the supply, the motor starts and disk starts rotating. 3. Now Run the “HallProbe.vi” file. 4. Click on the Run button, the data starts acquiring. 10 5. Now, observe the waveform graphs. 6. Observe the Hall voltage on the graph labeled as voltage. F Q 16. Note down your observations and inferences in your note books. F Q 17. What’s the difference you observe when using the Hall probe sensor from the solenoid? Write it down in your note book. F Q 18. EMF. How can we find the EMF from your Hall voltage observations? Sketch the curve of F Q 19. Can you come up with a method to measure the speed of the motor? Describe. Q 20. of you) 9.3 Can you find the angular speed of the disk motor? (HINT: All the information is in front Data Reading and Writing Operation This section illustrates how the data is read from a hard disk. F Q 21. Carry out the following procedure. 1. Place the magnets on the disk in some orientation, at the indicated positions. Such that it forms a letter. Your task is to identify what that letter is? 2. Align the Hall probe with the magnetic field of the magnets. 3. Turn ON the supply, the motor starts and disk starts rotating. 4. Now Run the “HallProbe.vi” file. 5. Click on the Run button, the data starts acquiring. 6. Now, observe the waveform graphs. 7. Observe the Hall voltage and the magnetic field. 8. Run Matlab and load letterRead.m file. 9. The stored letter is displayed. 10. Compare your result with the Table 1. Q 22. Change the orientation of the magnets and display a letter of your choice. References [1] Keith Gibbs, “Advanced physics, Second Eddition” (Cambridge, 1996) [2] Robert Kingman, S. Clark Rowland, and Sabin Popescu, “An experimental observation of Faraday’s law of induction”, Amer. J. Phys. 70, 595-98 (2002). [3] R.C. Nicklin, “Faraday’s Law– Quantitative experiments”, Amer. J. Phys. 54, 422-28 (1986). 11 [4] Martin Connors, “Measurement and Analysis of the field of disk magnets”, Phys. Teach. 40, 308-11 (2002). [5] Brad Hinaus and Mick Veum, “The Hard Drive: An Experiment for Faradays Law”, Phys. Teach. 40, 339-41 (2002). 12