Experiments in Electromagnetism

Transcription

1 Experiments in Electromagnetism These experiments reproduce some of the classic discoveries in electromagnetism, specifically those performed by Ampère and Faraday. These fundamental experiments were later generalised and famously summarised by Maxwell yielding the four equations that are commonly known by his name. In his derivation, Maxwell discovered that one additional term was necessary to add to Ampère s law in order to ensure full consistency between these equations. Later, when establishing the theory of Special Relativity, Einstein showed that the equivalence of apparently different electromagnetic phenomena were actually manifestations of the same physical process. Therefore the experiments you will perform have not only established electromagnetic theory, but also led to some surprising advances in modern physics. You may wish to review the relevant chapters of your textbook before attempting the experiments. Experiment 1: Parallel Plate Capacitor A capacitor is simply a pair of conductors that stores charge. The simplest configuration for a capacitor consists of two metal plates separated by a thin insulating layer. Figure 1 shows a capacitor with its plates connected by a wire, effectively short-circuited ensuring that the potential of both plates is the same. In this configuration free (electrons) and fixed (static positively charged metal atoms) charges are uniformly distributed throughout the circuit. Now if we add a battery supplying a voltage V, electrons are pulled from the positive terminal and pushed to the negative terminal. The charge distribution in the circuit now looks like the lower part of the diagram with an excess positive charge on one plate and excess negative charge on the other. This separation of charge across the insulating layer sets up an electric field in a direction opposite to that imposed by the battery trying to drive current around the circuit. When the two are equal there is no further flow of electrons. However the separation of charge represents stored electrical energy. If we removed the battery and replaced it with a short circuit electrons would flow around the circuit again (albeit in the opposite direction). This method of storing electrical energy is used in many applications, for example in a camera flash. Figure 1: Diagram showing charge distribution on an uncharged and charged capacitor. Blackett Laboratory, Imperial College London 73

2 The quantity of charge delivered to the capacitor plates is given by Q = CV where C is the value of the capacitor measured in Farads. The farad is a very large unit, so typically a capacitor would have values of pf or nf. The ability of capacitors to hold charge was discovered in the 18th century when early pioneers in electromagnetism inadvertently gave themselves electric shocks by touching a conductor that had been previously charged. In this lab experiment we will develop a quantitative and painless method for measuring capacitance and then investigate the properties of a capacitor. If a charged capacitor is suddenly connected to a resistor, current will flow from one plate to another through resistor R. Apparatus for performing this is shown in figure 2. Figure 2: Diagram showing apparatus for charging and discharging a capacitor. The current that flows through resistor R, will depend upon the voltage across the capacitor V. Recalling Ohm s law (V=IR) and the definition of capacitance above, we obtain: Recalling that current is the flow of charge in time we obtain the differential equation which if the capacitor has an initial charge Q0, has the solution Therefore, we can expect an exponential discharge from the capacitor with a time constant of 1/RC. Since Q=CV and we can measure voltage against time accurately with the oscilloscope, this will form the basis for measuring capacitance. Ultimately, the aim in this experiment is to determine the capacitance of the large parallel plate capacitor sitting on your desk, at different plate separations. By plotting a graph of capacitance against separation, it will be possible to estimate the universal constant ε0. Experiment 1a: Measuring the capacitance of a known capacitor: To develop your method for measuring capacitance, you will first measure the capacitance of a known ceramic capacitor, commonly used in electronic circuits. Rather than switching between a battery and resistor, as suggested in figure 2, you will use the function generator to perform this task. Use a 10V peak-peak square wave, and set the Offset dial so that the waveform starts at 0V and rises to +10V (rather than the default +5V to -5V). Set the frequency to approximately 5kHz and check the waveform using the oscilloscope. Your waveform should resemble that shown in figure 3a. (1) Now construct the circuit shown in figure 3b using the 470pF capacitor provided, and a resistor of approximately 100kΩ. Take care to ensure that the oscilloscope is connected with Blackett Laboratory, Imperial College London 74

3 the correct polarity. The negative terminals of the signal generator and oscilloscope are grounded, so must be connected with a common ground, as shown. On the oscilloscope you should observe an exponential rise and decay of the voltage across the capacitor. You may wish to use the second channel on the oscilloscope to superimpose the function generator signal over that measured across the capacitor 1. What effect does the frequency have on the trace that you observe? Try switching the resistor and capacitor, so that the oscilloscope now measures the voltage across the resistor. Why is this trace different? There are two methods that you can attempt for measuring capacitance. The first involves measuring the voltage across the capacitor and then using equation 1, which re-arranged and with appropriate substitution yields: V ln V 0 = t RC Taking measurements of V at various times t and measuring the peak voltage V0, will enable you to plot a straight line graph of ln(v/v0) vs t, which should have gradient -1/RC. Since you know the value of R, you can determine the value of C. Figure 3: (a) Pulse shape for charging and discharging the capacitor. (b) Circuit diagram showing the measurement of voltage across the capacitor. Alternatively, you can measure the voltage across the resistor and use this to determine the total charge that flowed onto the capacitor. Since the peak charge Q0 will correspond to the peak voltage V0 across the capacitor, the capacitance can be determined from: Q 0 = CV 0 where to determine Q0 it is necessary to take a series of current readings at various time intervals and integrate to give the maximum charge. Z T Q 0 = You will need to take a number of datapoints to obtain an accurate integral and you can choose the integration method, either trapezium rule or the simple graphical method of counting squares on standard graph paper. The accuracy of your value for Q0 and hence C will depend upon the number of datapoints you take and the integration method you use. 0 Idt 1 Blackett Laboratory, Imperial College London 75

4 Experiment 1b: Properties of the Parallel Plate Capacitor: Having established a method for measuring capacitance, you can now replace the ceramic capacitor with the parallel plate capacitor on the bench. From theory, we expect the capacitance to scale inversely with plate separation, it being related by the expression below. C = k 0A d where k is the dielectric constant, d is the plate separation, A is the plate area and ε0 is the permittivity of free space. Make a couple of measurements of capacitance at different plate separations and make a graph of capacitance against 1/d to determine the extent to which this expression holds. The dielectric constant of air is unity, so by making an estimate for the plate area, obtain a value for ε0 and compare it with the accepted value of 8.854x10-12 F.m -1 Experiment 1c (Extension) - Properties of dielectrics: On the lab bench, you will find some sheets of material. These have been chosen since they have different dielectric constants. Referring back to figure 1, we learned that charge will accumulate on the plates of a capacitor when an external voltage is applied. Placing a material between these plates can polarise the charge on this material which in turn attracts further charge to accumulate on the capacitor plates. The degree to which this occurs depends upon the charge density of the material and leads to a value for the dielectric constant > 1. Determine the dielectric constant of some of the materials on the bench and see if they correlate with known values. You should be aware that the dielectric behaviour of most materials only remains constant over specific frequency ranges. When driven by a continuously varying AC field, the charge will be continuously fluctuating and there will be certain frequencies where this frequency resonates with electronic transitions in the material. At these points the dielectric constant will vary abruptly, but away from these resonant frequencies, the dielectric constant will be largely frequency independent. You should bear in mind however that dielectric constants may be quoted at optical frequencies (10 14 Hz) and potentially quite different to those that you measure in your low frequency experiment. Blackett Laboratory, Imperial College London 76

5 Experiment 2: Ampère s Law The magnetic field associated with a flow of charge is described very generally by Ampère s law. In your electromagnetic lectures, you will consider the general form of Ampère s law and apply it to specific geometries. In this experiment we are only concerned with the magnetic field associated with electrical current travelling along a straight wire. In this case Ampère s law can be stated as: B = µ 0I 2 r where B is the magnetic field strength, I is the current, r is the radial distance from the wire and µ0=1.257x10-6 H.m -1 is the permeability of free space. The field pattern and geometry is illustrated above. Experiment 2a: Measuring the magnetic field associated with current flow. You will start by repeating work performed in 1820 by Ørsted who was among the first to notice that a compass needle was deflected by the passage of current along an adjacent wire. You will use the 2A power supply as a current source. Since the power supply can control either voltage or current, you should set the voltage at open-circuit to a large value, say 10V. This will ensure that the power supply limits the current flow, without restriction on the voltage necessary to achieve the specified current. Next, suspend a length of wire between the two retort stands provided. Place the compass on a stage below the wire enabling the B field produced by the wire to be determined from the deflection of the compass needle. You now want to position the compass so that it sits just below the wire (a couple of millimetres at most). This can be done using the plastic spacers to raise the compass above the stage. To achieve maximum deflection, you will want to arrange the experiment so that the magnetic field due to the wire (BF) is perpendicular to the Earth s magnetic field (BE.); the deflection angle can then be related to the relative magnetic field strength of the wire to the Earth s magnetic field. Once you ve decided on the direction, pull the wire taut, then wrap it round the clamp stand and tape it down. Ensure the wire is as straight and with as few kinks as possible. Having excess wire at each end is not a problem, so long as it is kept away from the point of measurement. Connect the wire through the multimeter via the 10A socket to the power supply. Turn on the power supply and set the current to maximum. You should observe a deflection of the compass needle. It may be necessary to gently tap the compass to encourage the needle to move. Experiment with your technique to find a method that yields consistent results. Turn the power supply off when you are satisfied that you can observe the effect. You now want to make some careful measurements of the deflection of the compass needle. Make a note of the initial angle of the compass needle and estimate the separation between the needle and the wire. Vary the current passing through the wire, ensuring to use the Blackett Laboratory, Imperial College London 77

6 maximum range available and record the deflected compass needle angle. For each measurement, tap the compass first to force the needle to move, as sometimes it gets stuck, and then measure the new needle position. Plot a graph of Tan(displacement angle) against current. If Ampère s law is obeyed you should obtain a straight line, the gradient of which will enable you to determine the Earth s magnetic field BE (with associated error). Experiment 2b: Measuring the magnetic pattern from the wire Ampère s law predicts that the magnetic field strength will scale with the inverse of the radial distance from the wire. You can measure this by using the same apparatus as above, but setting the current to maximum (2A) and varying the separation between the compass and the wire by removing the plastic spacers. If Ampère s law holds, a graph of magnetic field strength vs 1/r will yield a straight line. Consider carefully the measurement intervals for r since you will want roughly equal spacing of datapoints on a graph of field strength against 1/r. You can again estimate the value of BE and compare with your earlier value. Experiment 2c (Extension) - Magnetic Field of Coils: The magnetic field associated with a single straight wire is relatively weak, but can be enhanced by winding the wire into a coil; each coil effectively adding to the field strength of the last. Make a simple coil by winding wire around a pen or pencil and investigate the field strength and pattern by placing the compass at different locations around the coil. From your observations, try to establish a relationship between the number of coils and the magnetic field strength. Blackett Laboratory, Imperial College London 78

7 Experiment 3: Faraday s Law In 1831 Faraday performed what looks like a very simple experiment with a coil and bar magnet shown in figure 5. Faraday observed that a changing magnetic field induces an electric field, or in the case of this experiment, the motion of a permanent bar magnet results in a voltage appearing across the ends of a coil. We can express Faraday s findings in the simple expression: " = dφ dt where ε is the induced electromotive force (voltage) and dφ/dt is the rate of change of magnetic flux through a single loop. Figure 5. Schematic diagram showing Faraday s classic experiment The consequences of Faraday s experiments are profound, in particular the equivalence of moving either the magnet and coil which ultimately led Einstein to the foundation of special relativity some 74 years later 2. Einstein s work goes beyond what we can reasonably cover in a lab experiment but the experiment you are about to perform puzzled some of the finest minds of the 19 th century and ultimately led to one of the landmark achievements of modern physics. Experiment 3a: Repeating Faraday s experiment You will first reproduce Faraday s experiment shown in figure 5 using a coil, your oscilloscope and a permanent magnet. On the bench you will find two coils, a large 100 turn coil and a smaller 50 turn coil. Connect the smaller coil to the oscilloscope and try moving a permanent magnet through the coil. The time base setting is unimportant since you are simply observing the effect of the non-periodic motion of the magnet on the current in the coil. Observe the effect of moving the magnet faster or slower through the coil and does the orientation of the magnet matter? Experiment 3b: Electromagnet and Coil: To make a more controlled experiment it is convenient to apply your findings of Ampère s law to this experiment. We will replace the permanent magnet with an electromagnet and check that the relative motion of these two coils also obeys Faraday s law. Connect the larger coil to the power supply as shown in figure 6 and pass 2A of current though it. Try moving the two coils relative to each other and check that Faraday s law still applies. Next, with the coils arranged as shown in figure 6, observe what happens when you turn the power supply off. The current in one Figure 6: concentric coil arrangement with the primary coil connected to the PSU, the secondary coil connected to the oscilloscope. of the coils will drop abruptly and induce a large voltage momentarily in the second. You may be able to record a peak voltage on the oscilloscope but to obtain quantitative results, a more controlled means of controlling dφ/dt is required. 2 Introduction to Electrodynamics, D.J.Griffiths, 3rd Ed, Addison Wesley (1999). Blackett Laboratory, Imperial College London 79

8 The function generation provides a convenient means for controlling dφ/dt. Since we need to pass an appreciable current through the large, primary coil, the output of the function generator is fed into an amplifier and then passed into the coil via a 10Ω resistor 3. Set up the circuit shown in figure 7 and configure the function generator to ~5kHz, square wave, zero offset and about 1v pk-pk amplitude. You can use the oscilloscope to measure the output from the amplifier, the magnetic field produced by the primary coil and the induced voltage in the secondary coil. With the square wave, you should observe something similar to switching the power supply on and off, since dφ/dt is zero except when there are abrupt changes in primary coil current. The principle advantage of this arrangement is that the signal is now periodic and more easily measured on the oscilloscope screen. Switch to the triangle waveform and observe the waveform appearing across the secondary coil. Does the induced voltage that you measure correspond to what you would expect from Faraday s law? What is the effect of using a sine wave, does the shape of the waveform change? Experiment 3c: (Extension) - Investigating different magnetic cores Figure 7: concentric coil arrangement with the primary coil driven using a periodic signal supplied by the function generator. When measuring capacitance with the parallel plate capacitor, you observed that different dielectric materials can increase the ability of the capacitor to store charge. Similarly inserting a core into our coils, we can concentrate the magnetic field in the core. This arises since the magnetic field produced by the primary coil, serves to align the internal magnetic moment of electrons in the core material with the applied field. When the electrons align parallel to the field, the effect is called paramagnetism, when in opposition to the applied field the effect is called diamagnetism. Some materials exhibit persistent magnetisation parallel to the applied field and are called ferromagnetic. Inserting a core into the coil with a different magnetic material has the result that the magnetic field is concentrated in the core of the material. Inside the core (and therefore in its vicinity) the permeability in Ampère s law is no longer that of free-space but modified by the material to µ =Km µ0 The effect is small for paramagnetic and diamagnetic materials 4, but extremely large for some ferromagnetic materials where Km can exceed It is convenient to characterise materials using the magnetic susceptibility Χm=1-Km, some typical values are given in the table below. 3 Ω Ω 4 Blackett Laboratory, Imperial College London 80

9 On the bench you will find several different cores that you can insert into the smaller coil. Investigate the effect of inserting these different cores into your two coils. Do they behave as you would expect? If not, can you find an explanation for their behaviour? Returning to the method used to investigate Ampère s law, you can use the deflection of the compass needle as a means of assessing the magnetic field strength due to a coil wound around an iron core. A large nail is provided for this purpose. Try to estimate the magnetic field strength of your electromagnet with and without the core. Material Type Susceptibility Uranium Paramagnetic 4.0 x10-4 Aluminium Paramagnetic 2.1x10-5 Carbon Diamagnetic -2.1 x10-5 Copper Diamagnetic -9.7 x10-6 Iron Ferromagnetic 3000 Nickel-Zinc Ferrite Ferromagnetic 20-15,000 Blackett Laboratory, Imperial College London 81