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Physics Laboratory Manual PHYC 20080 Fields, Waves and Light Name................................................................................. Partner’s Name ................................................................ Demonstrator ................................................................... Group ............................................................................... Laboratory Time ............................................................... 1 2 Contents Introduction 1 1 Waves and Resonances 7 2 Interference and Diffraction 13 3 Studying Sound using an Oscilloscope 23 Measurement of the Focal Lengths of Lenses and a 4 Determination of Brewster’s Angle 31 5 Magnetic Breaking 37 6 Determination of the Resistivity of a Metal Alloy using a Wheatstone Bridge 43 3 Laboratory Schedule Students will be assigned to one of a number of groups and must attend at a specified time on Tuesday or Thursday afternoons. Check the notice-boards in physics to find out which group you are in. In total you will perform six experiments during the semester. Room 127 Wk 1 (Sept 12-17) Wk 3 (Sept 26-30) Wk 5 (Oct 10-14)th Wk 7 (Oct 24-28) Wk 9 (Nov 7-11) Room 131 Room 132 Magnetic Breaking Lenses Resistivity Waves Interference /Diffraction Wk 11 (Nov21-25) Sound 4 Introduction Physics is an experimental science. The theory that is presented in lectures has its origins in, and is validated by, experimental measurement. The practical aspect of Physics is an integral part of the subject. The laboratory practicals take place throughout the semester in parallel to the lectures. They serve a number of purposes: • • • an opportunity, as a scientist, to test the theories presented in lectures; a means to enrich and deepen understanding of physical concepts presented in lectures; the development of experimental techniques, in particular skills of data analysis, the understanding of experimental uncertainty, and the development of graphical visualisation of data. During the laboratory practicals, students will be paired off with a partner with whom they perform the experiment and collect the data. In order to make optimal use of your time in the laboratory, please read the manual and familiarise yourself with the experiment before the laboratory practical. This manual includes blanks for entering most of your observations. Additional space is included at the end of each experiment for other relevant information. All data, observations and conclusions should be entered in this manual. It is advisable to use a pencil when entering data in case of errors. Calculations are best performed in rough before entering the salient details in the space provided. Graphs may be produced by hand or electronically (details of simple computer packages will be provided) and should be secured to this manual on one of the blank pages at the end of each experiment. Your work will be collected and marked by your demonstrator and will contribute 23% of your final mark. Before leaving the laboratory, ensure that your lab demonstrator signs and dates the data you have taken. 5 Summary on combining uncertainties Suppose you calculate f which is some function of variables x and y. Both x and y have associated uncertainties δx and δ y respectively. δf = So long as x and y are uncorrelated, What is the uncertainty δf on f ? ∂f ∂f δx ⊕ δy. ∂x ∂y This expression is true in general, but let us summarise the three most usual cases. Case 1: f=Ax You just multiply x by a constant A; then δ f = Aδ x Case 2: f=x+y or f=x-y You add or subtract two independent measurements; then δ f = δx ⊕δy Just add the errors in quadrature! (Short cut: If one error is at least twice as big as the other, you can ignore the smaller error. So in the case that δ x > 2δ y then δ f ≈ δ x ) Case 3: f=xy or f=x/y δf You multiply or divide two independent measurements; then f = δx x ⊕ δy y Just add the fractional errors in quadrature! (Short cut: If one percentage error is at least twice as big as the other percentage error, you can ignore the smaller. So when δx x > δy y then δ f ≈ f δx ) x An alternative way to proceed An approximate, “quick and dirty” way to work out the errors is to move all the values to their upper bounds and calculate your answer. Then move them all to their lower bounds and calculate the answer. From this you know the likely spread in your answer. 6 Waves and Resonances Introduction This experiment produces waves in a stretched wire and looks at how the frequency of vibration is related to the tension of the wire, its length and the mass per unit length. The phenomenon of resonance is used in the investigation. Using Newton’s laws, it can be shown1 that the velocity, v, of a symmetrical pulse on a string, as shown on the right, is related to the tension of the string, T, and the mass per unit length, µ, by v= T µ (Eq. 1) Thus the velocity of a wave along a string depends only on the characteristics of the string and not on the frequency of the wave. The frequency of the wave is fixed entirely by whatever generates the wave. The w wavelength avelength of the wave is then fixed by the familiar relationship: v = fλ (Eq.2) Example: A string on a bass guitar is 1m long and held under a tension of 100N. If the string has a mass of 10g, what is the velocity of a wave on the string? stri 1 The portion of the wire a net vertical force of ∆l experiences a tension on each end. 2T sin θ ≈ 2Tθ = T∆ l R acts. The horizontal components cancel and Newton says this force = mass times µ acceleration. The mass is ∆l . Looking at the system from a frame where the pulse is at rest (and the wire moves with velocity v), ), the portion of wire moves along the indicated circle and the centripetal acceleration is v2 R . Thus T∆ l R = ( µ∆l )(v 2 R ) and the result follows. 7 If a wave with a frequency of 80Hz is sent into the string, what will its wavelength be? If a wave with a frequency of 50Hz is sent into the string, what will its wavelength be? A resonant frequency is a natural frequency of vibration determined by the physical parameters of the vibrating object. There are a number of resonant frequencies for a string which are the multiples of its length that allow standing waves to be formed. Thus λresonant = 2l / n . where n=1,2,3... What is the lowest resonant frequency (the fundamental) of the bass guitar string referred to above? What happens if a wave with a frequency which is the same as the resonant frequency enters the string? _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 8 What happens if a wave with a frequency which is different to the resonant frequency enters the string? _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ Experimental Procedure In this experiment we will attempt to confirm Eq.1 above by sending waves of different frequency into the wire and identifying the resonant frequency. The apparatus is shown above and consists of a stretched string held under tension by a hanging weight. In this configuration, the tension equals the force exerted by gravity. The wire passes over a bridge which defines a node thus changing the length in which a wave can resonate. A small horseshoe magnet should be placed half-way between the bridge and the pulley. An AC generator is connected to either side of the wire and can send electrical signals of selected frequencies down the wire. Choose eight different positions of the bridge (remember to move the magnet too), and find the lowest frequency at which maximum vibrations (resonance) occurs. This can be observed by placing a folded piece of paper on the wire and noting when it gets thrown off by the vibrations. Record your data below. 9 Wire tension (N) Resonant Frequency (Hz) Length (m) Mass/unit length µ= Eq.1&2 can be combined to give f =1 T λ µ . When this is at the resonant frequency f resonant = n 2l T µ (Eq. 3) Plot the resonant frequency against 1/l. What should the slope be equal to algebraically and numerically? What is the slope of a straight line fit to your data? 10 1/Length (m-1) Comment on whether you consider Eq.3 has been verified and whether a good agreement exists between theory and experiment. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ Now place the bridge in a fixed position. Vary the tension in the wire by adding or removing weights, and find the lowest frequency at which maximum vibrations occur. Length (m) Resonant Frequency (Hz) Frequency squared (s-2) Mass/unit length µ= Plot the square of the resonant frequency against the tension. What should the slope be equal to algebraically and numerically? 11 Wire tension (N) What is the slope of a straight line fit to your data? Comment on whether you consider Eq.3 has been verified and whether a good agreement exists between theory and experiment. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ If you have time, set up a resonant position and write down the frequency. Now increase the frequency until you see resonance again, and write down this frequency. What do you notice? With reference to Eq.3, explain what is happening. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 12 Interference and Diffraction Introduction The aim of this experiment is to investigate the interference of light. Shining light through a single slit or a double slit produces characteristic diffraction patterns whose features depend on the wavelength of the light and the size of the slit. When light passes through a slit, or around an obstacle, diffraction effects are produced if the dimensions involved are of the same order of magnitude as the wavelength of the light. If the diffraction ion pattern is viewed on a screen a set of alternate bright and dark fringes will be observed. The diffraction can be of two types (a) Fresnel, - where the object to screen distance is comparable to the object size and (b) Fraunhofer, where the light is parallel rallel and the distances involved are large. This is the easiest to treat mathematically. Investigation 1: Single Slit Diffraction The figure above shows the intensity pattern as a function of angle when a single slit of width a is placed in the path of a beam of light. Three different slit widths are shown. From left to right they correspond to a size 10 times larger than the wave length (λ) of the light, 5 times larger, and the same size as the wavelength of the light respectively. The width of the pattern depends on the width of the slit; as the slit width is reduced, the pattern broadens. 13 A schematic of the experimental arrangement is shown above. It can be seen that the minima of the patterns satisfy (Eq. 1) asinθ = mλ where θ is the angle the fringe makes with the beam axis and m is the order number. For the 1st minimum, m=1 and since θ is small, sinθ ≈ tanθ = Y / D where Y is the distance from the central maximum to the first minimum and D is the distance from the pattern to the slit. Thus a = λ D/Y (Eq. 2) Apparatus A laser is used as the light source since it provides a source of monochromatic coherent light. The light passes through the slit and onto a screen. Explain why we use monochromatic source. a coherent and _______________________________________ _______________________________________ _______________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ 14 Procedure Align the laser beam so that it passes through the slit and strikes the viewing screen. Make the distance, D, from the slit to the screen as large as possible in order to maximise the size of the image and reduce your measurement error. The diffraction pattern should look something like the figure here. Identify the central maximum and the first order minima around it. The distance between these two minima is 2Y. Given that the wavelength of the light is 692.8nm, you should work out the width of the slit. 1. 2. 3. Record your data below and find a value (with an uncertainty) for the width of the slit. Is it possible to measure the slit width using the second order minima? If so, what value do you get and how does it compare? If time permits, repeat these measurements for a second slit. ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ 15 ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ 16 You can use this technique to measure very small distances with high precision. If an object which presents a rectangular aspect to the beam (such as a wire or human hair) is placed in front of the laser, a diffraction pattern will be produced identical to that produced by a rectangular aperture of the same dimensions. Find the width of a human hair. Record your observations below. ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ 17 ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ 18 Investigation 2: Double slit Diffraction (Young's slits) If a double slit is placed in the path of the beam, a Young's slits interference pattern is obtained, which is superimposed on the single slit diffraction pattern as shown in the diagram above. If d is the separation of the slits, constructive interference (a bright fringe) is obtained when nλ = dsinθ (Eq.3) When the angle is small this can be written as nλ λ=dθ (Eq. 4) Therefore the angular separation between adjacent maxima is λ/d = y/D (Eq. 5) where y is the linear seperation between adjacent maxima. Procedure Replace the single slit by the double slit. Move the slit carrier sideways until the laser light passes through one slit ,when you will get a single slit diffraction pattern on the screen. As you continue to move the slit carrier, the laser light will pass through both slits and the diffraction will break up into an interference pattern. Use Eq.5 above to estimate the slit separation. To do this, you will need to measure the separation between the adjacent maxima which are very close together. To increase your precision therefore, measure a number of bright fringes and divide by the number of separations. (Careful: 2 fringes delineate just one gap.) Record your observations below and make an estimate of the slit separation. 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______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ 21 22 Studying Sound using an Oscilloscope Introduction Sound is a longitudinal wave that propagates in solids, liquids or gases. In this experiment you will measure the speed of sound in air by first measuring the frequency, f, using an oscilloscope loscope and then using a combination of the resonance tube and the oscilloscope to measure the wavelength, λ, of the sound. The product of these two quantities gives the speed of sound in air, c,, as shown by the equation: c = fλ (Eq. 1) As sound passes a point in air, the pressure rises and falls in an almost periodic fashion. For this experiment we will assume that the pressure changes are periodic. The sound is generated by a tuning fork. When the tuning fork is struck off the bench tthe ends vibrate at a fixed frequency. As the ends vibrate, they alternately compress and rarefy the air next to them, setting up a pressure wave in the air. The pressure wave is picked up by the microphone which acts as a transducer, changing the sound ound signal into an electrical signal. This can be displayed on the oscilloscope and the frequency directly measured. 23 The Oscilloscope An oscilloscope is a device for looking at electronic signals that may change in time. It is a bit like the zoom lens on a camera: you can change the magnification on both the time and voltage axes and see the signal in more detail. But changing the magnification (or gain, to give it its proper title) does not change the signal itself. The oscilloscope consists of a cathode ray tube (a TV tube) and four main electronic circuits to control the screen all housed in one unit. We will now look at how to use an oscilloscope; for details of its internal workings see the appendix . The X-axis on the oscilloscope screen is the time axis; the Y-axis is a voltage axis. The trace that appears on the screen may be measured by counting the number of divisions between features e.g. peaks. In this case there are 4 divisions between peaks, and the oscilloscope is set to 2 ms per div. With reference to the image here: What is the period (T) of the wave? What is the peak-to-peak voltage? From the value you have obtained for the period of the wave, you can calculate a frequency using the relationship f = 1/T. What is the frequency of the signal displayed in the image above? (Note: convert the period to seconds before calculating he frequency). 24 The oscilloscope can display waveforms for two signals which are input on one of two channels (CH1, CH2). You can alter what you see using the controls on the front panel. 1 2 3 Position Control Input AC-GND-DC 4,5 6 7,8 9 10 11 12 Volts/Div Mode Volts/Div AC-GND-DC Input Position Control Pilot Lamp 13 14 15 16 Power Switch Intensity Control Focus Control Ext. Trig 17 18 19 20 21 22 23 Source Sync Level Pull Auto Switch Position Sweep Time/Div Cal 1Vpp Controls vertical position of spot for CH1 Input jack for CH1 If set to AC, only the AC component of the signal is displayed. If set to GND, gives the zero voltage level. If set to DC gives AC signal + any DC offset. Controls vertical attenuator for CH1 select whether CH1,CH2 or both are displayed As for 4,5, but for CH2 As for 3, but for CH2 As for 2, but for CH2 As for 1 but for CH2 Light when power switched on. Note, after switching off, light remains on for a few seconds. Switches on the unit Adjusts the brightness of the spot on the screen Brings the trace into focus Connect to external trigger source when the source switch (17) is EXT Trigger for sweep is internal (INT) or external (EXT) Determines point on waveform where sweep starts Sweep generated without a trigger Controls horizontal position of trace Horizontal sweep time selector. Provides 1V square wave pulse for calibration 25 Measuring the frequency of the sound Turn the calibration knobs on the oscilloscpe fully clockwise. This ensures that the settings for voltage and time are as indicated on the adjustable knobs. Connect the microphone to Ch.1 and set the mode switch (6) to Ch. 1. When the tuning fork is struck and held close to the microphone a sine wave will be displayed on the oscilloscope. This will slowly drop in amplitude over about 20s. Why is this? ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ Measure the period four times at one timebase setting (dial 22) and four times at a second setting. Each time the four readings should be taken at the following places: 1) any peak to any other peak; 2) any trough to any other trough; 3) any rising edge across the 0 V axis to any rising edge; 4) any falling edge to any falling edge. The period is found by counting the number of divisions between any feature on the wave, multiplying by the sweeptime/div setting on the oscilloscope to get the total time, and dividing by the number of wave cycles between these points to get the time for one cycle. Tabulate your data below and calculate the frequency of the sound. Feature No. of Divs Sweeptime/div. Time for N wave cycles (s) Period, T (s) Peak – Peak Trough – Trough Rising @ 0 V Falling @ 0 V Best value for T (average of these 8 readings) (standard deviation gives uncertainty) ± ± Frequency, f = 1/T 26 Measuring the wavelength of the sound We will use the fact that sound resonates in tubes. In this case, the tube is closed at one end. This forces a node to occur at the closed end and an anti-node node to occur at the open end. The longest wave (the fundamental) that can exist in a tube like this is shown here. You can see that when this resonant wave is present in the tube, its wavelength is 4 times the he length of the tube. When using the tuning fork as a sound source the wavelength is fixed. So we need to adjust the length of the tube until the condition λ = 4L is met. You will be able to tell that this is the case as the amplitude of of the sound will be at a maximum due to the air in the tube resonating in addition to the vibrations around the tuning fork. With the microphone close to the open end of the tube, sound the tuning fork. Start with no air in the pipe and increase the length length of the open pipe until you have maximised the amplitude of the signal on the oscilloscope. You should also hear the sound grow louder as you approach the resonant condition. At the position of greatest amplitude, measure the length of the air column, L,, in the open tube. What is the wavelength of the sound? ± If you increase the length of the tube still further, can you find any other positions where resonance occurs? If so, why is this? Can you use these positions to calculate the wavelength? Try it, and record your observations. ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ 27 Measuring the Speed of Sound Use Eq. 1 to calculate a value (with its uncertainty) for the speed of sound in air. Comment on how it agrees with the accepted value of 332 ms-1 at 0C at sea level. ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ Appendix :The Cathode Ray Tube (CRT): Fig. 1 Schematic Diagram of Cathode Ray Tube A CRT consists of an evacuated glass tube. A hot filament (1) emits electrons. These are accelerated and focused by the electron gun (2). The accelerated electrons (5) strike the fluorescent screen (6) with sufficient energy to make it fluoresce, creating a spot of light. On their way to the screen the electrons pass through two pairs of metal deflecting plates (3) and (4). A voltage variation applied to the X-plates (4) causes the electrons to deflect in the X- direction, the deflection being directly proportional to the voltage. 28 Circuitry: The four main circuits are the time base generator, the synchronising amplifier, the display unit and the vertical amplifier. (1) Time base generator produces a saw-tooth waveform. This potential difference is applied across the X-plates. During the rising part of the saw-tooth the electron spot is swept across the screen at a constant speed. During the “fly back” of the waveform the electron gun spot returns swiftly to the start again. The frequency of the saw-tooth is variable and hence the speed at which the spot travels across the screen is also variable. The time base generator therefore produces a time scale in the horizontal or X-direction controlled by the Sweeptime / div knob. (2) The Synchronisation Amplifier ensures that the time base sweep and input signal to the Y plates commence at the same time. (3) The display unit controls the size of the spot and the overall brightness of the spot on the screen. (4) The vertical amplifier controls (a) the degree of amplification of the signals being fed to the Y-deflection plates of the tube and (b) the vertical position of the spot on the screen. The oscilloscope used in this experiment is a two beam oscilloscope i.e., there are two spots which can move across the screen either one at a time or both together allowing two traces to be displayed simultaneously if required. Free Running Spot: If the saw-tooth waveform is applied continuously in “free run” or “auto” mode this voltage waveform makes the spot scan continuously across the tube face. Internal Triggering: A single cycle of the saw-tooth waveform may be initiated by triggering the oscilloscope. The signal trigger can be derived from any signal applied to the Y-input or vertical amplifier by the ‘scopes internal triggering circuitry. This causes a single sweep of the beam displaying the time variation of the Y input signal for the duration of this sweep. External Triggering: Alternatively, the signal trigger can be derived from some external source applied via the external trigger input. This initiates a single sweep irrespective of what is applied to the Y amplifier. This is used in the second experiments, which involve the use of an oscillator ( or signal generator). 29 30 Measurement of the Focal Lengths of Lenses and a Determination of Brewster’s Brewster’s Angle This experiment investigates the behaviour of light crossing from one medium to another. In the first part you will determine the focal lengths of a concave and convex lens. In the second part you will see how the reflected and transmitted transmitted rays depend on the polarisation of the light and determine Brewster’s angle, the angle at which no parallel light is reflected. Part 1: Determining the focal length of Lenses and Mirrors. The phenomenon of refraction (or bending) of light at the interface interface between two materials can be used to make optical components to form images. Parallel light hitting a spherical material will either converge or diverge depending on whether the surface is convex or concave - to remember which shape is which, recall the entrance to a cave. Thus both a convex lens and a concave mirror are converging. For spherical lens and mirrors the distance of the object from the lens, u,, and the distance of the image, v, are related by 1 1 1 + = u v f (Eq.1) where f is a constant for a particular lens or mirror, called its focal length length. (1) To determine the focal length of a convex lens Set up the apparatus as shown here with the light source passing through the lens forming an image on the screen. The object is a cross s drawn on the frosted glass of the projector. Place the source 50cm from the lens and move the screen until you obtain the sharpest image. Adjust the equipment so that the object and image are both at the same height i.e. the optical axis is horizontal and nd parallel to the edge of the bench. 31 Make at least five different measurements of source and image distances and enter them in the table below with their associated uncertainties. For each pair of measurements, calculate the focal length and enter it in the last column. Find the mean and standard deviation of these focal lengths which are roughly equal to the best estimation and its uncertainty. u (cm) δ u (cm) v (cm) δ v (cm) f (cm) Mean of the separate focal length determinations Standard deviation of these determinations Graph 1/v against 1/u. What should the slope and intercept of the graph be? ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ Using the graph, calculate the best value of the focal length of the lens with its associated uncertainty. ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 32 ____________________________________________________________________ (2) To determine the focal length of a concave lens Substitute the concave lens for the convex lens. Why can’t you proceed as you did for the convex lens? ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ To overcome this problem, use the concave lens together with the convex lens. This combined system will have a focal length, f, given by 1 1 1 = + (Eq. 2) f f1 f 2 where f1 and f2 are the focal lengths of the individual lens. Proceed as before and fill in the table below. u (cm) δ u (cm) v (cm) δ v (cm) f (cm) Mean of the separate focal length determinations Standard deviation of these determinations Use Eq.2 to find the focal length of the concave lens (with its associated error.) ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 33 Part 2: Determination of Brewster’s Angle Light is an electromagnetic wave so it consists of electric and magnetic fields which have an interesting property in that they are always perpendicular to the direction in which the light is travelling. Usually the direction of the electric field is orientated in many planes (each perpendicular to the direction of motion.) The light is said to be unpolarised. Light is coming towards you. Electric field oscillates as shown by the arrows in the plane perpendicular to you. However, it is possible to restrict the field to a single plane when the light is said to be plane polarised. θi When light passes from one medium to another, some of the light is transmitted and some is reflected. The angles of θr incidence θ i and reflection θ r are equal while the angle of incidence and transmission are related by Snell’s law: n1 n2 n2 sin θ i = n1 sin θ t θt (Eq. 3) How much light is transmitted and how much is reflected depends on the polarisation of the light. Considering light polarised in the plane parallel to and perpendicular to the plane of incidence separately, the ratio of the intensity of the reflected to the intensity of the incident light is given by R parallel tan 2 (θ i − θ t ) = tan 2 (θ i + θ t ) and R perpendicular sin 2 (θ i − θ t ) = sin 2 (θ i + θ t ) (Eq.4) so that if R=0 no light is reflected while if R=1 all the light is reflected. The apparatus consists of a table on which the object to be measured is placed, and two arms that are free to move. One arm holds the a laser whose plane of polarisation is either known or can be set by using a polaroid, the other holds a photoresistor detector whose resistance decreases the greater the intensity of light falling on it. Thus a voltage measured across the resistor will increase in proportional to the intensity of light. The angles of incidence and transmission can be read from a scale on the table. Set the plane of polarisation of the light perpendicular to the plane of incidence. Vary the angle of incidence and measure the intensity of the reflected light. Record your data in the first two columns below and graph the results. Repeat your measurements for light that is polarised parallel to the plane of incidence. 34 Light polarised perpendicular to the plane of incidence Angle of Incidence Voltage measured Calculated angle of (degrees) (V) transmission (degrees) Theoretical value for Rperpendicular Light polarised parallel to the plane of incidence Angle of Incidence Voltage measured Calculated angle of (degrees) (V) transmission (degrees) Theoretical value for Rperpendicular 35 Comment on the curves you have obtained and find a value for Brewster’s angle θB, the angle of incidence at none of the parallel polarised light is reflected: Rparallel = 0. Calculate also the refractive index, n, of the material, related to Brewster’s angle by n = tanθ B (Eq. 5) ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ If time permits, use Eq.3 to fill in the third column of the tables above (the refractive index of air can be taken to be 1). Then use Eqs. 4 to fill in the fourth column. Compare these theoretical values to your experimental values. If possible, overlay them on the same graph. Comment on the agreement. ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 36 Magnetic braking by electromagnetic induction Introduction When a magnet is dropped through a metallic pipe, electromagnetic induction produces eddy currents, which in turn produce magnetic fields retarding the fall of the m magnet. The present experiment (1) investigates and quantifies this phenomenon and (2) also finds a value for the magnetic moment of the magnet. Background Theory: Normally, when an object is released from rest and allowed to fall, it accelerates downwards ds at a rate equal to g,, the gravitational acceleration. However, when a magnet is released from rest and is allowed to fall through a metallic tube, forces other than gravity are called into action. From Faraday's law a changing magnetic field passing through hrough a conducting loop induces electric currents, called eddy currents, in the loop. From Lenz's law the action of these induced currents is such that the magnetic field produced by the current opposes the change in the magnetic field that produces the current. Thus, the falling magnet produces a changing magnetic field which exerts a force on it. This force, which proportional to the velocity of the magnet, acts upwards against the weight of the magnet. The falling magnet quickly reaches a terminal velocity velocity (when the upward force becomes equal to the downward force, resulting in zero acceleration) and the currents generated reach a steady state value. The magnet then falls through the remainder of the pipe at a constant velocity. Theory: Consider a magnet travelling at constant velocity, v,, through the pipe. By calculating the magnetic force generated by the eddy currents (which are induced by the falling magnet) and equating this force with the weight of the magnet, a value for the terminal velocity, v,, can be found. The terminal velocity is given by the equation v = 22.756 m g R4 / σ d µ0 2mB2 Eq. (1) The other quantities are the mass, mass m,, of the magnet, the annular radius, R, of the tube (= 7.1 mm), the tube wall thickness, d (= 0.8 mm); mB is the magnetic moment, moment µ0 is -7 -2 the magnetic permeability of air (=4π×10 (=4 NA ) and the conductivity of the metal tubing is σ = 5.964×107Ω-1m-1. 37 Figure 1. Effects of dropping a magnet through a conducting tube. The value of the velocity, v, can be measured in the experiment, while all the other quantities are well defined apart from the magnetic moment, mB, of the magnet. For a point on the axis of the magnet, the axial magnetic field is Bz(z )= µ0 mB / 2 π z3 Eq. (2) where z is the distance along the axis from the centre of the magnet. Procedure The experiment consists of two parts. The first measures the terminal velocity of the magnet through the tube. The second measures the field strength of the magnet in order to find a value for mB or µo mB . Apparatus Copper pipe and support, magnet, magnetic field probe, timer and ruler. Figure 2. Apparatus for measuring velocity of falling magnet and its magnetic moment. 38 Part 1. Measure the terminal velocity of the magnet through the tube The apparatus consists of a vertically suspended copper tube and a magnet. Measure the height, h, of the copper tubing, and take about 30 measurements of the time, t, taken for the magnet to travel the length of the tube and record in Table 1 below. This will give an accurate measurement of the average velocity, vave, of the magnet through the tube which will be approximately equal to the terminal velocity, vt. The annular radius, R, of the tube is the average of the inner and outer radii of the tube while the wall thickness, d, of the tubing is half the difference of these radii. The values of these are given below. Theoretical Time through Time outside Time of plastic Measured Velocity, vmeas Velocity, Vcalc . tube Tube through tube [ms-1] [ms-1] [s] [s] [s] Use Eq. 1(see later) Mean = Mean = Mean = 39 Length of tube [m] = Part 2: Determine the magnetic moment (mB )of the magnet To measure the dipole moment, mB, of the magnet, place it on the small adjustable slide. Set the multimeter on the DC 2000mV range. Move it the furthest distance from the magnetic field probe and use the variable resistor to zero the reading on the multimeter . Slide the magnet closer to the magnetic probe in small steps and record the distance from the probe to the centre of the magnet, z, and the multimeter reading in the Table below. This is converted to field strength, Bz using the conversion factor 1mV = 1.25 10-5 Tesla. Repeat this for up to 10 positions. Position. z [m] Meter reading. [mV] Bz [Tesla] 1 / z3 [m-3] 10 readings Mean = ? Plot a graph of Bz against 1 / z3. From Eq. 2, the slope of this graph is equal to µ0mB/2π so that mB or µo mB can be evaluated. Slope of Bz against 1 / z3 : the value of the slope = Use this space to calculate mB and µo mB 40 ± mB = ± µo mB = ± In measuring the mass of the magnet, to avoid a contribution from the magnetic attraction between the magnet and the balance, the magnet should be placed on a teflon cylinder of suitable height. Use the value for mB you just calculated to find the velocity as predicted by Eq 1 vcalc = 22.756 m g R4 / σ d µ0 2mB2 = ± Compare this value to the measured result (vmeas) in table 1. Calculate, using the standard laws of motion: 1. the average and final velocities of the magnet if it were dropped from a height, h , but not through a conducting tube. v final = 2 gh vav = = ± vinit + v final = 2 ± 2. the final kinetic energies (KE) of the magnet after falling through the height, h, KE final = 1 2 mv 2final = ± 3. Use vmeas (from table 1) to find the KE after it emerges from the conducting pipe KEmeas = 1 2 2 mvmeas = ± 41 Why are these energies different and where does the missing energy go? ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 42 Determination of the Resistivity of a Metal Alloy using a Wheatstone Bridge Introduction This experiment uses a simple but clever piece of apparatus, called the Wheatstone bridge, to measure an unknown resistance to high precision. The Wheatstone bridge is a circuit set up in a delicate balance so that no current flows through a galvanometer (which is a very sensitive current measuring device). Any deviations from zero are readily detected, so when the wheatstone bridge is balanced, you know to high precision that no current is flowing in the galvanometer and this leads to high precision in determining the resistance of an unknown substance. Wheatstone bridge circuits are often at the centre of many electronic measuring devices, remote sensing equipment and switches. The resistance, R,, of a substance is directly proportional propo to its length, L, L and inversely proportional to its cross-sectional sectional area, A,, as well as depending on the material it is made from. Thus, R = ρL A (Eq.1) where the resistivity, ρ, depends on the material. If you double the length of a resistor what happens to its resistance? If you double the radius of a cylindrical resistor, what happens to its resistance? 43 Wheatstone Bridge circuit The Wheatstone bridge is shown here and has two known resistances R1 , R3 , one variable resistor, R2 , and the unknown resistor R X . When the meter is balanced, i.e. when the current through the galvanometer RX R3 = R2 Rg is zero: R1 (Eq. 2) (We will prove this later) Apparatus The apparatus used to measure the resistance of the alloy is shown on the right and schematically below. Note the similarity with the Wheatstone bridge above. Instead of measuring R1, R2 however, we take advantage of the fact that the resistance of a resistor is proportional to its length and so R2 R1 = L2 L1 (Eq. 3) RX R3 L2 L1 Method 44 Connect the apparatus as shown above, taking particular care that the contacts of R3 and RX are tight. Make sure that the longest length of RX is used, i.e. make the contacts with the shortest possible lengths at each end and ensure that the wire cannot short itself. Use the shortest possible lengths of wire to connect the known resistors to the bridge. Press the contact maker against the wire and find two points, which give deflections of the galvanometer in opposite directions. Between these points locate accurately the point of contact which gives no deflection. Measure L1 and L2, noting that L2 is the length opposite RX and L1 is the length opposite R3. Interchange R3 and RX and repeat the experiment. (Lengths L1 and L2 will also swap: note that L2 is always the length opposite RX. If the values of L1 and L2 for the second balance point differs by more than 1 cm from the original, consult the demonstrator before proceeding. The discrepancy may be due to bad contacts on the meter bridge. Record your values in the table below. (Don’t forget to include errors.) R3 L1 L2 RX Average value for the unknown resistance of the alloy Repeat the measurements for different values of the fixed resistor, R3. R3 L1 L2 RX Combine all your measurements to get the best value for the resistance of the alloy. 45 Detach the sample wire and uncoil it carefully so as not to produce any kinks. Measure the length making allowance for the short lengths used at each end which were not included in the resistance measurement. Measure the diameter by means of a micrometer screw at about six places uniformly distributed along the wire. Record you measurements below and use Eq. 1 to calculate the resistivity of the metal alloy along with its error. 46 Finally, prove what we have assumed all along, that RX R3 = R2 R1 Remember, when balanced, no current flows in the leg DB. The solution is outlined in the seven steps below. 1) 2) 3) 4) 5) What current flows along DB? What can you say about VDB, the voltage difference between D and B? What can you say about VAD compared to VAB? Write an equation to express this. What can you say about VDC compared to VBC? Write an equation to express this. Let IL be the current that flows along the left hand line A->D->C and IR the current that flows along the right hand line A->B->C. (Remember nothing flows from D->B) 6) Express each of VAD , VAB ,VDC , VBC as a current times a resistance using Ohm’s Law. 7) Substitute into the two equations in steps 3&4 and divide one equation by the other. 47 48