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Vol-12, No-3, Sept.-2012 Stewart & Gee Experiment-373 S MEASUREMENT OF MAGNETIC FIELD ALONG THE AXIS OF A CIRCULAR CURRENT CARRYING COIL USING DIGITAL STEEWART AND GEE APPARATUS Jeethendra Kumar P K, Santhosh K and Ajeya Padmajeeth KamalJeeth Instrumentation & Service Unit, No-610, Tata Nagar, Bengaluru-560092. INDIA. Email: [email protected] Abstract Using a search coil, microcontroller and an LCD display, the magnetic field (flux) and induced emf produced at the center of a circular copper coil carrying current are recorded and verified vis a vis the values obtained from the relevant theoretical equations. The permeability of free space is calculated and the variation of magnetic field along the axis of the coil is also studied. Introduction Balfour Stewart and William Winston Haldane Gee designed an instrument to study the variation of magnetic field along the axis of a circular coil. It consists of a coil with specified number turns of copper wire and a compass that can be moved along the axis of the coil. This simple apparatus is still in use in most physics labs. The problem with this apparatus is, however, poor quality of the magnetic compass used in it. Hence, the magnetic field calculated using the deflection of the compass needle is not accurate. In this experiment we have used a search coil which can be moved along the axis of the coil. The voltage induced in the coil is noted, from which the magnetic field produced by the circular coil is calculated. A microcontroller is used to record the induced voltage and a program is written to calculate the magnetic flux (B) value. The display shows both the induced voltage and the magnetic field values. One can verify the experimental results using the values computed from the relevant theoretical equation [1]. Magnetic field along the axis of a current carrying coil A stationary electric charge produces an electric field whereas a moving electric charge produces magnetic field around it. This is the basic principle of electro-magnetic induction. The magnetic field at the center of a coil carrying current I is given by B= where I the current flowing through the coil, R is the radius of the coil, and …1 Lab Experiments 1 Kamalj Kamaljeeth Instrumentation and Service Unit Vol-12, No-3, Sept.-2012 Stewart & Gee µ o is permeability of free space. If the number of turns in the of coil is ‘N’, the magnetic field, B, at the center of the coil is given by B=N …2 As one moves away from the center of the coil, the magnetic field decreases. Hence the magnetic field varies along the axis of the coil. A The magnetic field at the point A at a distance Z from the center of the coil is given by B= where …3 / I the current flowing through the coil, R is the radius of the coil, µ o is permeability of free space, N is number of turns in the coil, Z is the radial distance from the center of the coil to the point of observation. Figure-1 shows the geometry of the field. Using Equation-3 one can determine the field along the axis of the coil which decreases inversely with the square of the distance. By R Axis I B O Z A BZ Figure-1: Directions of magnetic field along the axis of the coil Search coil detector The above equations (Equation-3) hold good when a battery or DC source is used to energize the coil. In the present experiment we have used an AC source and a search coil. The alternating current flowing in the field coil induces an electromotive force (emf) in the search coil. The voltage generated in the search coil due to electromagnetic induction is governed by the Faraday’s law of induction. Lab Experiments 2 Kamalj Kamaljeeth Instrumentation and Service Unit Vol-12, No-3, Sept.-2012 Stewart & Gee The search coil used in this experiment is smaller (with 2.2cm square bobin size) compared to the filed coil (with radius R = 9.75cm), hence the magnetic field at a given instant of time is nearly uniform over the area of the search coil. The magnetic flux Φ corresponding to the magnetic field vector B generated by the search coil is defined as the product of the area A of the coil and the magnetic vector B normal to the plane of the search coil Φ = AB cos α. …4 where α is the angle between the planes of B and Ф. Since the search coil is also placed with its plane parallel to the field coil α = 0, cos 0 =1, so that Equation-4 becomes Φ = AB …5 The search coil is fitted to a half meter long rectangular iron rod, which acts like a core and is placed at the axis of the field coil. The coil can be moved along the axis. The iron core increases the emf induced in the search coil due to its high relative permittivity [2]. Hence the total flux is given by Φ = A B µr …6 where A is cross sectional area of the core of the search coil, B is the magnetic field intensity, and µ r is the relative permittivity of the material of the core of the search coil. The ratio of inductance of the search coil with air as the core and inductance of the search coil with iron as the core (in the form of a rod on which the coil is fixed for movement) gives the relative permittivity of air with respect to the material of the core µr = …7 ` Faraday’s law of induction gives the voltage induced per turn of the search coil as V = -n Ф …8 where n is the number of turns in the search coil, V is the induced emf in the search coil, and Ф is the flux. Substituting for flux (Ф) from Equation-6, one can rewrite Equation- 8 as V = -n !" V = -n A µ r # "# …9 …10 Lab Experiments 3 Kamalj Kamaljeeth Instrumentation and Service Unit Vol-12, No-3, Sept.-2012 Stewart & Gee Due to the current induced in the search coil it produces a magnetic field in the opposite direction as indicated by the minus sign in Equation-10. The magnetic field produced by the search coil opposes any change in the magnetic field produced by the field coil, as described by the Lenz’s law. The current I flowing through the field coil vary sinusoidally with time so that the magnetic field B also varies in the same manner. The magnitude of the magnetic field B produced by the field coil can be written as B= "$$ cos ωt where …11 Bpp is the peak to peak value of the magnetic field, and ω is the angular frequency of AC signal in the field coil. This is the general equation representing sinusoidal voltage (or current) produced by the field coil. Substituting for B in Equation-10, one obtains "$$ % V = -n A µ r V = -n A µ r ",, - cos ωt+ .# ...12 By differentiating cos (ωt ) and substituting it in Equation-12 we get V = -n A µ r "// (-ω sinωt) "// V = nAω µ r sinωt ...13 …14 This is the instantaneous (time varying) voltage induced in the search coil. Hence the peak to peak voltage is given by VPP = nAω µ r "// …15 The RMS value can be written as Vrms = !ωµ01 √ // = !ω3µ01 4√ // …16 Hence the magnetic field can be determined by measuring the induced emf in the search coil. The magnetic field is given by BPP = 4√ 5067 .! 0 Substituting for ω =2πf, we get Lab Experiments 4 Kamalj Kamaljeeth Instrumentation and Service Unit Vol-12, No-3, Sept.-2012 Stewart & Gee BPP = BPP = 4√ 5067 8! 0 , or √ 5067 8! 0 …17 Using this equation one can calculate the magnetic field at the center of the coil as well as along its axis on either side. For a give set of search coils, the parameters n, A, f, µ r are constants, hence the magnetic field is proportional to the induced emf, as given by Equation-17. In the present work, instead of using a magnetic compass we have used a search coil. Hence AC power supply is used to energize the coil. A rectangular iron rod, with dimensions 1.65 cm x 1.65 cm, which acts as the axis of the coil is provided with the set-up. The search coil is fixed in this rectangular bar which acts like a core for the search coil. A pointer fitted on the search coil is used to read the distance from the center of the coil, as shown in Figure-2. Figure-2: Search coil fitted along the axis of the coil and the pointer for reading its position Determination of permeability of free space Since a core is placed along the field axis, by changing the material of the core the relative permittivity of the core material and permeability of free space can be determined using this instrument. The magnetic field at the center of the coil is given by Equation-2. In the presence of the core, the magnetic field is multiplied by µ B=N μ …18 A plot of B versus I provides the average value of B/I from which knowing the values of N, R and µ permeability of free space can be determined from the equation µo = " 0 = 0 X Slope , the …19 Lab Experiments 5 Kamalj Kamaljeeth Instrumentation and Service Unit Vol-12, No-3, Sept.-2012 Stewart & Gee Apparatus used Digital Stewart-Gee apparatus, AC power supply 2-12V, AC ammeter 0-2A, Rheostat 100Ω, digital vernier, and LCR meter. The complete experimental set-up is shown in Figures-3 and 4. Experimental procedure The experiment consists of two parts: Part-A: Determination of permeability of free space (µo) Part-B: Variation of magnetic field along the axis of the coil The number of turns and diameter of the field coil are noted as Number of turns in the field coil, N = 50 Diameter of the coil including its thickness, d = 19.5cm Hence radius of the coil, R = 9.75x10-2m. Frequency of AC power supply =50Hz The number of turns, core area and relative permittivity of the search coil are noted from the Stewart- Gee apparatus as Number of turns in the search coil (n) = 2751 @.4BC Relative permeability, µ = @.DD@ =4.42 Area of the search coil core (16.5mm x 16.5mm) = 2.73x10-4 m2 These data are provided by the manufacturer of the Stewart- Gee apparatus. Lab Experiments 6 Kamalj Kamaljeeth Instrumentation and Service Unit Vol-12, No-3, Sept.-2012 Stewart & Gee Figure-3: Stewart - Gee apparatus with the search coil Figure-4: Microcontroller based magnetic field and induced voltage indicator Part-A: Determination of the permeability of free space (µo) 1. The Stewart-Gee apparatus is connected to the AC power supply in series with a digital AC current meter and rheostat, as shown in Figure-5. (There is no need to align the instrument along the magnetic meridian.) 2. The search coil cable is connected to the digital field along the axis of the instrument and it is switched on. 3. The AC voltage in the power supply is set to 10V. The current through the field coil is adjusted to the value 0.1A using a rheostat. Rheostat 100 Ohms 0-2A Coil 2-12V AC power supply Figure-5: Circuit connections 4. The magnetic flux (Bpp) and the induced emf (Vrms) are recorded from the LCD display and are given in Table-1. Lab Experiments 7 Kamalj Kamaljeeth Instrumentation and Service Unit Vol-12, No-3, Sept.-2012 Stewart & Gee Vrms = 0.07V; Bpp = 3.02 Gauss; Brms = "$$ √ = 1.06 Gauss The readings obtained are compared with the value obtained from Equation-17 Bpp = √ 5067 π ! µ0 = √ F@.@G π FH@ F GHDF 4.4 F .GIFD@JK = @.DLB H D.4I = 0.000379 Tesla = 3.79 Gauss This value is slightly higher than the value (3.02Gauss) read by the meter. 5. The experiment is repeated by varying the field coil current in suitable steps up to the maximum value of 1A. In each case the flux and Vrms values are recorded in Table-1 and verified vis a vis the value obtained from Equation-17. 6. A graph is drawn taking field coil current along the X-axis and the magnetic flux along the Y-axis, as shown in Figure-6. The slope of the straight line is given by 4.C S Slope = G.C S @.D = 12.10 In the SI unit, magnetic flux is specified in Tesla, therefore converting flux into Tesla unit, the value of slope becomes D .D Slope = D@@@@ = 12.1V10W4 µo = " µ0 = FL.GHFD@J [email protected] 12.1X10W4 = 1.06X10WC This value is about 15% lower than the standard value of µo of free space which is 1.256x10-6. Table-1 Field coil current Induced emf Vrms Magnetic Flux (BPP) Gauss (A) (V) (Bpp) Meter Brms From Eq.-17 0.070 3.02 1.06 3.97 0.101 0.130 7.18 2.54 7.38 0.208 0.180 10.18 3.60 10.22 0.298 0.250 14.20 5.02 14.20 0.408 0.300 16.30 5.76 17.04 0.500 0.360 20.36 7.20 20.45 0.595 0.450 25.46 9.00 25.56 0.757 0.520 29.52 10.44 29.54 0.868 0.560 31.56 11.16 31.81 0.919 0.560 31.56 11.16 31.56 0.926 0.810 38.64 13.66 39.20 1.137 Variation of magnetic field at the centre of the coil Lab Experiments 8 Kamalj Kamaljeeth Instrumentation and Service Unit Vol-12, No-3, Sept.-2012 Stewart & Gee Part-B: Variation of magnetic field along the axis of the coil In this part of the experiment the search coil is moved along the axis of the coil and magnetic flux is determined. 7. The current in the field coil is now adjusted to 1A and voltage is set to 12V. The field at the centre of the coil is recorded from the LCD display as At the center (Z =0), Vrms = 0.60V; Bpp = 33.60 Gauss The search coil is now moved to the left of the centre by 1cm and the flux and the induced emf are noted as Z = 1cm; Vrms = 0.60V; BPP = 33.60 Gauss Magnetic flux Brms (Gauss) 16 14 12 10 8 6 4 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 Field coil current (A) Figure-6: Variation of magnetic field with current at the center of the field coil 8. The experiment is repeated by moving the search coil in steps of 1cm, recording the emf and the flux each time. The readings obtained are tabulated in Table-2. 9. A graph is drawn with distance along the X-axis and magnetic flux along Y-axis, as shown in Figure-7. 10. A line parallel to the X-axis is drawn where the curvature of the curve changes. These two pints (P and Q) are known as the inflexion points. 11. The distance between the two inflexion points is noted as Distance between P and Q = 12cm, which is slightly greater than the diameter of the field coil (9.75cm) Lab Experiments 9 Kamalj Kamaljeeth Instrumentation and Service Unit Vol-12, No-3, Sept.-2012 Stewart & Gee Further, the curve is not exactly symmetrical about the vertical axis. This may be due the slight realignment of the search coil axis. Results and conclusions Permeability of free space (µ o) is found to be 1.06x10-6 (= 3.21π x10-7), the standard value being 4π x 10-7. Table-2 Distance Induced emf Magnetic Distance Induced emf Magnetic from the Vrms (V) Field from the Vrms (V) Field center of the Bpp (Gauss) center of the Bpp(Gauss) field coil field coil Z (cm) Z (cm) Left hand side of the coil Right hand side of the coil 0.60 33.60 0 0.61 35.62 0 0.60 33.60 1 0.61 35.62 1 0.60 33.60 2 0.60 33.60 2 0.60 33.60 3 0.56 31.56 3 0.59 33.54 4 0.54 29.58 4 0.57 31.52 5 0.50 28.50 5 0.54 30.54 6 0.47 25.46 6 0.52 28.56 7 0.43 24.48 7 0.48 26.48 8 0.40 22.30 8 0.45 24.44 9 0.36 20.36 9 0.41 22.46 10 0.33 19.34 10 0.38 20.32 11 0.30 16.30 11 0.35 19.34 12 0.28 15.28 12 0.31 17.36 13 0.25 14.20 13 0.29 15.28 14 0.23 12.28 14 0.26 14.26 15 0.21 11.26 15 0.24 12.26 16 0.19 10.14 16 0.22 11.26 17 0.17 9.12 17 0.19 10.14 18 0.16 8.10 18 0.17 9.12 19 0.14 7.14 19 0.16 8.10 20 0.13 6.12 20 Magnetic flux variation along the axis of the coil Lab Experiments 10 Kamalj Kamaljeeth Instrumentation and Service Unit Vol-12, No-3, Sept.-2012 Stewart & Gee Magnetic Flux Bpp (Gauss) 40 35 Q 30 P 25 20 15 10 5 0 -25 -20 -15 -10 -5 0 5 10 15 20 25 Distance from the center of the field coil Figure-7: Magnetic flux variation along the axis of the coil Based on this work, the following conclusions may be drawn: a. The search coil method of measuring magnetic field is found to be quite accurate, as evident from the first part of the experiment. b. The permeability of free space is measured to an accuracy of about 10%. c. The digital microcontroller based measurement avoids lengthy calculations involved in determination of the magnetic flux as per Equation-3. d. Based on this, it is planned to design experiments in future to determine permeability of free space using core of different materials in the search coil. e. The microcontroller based instrument is found to be more accurate and reliable. Reference [1] http://physics.nyu.edu/~physlab/GenPhysII_PhysIII/MagFieldCoil.pdf [2] Dr Jeethendra Kumar P K and Dr J Uchil, Magnetic Hysteresis, LE Vol-6, No-4, Page-296, 2006. Lab Experiments 11 Kamalj Kamaljeeth Instrumentation and Service Unit