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Course 1 Laboratory Second Semester Experiment: Magnetic Fields 1 Magnetic Fields 1 Introduction The purpose of these experiments is to determine the magnetic flux density, B, in the neighbourhood of current carrying coils. These coils will be carrying an alternating current and the induced voltage (commonly referred to (incorrectly) as the electromotive force emf) in a search coil, of known area, placed at a point can be used to calculate the magnetic field. The experimental measurements of B can be compared with values calculated from the theoretical formulae for coils of known dimensions carrying a known current. 2 Theoretical Background Starting with the magnetic field of a short element (Biot-Savart law (Lectures 13&14 of EMI) it is possible to derive the equations describing the magnetic field distributions for any geometry. One of the simplest geometries is the axial fields of simple coil configurations, e.g. solenoids or flat coils. Length 2.1 Solenoids Figure 1 (a) schematically shows the cross-section of a solenoid. At any point, P, on the axis of a solenoid, the magnetic field vector B is parallel to the axis and its magnitude is given by B= µ 0 nI (cos α − cos β ) 2 (1) β α P B-field where the angles are defined in Figure 1, n is the number of turns per unit length of the solenoid, I is the current, and µ0 is the magnetic constant, which has a value of 4π×10-7 T m A-1. Figure 1 (b) shows the variation of the B-field along the axis of the solenoid. 2.2 Plane Circular Coils Figure 2 (a) is an illustration of a coil of N turns, radius a carrying a current I. At a point P on the axis of this circular coil the magnetic field vector is parallel to the axis and has a magnitude given by Figure 1. (a) The angles defining the magnetic field for a solenoid and (b)the associated magnetic field. a P x I Figure 2. (a) A plane circular coil of N turns. 2 Distance B= µ 0 NI a2 2 ( a 2 + x 2 )3 / 2 (2) B-field where x is the distance of P from the plane of the coil. Figure 2 (b) shows a plot of the field profile for such a coil. Equations (1) and (2) show that it is possible to simply calculate the magnetic field strength for either of these geometries if the details of the coil or solenoid is known. In the present experiment an alternating current is used. The coils do not contain any magnetic material, so B is a linear function of I. Your measuring instrument displays the root-mean-square value of a sinusoidal input so the simplest thing is to calculate Brms, the root-mean-square value of B x Figure 2. (b) Schematic of the magnetic field profile on the axis of a plane coil as a function of distance. 2.3 Finding the B-field from the magnitude of the emf induced in the search coil If a search-coil consisting of n turns, each of area S is placed with its plane perpendicular to a uniform magnetic field of magnitude B, the magnetic flux linking the search-coil is φ, where φ = BnS (3) If B depends on time, an emf ε is induced in the coil. This is given by ε=− dφ dB = −nS dt dt (4) If the current responsible for the production of the magnetic field is a sinusoidal function of time, with frequency ω/2π, and if the circuit does not contain any magnetic material, B can be written as B = B0 sin( ωt ) ε = − ωnSB0 sin( ωt ) εrms = − ωnSBrms (5) (6) Thus if εrms, ω, n and S are measured Brms can be calculated. 3 Apparatus and Experimental Procedure An oscillator producing a high quality sinusoidal output feeds a signal of approximately 10 V (rms) to each of three experimental stations, where is passes through a resistor of 28 Ω and appears at the blue and green terminals on a black box. The resistor limits the current that you can draw, thus ensuring that one experimenter does not affect another. The stable frequency of the signal is recorded on the black box and can be checked with a frequency meter if required. You are also given a 0.01 Ω four terminal standard resistor labelled with its exact value. The electronic voltmeter, Levell type TM3B, is battery operated so please remember to switch off at the end of your measurements! Make sure that you understand the controls on the TM3B voltmeter ask a demonstrator if you are unsure. Set the controls of the voltmeter as follows: • B.W. (Bandwidth) 10 Hz – 10 kHz • 5 mV (Full scale deflection) 3 Now wire up the circuit shown in figure 3. Assuming that you have done everything correctly you should see a reading of about 2 mV on the meter with the standard resistor in circuit; that is, with the switch (figure 3) in position “a”; what is the current in the solenoid? Now switch the meter from the standard resistor to the search coil. On inserting the search coil into the solenoid you should see a reading of a few mV. It may be necessary to change ranges on the voltmeter from time to time and you should check the current through the solenoid at intervals to see whether it is remaining constant. 4 terminal resistor Bench Power Supply 433 Hz C1 Green Terminal C P1 P Solenoid Switch box b a Search Coil Coaxial cable to TM38 input Figure 3. The experimental arrangement for the connections to the solenoid and search coil. 3.1 Experiment 1 Use the search coil to investigate the variation of B, the magnetic flux density, along the axis of the solenoid. Do a preliminary run to decide how many observations you need; where the field is varying slowly observations can be fairly widely spaced but you need more where it is changing rapidly with position. Take care over the measurement of the distance of the search coil from the ends of the solenoid using the callipers provided. It is a sensible precaution to remove all metal objects e.g. rulers and callipers from the vicinity of the coils before taking the voltmeter readings. Why? Information about all the coils used in this experiment can be found in section 4. Please read and observe the precautions mentioned at the beginning of section 4 when using metal measuring instruments on any of the coils. Measure the appropriate dimensions of the solenoid (see section 4) and use these, together with the measured value of the current, so calculate values of Brms for each 4 position of the search coil. Use the Mathcad programme solenoid to calculate the Bfields from your measured emfs. But before using the programme you must make two “hand” calculations – one for a position near the centre of the solenoid and one for a position just outside the windings. If these show reasonable agreement with the search-coil results, proceed with the computer programme and make sure that you are using it properly. In an ideal situation, values of Brms determined experimentally and theoretically should, of course, agree with each other to within the limits of experimental error; but practical arrangements do not always conform to ideal specifications so you should be on the look-out for, and try to explain, any significant discrepancies between the results obtained by the two methods. 3.2 Experiment 2 Replace the solenoid by one pair of plane coils and investigate the magnetic field along its axis. Modify the Mathcad programme solenoid to calculate the magnetic field for this case. 3.3 Experiment 3 (only if time allows) Connect both plane coils in series, with the current circulating in the same sense in both. Make investigations of this well-known (Helmholtz) arrangement. What useful properties does it have? What happens if you reverse the connections to one coil? Again, modify the Mathcad programme solenoid to calculate the magnetic field for the two coils. 4 Dimensions of Coils These coils are easily damaged by metal measuring instruments. Please ensure that your measurements do not remove the enamel or distort the wire! Search coil data: The search coils were wound on a former of diameter 17.32 mm. You may take this as the internal diameter of the coil. The coils have a single layer winding, which has 10 turns (34 standard wire gauge (s.w.g.), diameter 0.2337 mm, enamel covered). Solenoid: You may determine all you need to know by direct observation. The solenoid has a single layer of wire (30 s.w.g., diameter 0.3150 mm, enamel covered). Plane coils: Each of the plane coils was wound on a former of diameter 20.0 cm. Note the relationship between this and the separation of the two coils as mounted. The number of turns on each coil is 72, wound in three layers (24 s.w.g., diameter 0.5588 mm, enamel covered. 5