#6A& Mgnetic Field Mpping Gol y performing this lb experiment, you will: 1. use mgnetic field mesurement technique bsed on Frdy s Lw (see the previous experiment),. study the mgnetic fields generted by vrious conductor geometries (different cses ech week). Reding Sources of mgnetic fields nd the iot-svrt lw re discussed in Young nd Freedmn, Sec. 8.1-3 nd 8.5 in the 1 th edition. Overview 3 bsics 1. Current in wire cretes mgnetic field in the vicinity of the wire. A long stright conductor is surrounded by concentric mgnetic field lines, whose mgnitude depends on r, the distnce from the wire. As indicted in Fig. 1(), the direction of is given by right-hnd rule.. The mgnetic field is vector quntity, so t ech point in spce the field hs mgnitude nd direction. As with ny vector, the mgnetic field cn be resolved into three mutully perpendiculr components. 3. We will mesure this field using probe coil. As we lerned from Frdy s Lw, the probe is sensitive to the field component prllel to the xis of the coil. The voltge signl seen on the oscilloscope is proportionl to the mgnitude of this mgnetic field component. (n our experiment n oscillting field is generted by n lternting current (c) signl from function genertor, becuse we cn mesure n oscillting field with this method bsed on Frdy s lw.) Theory A. Genertion of Mgnetic Fields The mgnetic field d produced t displcement r from smll current segment dl is given by the iot-svrt lw: d = μ 0 dl r 4π r 3. The totl mgnet field produced by complete current-crrying circuit is obtined by integrting this expression over the entire circuit. A long stright conductor is surrounded by mgnetic field lines of mgnitude = μ 0 /πr, where is the current nd r is the rdius of the circle, i.e., the distnce from the conductor. As result of the vector product in the iot- Svrt lw, the direction of is given by the right-hnd rule. Figure 1(b) shows the mgnetic field lines produced by circulr current loop, while Fig. 1(c) shows two identicl circulr coils rrnged in Helmholtz configurtion, where the seprtion of the coils is equl to their rdii. The Helmholtz configurtion is distinguished by the uniformity of the mgnetic field in the region between the two coils. The mgnetic fields generted by these nd other geometries will be mesured in this lb. Formule describing some properties of these fields re given in the Appendix. () Stright conductor (b) Circulr current loop (c) Helmholtz coils Fig. 1: Mgnetic field configurtions for three of the circuit geometries studied in this lb. Unlike electric field lines, which strt t positive chrges nd end t negtive chrges, mgnetic field lines must form closed loops. Note then tht most of the field lines indicted in (b) nd (c) re incomplete. 1
. Mgnetic Field Detection & Mesurement The fields will be mesured using Frdy probe consisting of coil mounted on non-mgnetic rod. The opertion of the probe relies on Frdy s lw of mgnetic induction, which sttes tht chnging mgnetic field produces n electric field. Specificlly, if the totl mgnetic flux Φ = da through circuit chnges, n electromotive force (emf) E will be produced. This emf is effectively equivlent to potentil difference. Formlly, E = n dφ, (1) dt where the mgnetic flux Φ is obtined by integrting the field over the cross sectionl re A of single coil loop, nd n is the number of loops or turns in the probe coil. According to the vector definition of re, A is directed norml to the coil plnes, i.e., prllel to the xis of the coil. Let this be the x-direction, s shown in Fig.. Thus the integrl Φ = da = x A, where x is the x-component of the mgnetic field, verged over re A. f sufficiently smll, the coil probes region of spce where x is effectively constnt. According to Eq. (1), n emf will only result if the mgnetic flux Φ chnges with time. This cn be ccomplished if the mgnetic field is produced by sinusoidl current, becuse, ccording to the iot-svrt lw the mgnetic field is proportionl to A x the energizing current, i: x = β x i, where the constnt β x depends only on geometry. Fig. : Projection of field f the current is sinusoidl, i.e., i = 0 cos(ωt), then the field nd flux oscillte onto re vector A in sinusoidlly, so tht definition of mgnetic flux. E() t = na d x di = naβ x dt dt = naωβ x 0 sin ωt The induced sinusoidl emf E hs mplitude naωβ x 0 = naω 0 x, reltion tht enbles bsolute, not just reltive, mesurements of the field component 0 x. For mximum experimentl sensitivity to 0 x the number of turns n, ngulr frequency ω, nd current mgnitude 0 should ll be lrge. Apprtus Vrious conductor geometries re vilble for study: solenoid, long stright wire, circulr loop, circulr coil, nd pir of Helmholtz coils, toroid. Use sinusoidl current derived from the function genertor to produce oscillting mgnetic fields. The Frdy probe on your work bench hs the coil xis prllel to the rod. t is therefore sensitive only to the mgnetic field component directed long the rod. Also vilble for your use re probes tht hve the coil xis perpendiculr to the rod xis. Fig. 3. Circuit for ll mesurements to be mde in this lb. R = 10 Ω
Mesurement Overview (lso see report sheet) 1. First fmilirize yourself with the pprtus. Wire the circuit bove with R = 10 Ω, nd the solenoid s the source coil. Adjust the function genertor to produce 5 khz sine wve with mximum output mplitude. nspect the voltge cross the 10 Ω resistor to ensure tht it forms good sinusoidl oscilltion. f not, djusting the offset control or reducing the function genertor output mplitude my improve it.. t is prticulrly importnt to keep the unshielded wires wy from the probe coil, especilly with the long stright wire, where the mgnetic field is smll. (Why?) 3. We will spend some time setting up the oscilloscopes s group for these mesurements. We will use verging to minimize the effect of rndom high frequency electronic noise in the mesurements. 4. Fmilirize yourself with the Frdy probe by moving it both inside nd outside the solenoid, nd chnging its orienttion. Observe the emf induced in the probe coil. Develop strtegies for positioning nd orienting the probe. Do this crefully for ech new cse before mking serious mesurements. 5. Next mke mgnetic field mesurements for severl different mgnetic source circuits, s indicted below. n some cses you re sked to mesure nd plot. n other cses you re sked to explore nd then to comment on, or to compre, your observtions. Frdy s lw implies tht, t constnt genertor current nd frequency, the induced emf in the probe coil is directly proportionl to the mgnetic field component long the probe coil xis. Thus, you cn record the pek-to-pek mplitude of the induced emf s mesure of the mgnetic field. 6. Although we try to minimize this effect by signl verging, the trce of the induced emf signl on the oscilloscope screen my look fuzzy becuse of rndom electronic noise, especilly noticeble when the signl is smll. The mesurement of pek-to-pek mplitude includes the rndom noise, but mesurements fr from the source circuit provide bseline for determintion of the emf corresponding to zero field. Reltive Field Mesurements for week 1: 1) Solenoid (r is the distnce from the xis, nd x is the distnce long the solenoid xis, mesured from the geometric center.) Mesure the emf induced in the probe coil by the xil field t the geometric center of the solenoid. Mesure nd plot x vs x, for x > 0. On your grph, indicte () the physicl end of the solenoid, (b) the point t which x flls to 90% of the vlue t the center, nd (c) the point where x is equl to 50% of the vlue t the center. Explore x (x =0) vs r both inside nd outside the coil. Wht do you find? Explore r outside the coil. Where do you find the lrgest vlue of r? Explore r inside the coil. Wht do you find? How does the mgnitude compre with x (x =0)? ) "nfinitely long stright conductor (r is the distnce from the center of the wire.) The induced signl is smll for this cse, but the sensitivity my be improved by incresing the frequency. Mesure the tngentil field vs r nd plot s function of 1/r. This is expected to yield stright line. (Use the vernier cliper or micrometer to mesure the rdius of the wire nd of the probe coil.) Explore the other components of, i.e., the rdil component nd the component prllel to the conductor. Wht do you find? s this wht you expect? Reltive Field Mesurements for week : nstructions for dditionl mesurements of the mgnetic fields produced by single multi-turn coil, nd by Helmholtz coil pir, nd other configurtions will be provided with the report sheet for week. 3
Appendix: Mgnetic field formule Note tht in ll these equtions the current does not necessrily hve to be constnt in time. t could, for exmple, be oscillting sinusoidlly, s in the experiment. nfinitely long stright conductor Tngentil field t rdius r from xis: r ()= μ 0 π r. f the current nd mgnetic field vry in time, s in this experiment, we cn write this s: μ ( ) 0 ( t) r, t =. π r For simplicity, we will not explicitly indicte the time dependence. Circulr coil Field t distnce x long xis of coil with rdius nd N turns: x ()= μ 0 N [ ] 3/ x + ( ) x 1 = 0 1+ x 3/ where 0 = x= ( 0)= μ 0 N. Solenoid coil Field long xis of solenoid tht hs rdius, length L, nd N turns: x ()= μ 0 N x + L x L x= ( 0)= L ( x + L /) + ( x L /) + Helmholtz coils μ 0 N L + 4. Field long xis of Helmholtz coil pir, ech with N turns nd rdius. The coils re lso seprted by distnce, nd x is the distnce of the point on the xis, midwy between the two coils: x ()= μ 0 N x 3/ + x + 5 + x 3/ x + 5 x= ( 0)= 0.716 μ 0 N 4 4 L / / x x O x circulr coil solenoid Helmholtz coil 4
#6A Lbortory Report Sheet Mgnetic Field Mpping - A Nme: Prtner: Lb Section: Dte: Mke mgnetic field mesurements for the circuits, referring to the lb hndout for the detils. Use the pek-to-pek induced emf s mesurement proportionl to the mgnetic field component. Two grphs re required. n ddition, quntittive nd qulittive observtions re required. n ech cse crefully describe exctly wht you re mesuring. A smll sketch of the geometry might help in some cses. Mke sure you mesure the mgnetic field component tht is requested! The mesured mgnetic field component depends on the Frdy probe nd on how you hold it. 1) Solenoid Attch your grph of x vs x, for x > 0, where x is the distnce long the solenoid xis from the center. On your grph, indicte () the physicl end of the solenoid, (b) the point t which x flls to 90% of the vlue t the center, nd (c) the point where x is equl to 50% of the vlue t the center. Comments: Comment on your observtions of x (x =0) vs r both inside nd outside the coil. Where do you find the lrgest vlue of r? (Use sketch if convenient.) Wht do you find for r inside the coil? (Use sketch if convenient.) How does the mgnitude compre with x (x = 0)? How did you mke this comprison? 1
) nfinitely long stright conductor e creful to keep the unshielded wires wy from the probe coil. Mesure the tngentil field vs r nd plot s function of 1/r. R (wire) = R (probe) = r (min) = R (wire) + R (probe) = Trget r (cm) Actul r (cm) r (min) 1.0 1.5 1.5.0.5 3.0 4.0 5.0 10.0 0.0 1/r (cm -1 ) E (mv) Comment on your grph of s function of 1/r. Wht did you lern bout the other components of, i.e., the rdil component nd the component tht is prllel to the conductor? s this wht you expect? Why were you cutioned to keep the unshielded wires wy from the probe coil?