Finding the Earth s magnetic field

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1 Labratry #6 Name: Phys Dr. Cristian Bahrim Finding the Earth s magnetic field The thery accepted tday fr the rigin f the Earth s magnetic field is based n the mtin f the plasma (a miture f electrns and psitive ins) lcated in the ht cre f the Earth. This plasma creates cnvectin currents due t the rtatinal mtin f the planet Earth abut its wn ais. (This spinning mtin creates the cycle day-andnight.) The charge-carrying cnvectin currents mve n circular paths in such a way that the assciated angular mmentum ppses t the angular mmentum due t the spinning mtin f the Earth. This is a manifestatin f the cnservatin f the ttal angular mmentum fr Earth as an islated system. Yu will measure the Earth s magnetic field in Beaumnt s area, using a circular current-carrying cil with N turns. The current-carrying cil prduces at the center a magnetic field f magnitude µ N B = i [1] 2R where i is the electric current thrugh the cil f radius R. The directin f the magnetic field vectr, B, is perpendicular t the area f the cil (use the right-hand rule). At the center f the cil, the field vectr B adds up with the magnetic field f the Earth, B e, and creates a net magnetic field B net = B + B e [2] The Earth s magnetic field B e can be decmpsed alng tw aes as shwn in Fig.2 B e = Bh j + Bv k [3] where B and B are the vertical and the hrizntal cmpnents, respectively, and v h j and k are the unit vectrs fr the y-ais and z-ais, respectively, accrding t Fig.2. Alignment f the setup: Let n current thrugh the cil. Lcate the cmpass needle at the center f the cil and in a plane parallel with the grund. The needle is ging t be riented in the directin f the Earth s magnetic field. Carefully align the cil s that the needle is parallel with the diameter f the cil. The set-up is nw in such a way that Bh j is perpendicular n the magnetic field B created by the current-carrying cil (see Fig.1). First, yu will measure the hrizntal cmpnent f the Earth s magnetic field. Frm the vectr diagram shwn in Fig.1, we get B B h =. [4] tanθ 1

2 A cmpass needle riented parallel with the grund pints in the directin f the net B net (Fig.1). If yu rtate the cmpass needle in a vertical yz-plane magnetic field (Fig.2 is a view frm the right-hand side f Fig.1) the needle will pint in the directin f the Earth s magnetic field B e. Frm Fig.2, we find that the magnitude f the Earth s magnetic field is Bh B e = [5] sinα Bh j y θ z B B net z B e Bv k α y B h j side view f Fig. 1 Fig.1 Fig.2 Using the frmulas [1], [4] and [5], yu can find the Earth s magnetic field, B e, after measurements f the angles θ and α, which crrespnd t the rientatin f the cmpass needle in the hrizntal y- and vertical yz-planes, respectively, fr different values f the electric current, i, thrugh the cil. Measurements and data analysis: 1. Check the set-up: A cil with N turns is cnnected thrugh an ammeter A t a battery (see Fig.3). On the hrizntal table f the cil yu shuld have a cmpass needle. In the absence f any current thrugh the cil, the cmpass-needle pints in the directin f the hrizntal cmpnent f the Earth s magnetic field. Rtate smthly the cil until the cmpass-needle is aligned alng the diameter f the cil. A gd alignment is achieved when the cmpass-needle is riented alng the degrees ais and it is als parallel with the diameter f the cil. DC Pw er supply + -- Cil with N turns A cmpass R 2 Fig.3

3 2. Data acquisitin. Vary the electric current frm a DC pwer supply and read its value with an ammeter. When the current i in the cil increases, the magnetic field B increases accrding t frmula [1], and the cmpass needle will rtate in the clckwise directin. Yu have t adjust the current in the cil s that yu can read the angles θ indicated in Table 1. In rder t get an accurate reading fr angles yu need t lk vertically dwnward twards the middle pint (the supprt) f the needle, and t see simultaneusly three pints: the middle f the needle and the 0 and 180 degrees marks n the cmpass scale. Duble-check yur measurements fr i by decreasing smthly the current in the cil. 3. Calculatin f the Earth s magnetic field by using a linear fit prcedure. Using frmulas [1] and [4] yu can calculate the hrizntal cmpnent f the Earth s magnetic field, B h, as the slpe f the fllwing linear dependence: 2 R i = Bh tanθ, [6] µ N 2 R where tanθ is the independent variable. Fr this purpse yu can use the µ N Micrsft Ecel prgram (fr details see part 4, belw). T make a linear fit means t use a functin y = m + b, where is the independent variable, y is the dependent variable, m is the slpe and b is the intercept with the y-ais (fr = 0). Measure and recrd the radius f the cil: R = m. Yu need t fill ut Table 1, where = Table1 2 R µ N tanθ and µ is 4 π 10-7 T.m/A Trial θ(deg) N = 5 turns N = 10 turns N = 15 turns

4 4. Yu have t run a linear regressin f the data in Table 1 and get the slpe B h (accrding t equatin [6]). Fr that, yu need t use the Micrsft Ecel prgram and fllw the steps: 1) Open Micrsft Ecel. 2) Frm Table 1, fr eample take the N = 5 turns case and write the data n tw clumns f a wrksheet: in clumn A yu write i and in clumn B yu write. 3) Open Tls, Data Analysis, Regressin. 4) A windw called Regressin will appear n the screen. i. G with the cursr n Input Y range and select the clumn f data fr the dependent variable. This shuld be the current i. ii. G with the cursr n Input X range and select the clumn f data fr the independent variable. This shuld be variable. iii. Set Cnstant t zer (this means the intercept is zer). iv. Click n Output range and the empty cell net t it, and net, n the wrksheet in an empty cell smewhere belw the table. v. Click OK. 5) Yu get a reprt. Identify n the last line X variable the number belw Cefficient. This number is the slpe (which is actually the hrizntal cmpnent B h ) in frmula [6]. 6) Repeat the steps 2-6 n tw different wrksheets (i.e. Sheet 2 and Sheet 3) fr the ther tw values f the number f turns N f the cil. 5. Reprt: Fr N = 5 turns, B h = ( Fr N = 10 turns, B h = ( Fr N = 15 turns, B h = ( Calculate the average value fr the hrizntal cmpnent f the Earth s magnetic field Reprt:: B h = ( ) 10-5 Tesla Measure the deviatin f the cmpass needle frm the vertical ais α = deg. (Fig.2) 4

5 Calculate : the average value fr the Earth s magnetic field (using frmula [5]): B e = ( the vertical cmpnent f the Earth s magnetic field (using the vectr diagram in Fig.2) B v = ( Questin: Which cmpnent is larger? B h r B v 6. Check if the data cllected in Table 1 fllws a linear trend (Only in this case the linear regressin yu have used in rder t get B h is fully justified!). Yu need t plt i versus fr each case : N = 5, 10 and 15 turns. Fllw the steps: a. Under Micrsft Ecel, g n the Sheet# where yu have the data yu chse fr pltting and select the clumn fr the electric current, i. b. G n Insert, Chart. It will pen the Chart Wizard windw. c. Select XY (Scatter) fr Chart Type and click Net. d. The Data Range fr the vertical ais f the plt is already selected. e. On the same windw with Data Range click n Series and g t X Values. Select the clumn fr and press Net. f. Press Finish. Shw the plts t yur instructr. 5

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