Petter Lundberg 2015-03-16

Oscillating Magnetic Dipole in an Inhomogeneous Magnetic Field - an instruction for the experimental lab

1

Introduction

Scientific work presented in journals are a common source of knowledge and inspiration. This experimental lab is inspired by the article ”Oscillation of a dipole in a magnetic field: An experiment”, published in Am.J.Physics 58(9), 1990, and corresponding erratum (correction)1 : both of which are based on the work carried out by Bisquert, J, et al.,[1, 2]. As the title of the article suggest, it considers an experiment of small ”frictionless” oscillations, so that the concept of a simple harmonic oscillator can be applied, along the axis of a circular coil carrying an electric current. This oscillator consists of a parallelepipedal permanent magnet attached to a glider, which is allowed to oscillate close to friction free, on a linear air track surrounded by a circular coil of rectangular cross section. The described system is analyzed both theoretically and experimentally in the presented article, where the authors aim to provide a clear example of how a ”judicious use of approximations are essential in order to apply the general theory to a real world problem”. This is motivated by the use of different theoretical models for the angular oscillation frequency: one deduced under the assumption of an infinitely thin coil and neglecting the finite size of the magnet (ωI ), whereas the second model (ωII ) takes these assumption, somewhat, into consideration. Finally both these models are compared to an experimentally deduced value of the oscillation frequency (ωexp ). As you will notice when reading the article, even though the presented experiment is suggested to be suitable as a laboratory project at an intermediate/advanced undergraduate level, it contains some troublesome calculations and derivations2 . Therefore you will be given additional theory, see section 2, as a complement to the theory given in the article, as well as a more clearly stated experimental procedure, see section 3. However, before coming to the laboratory, you are strongly urged to read both the article, its erratum,3 and the upcoming theory section (section 2) thoroughly. You are required to complete the preparatory exercises given in section 2.3. It is of course also, as always, important to read through the entire lab instruction, before the lab, so that you have a clear picture in mind of the work at hand. The examination criteria (report structure, deadline etc.) for this experimental lab are given at the end of this instruction, see section 4.

1.1

Aim

The magnetic dipole and harmonic oscillator are two very important, and commonly used, physical models. The aim of this experimental lab is therefore to enhance your understanding of these models, their underlying approximations, and how these approximations limits their practical applications. Since the exercises in this lab are inspired by the work carried out by Bisquert, J, et al. [1, 2], an additional aim is to introduce you to scientific reading. 1

See attached article and erratum at the end of this lab instruction. Some which even the authors got wrong initially and hence the erratum. 3 Note that eq.(7) and eq.(9) in [1] should be replaced by eq.(1) and eq.(2) in [2], respectively. 2

Ume˚ a University Department of Physics

2

Oscillating Magnetic Dipole in an Inhomogeneous Magnetic Field - an instruction for the experimental lab

Theory

The majority of underlying theory for this lab is given in [1]. However, as mentioned above, this section aims to facilitate your reading and understanding of the work described therein, and thereby also your own experimental work.

2.1

Magnetic dipole moment

When examining the magnetic field around a small permanent magnet, it shares many attributes with those of a magnetic dipole. A magnetic dipole can be viewed upon as a very small loop, carrying a current I . The loop radius is very small, in relation to all other distances. The dipole can be described by its magnetic moment m, according to m = nI Aˆ n,

(1)

ˆ where n is the number of loops around the edge of an certain cross sectional area A, with n being the normal direction. The magnetic field of a dipole Bdipole may, after quite troublesome calculations, using the relation between magnetic field and vector potential, be expressed as Bdipole (r) =

µ0 ˆ ˆ (3 (m · r ) r − m , 4πr3

(2)

where µ0 is the permeability of free space and r = rˆr is the radial distance to the dipole. The magnetic field lines for such a current loop is illustrated in figure 1, where the current ˆ, goes in to (×), and out from (·), the page. If one introduces polar coordinates, with zˆ along n and only considers points along the axis r = zˆz, the magnetic field strength, Bdipolez (z), is given by µ0 m , (3) Bdipolez (z) = 2πz 3 where m is the magnetic moment in ±ˆz. Due to the resemblance between a dipole and a permanent magnet, a magnet can be seen as a huge number of microscopic dipoles, which are more or less parallel. However, since it is impossible to measure I , in the loop around A = π2 , on an atomic level, one has to determine the total magnetic moment mp for the entire permanent magnet. Using superposition, mp can be expressed as the vector sum of all the dipoles inside the magnet. There are a number of alternative procedures of measuring mp . One method, which will be used in this lab, is to measure the magnetic field created by the magnet along its symmetry axis Bexpz and compare it with a theoretical deduced expression, Bcalz . This theoretical expression will be expressed as a function Figure 1: A sketch over the of mp and the dimensions of the magnet. The way of deriving magnetic dipole field of a small this theoretical expression may differ, dependent on which ascurrent loop, figure from [3]. sumption one takes. The simplest approach is to assume that one measures the field from a far enough distance so that the entire magnet may be seen as a single dipole. This enables one to neglect its finite size and hence express Bcalz = Bdipolez . Note however that this model only is valid for distances much larger than the magnet size, i.e. at distances where the magnetic field is very weak and hence hard to measure. Therefore, to get a reasonable accurate expression for Bcalz , close to the magnet, one should take the magnets dimensions into consideration.

2

Ume˚ a University Department of Physics

Oscillating Magnetic Dipole in an Inhomogeneous Magnetic Field - an instruction for the experimental lab

In [1] they made use of the magnet-solenoid equivalence under the assumption of uniformed magnetization of the magnet, when taking the magnets dimensions into consideration. As described, they carried out the derivation for a parallelepipedal-shaped magnet. However if changing the geometry of the magnet, one will have to consider another expression. If for example the magnet instead had a cylindrical shape, with length L and radius a, as illustrated in figure 2, one can express the strength of the magnetic field at the axis of symmetry, assuming it is homogenous magnetized, by integrating the magnetostatic potential Ap over the entire magnet. This integration can be simplified by using cylindrical coordinates ({ρ, φ, z}) and the symmetry of the system. When integrating for the potential along zˆ, the integration can be reduced to an integration over two magnetic monopolar disks, separated by the distance L. Then since Bpz = −∂Ap/∂z the final expression L L z+ 2 z− 2 µ0 m p , q q (4) − Bpz = 2πa2 L L 2 L 2 2 2 z+ 2 +a z− 2 +a is derived, where z originates from the center of the magnet. y a

mp ρ

(0, 0, zM P )

φ L

z

x

Figure 2: A permanent cylindrical magnet of length L, radius a and mp = mp zˆ, built up of numerous of atomic current loops, has an certain magnetic field strength at the position (0, 0, zM P ) along its symmetry axis. Within this lab you will also be using a larger current loop, a so called solenoid. If viewed from a far away distance it could also be approximated as a magnetic dipole. However, since you are going to investigate properties of an oscillating movement along its central axis, such an approximation is invalid and hence is the coil dimensions non-neglectable. Two ways of expressing the field strength of the coil along its central axis Bcoilz were considered in [1]. The first method uses Biot-Savart law, assuming an infinitely thin coil of radius R and N number of turns carrying the current I, which after some calculations (see exercise 1b, in section 2.3), gives the following expression µ0 N IR2 , (5) BcoilIz (z) = 2(R2 + z 2 )3/2 for the magnetic field strength as a function of distance [5]. Note that if R → 0 eq.(5)→eq.(3), I and also that Bcoil (z) has its maximum at z = 0. z The second approach described in [1] considers a coil of finite thickness by approximating the cross-section of the coil to be rectangular with the dimensions 2l1 × 2l2 . By parametrization of the whole coil section, introducing the two coordinates y1 and y2 with their origin at the center of the coil, located R from the coil center and limited by ±{l1 , l2 }, it is possible to compute the current contribution dI from every infinitesimal area dA = dy1 dy2 . As it turns out, this approach is equivalent to integrating eq.(5) over the entire cross section while letting {R, z, I} → {R + y2 , z − y1 , dI}. The final expression, for the magnetic field strength of a coil of finite size

3

Ume˚ a University Department of Physics

Oscillating Magnetic Dipole in an Inhomogeneous Magnetic Field - an instruction for the experimental lab

and rectangular cross-section BcoilII (z), after integration is z µ0 N I (z − l1 ) ln c1 − (z + l1 ) ln c2 , 8l1 l2 p R + l2 + (R + l2 )2 + (z − l1 )2 p c1 = , R − l2 + (R − l2 )2 + (z − l1 )2 p R + l2 + (R + l2 )2 + (z + l1 )2 p c2 = , R − l2 + (R − l2 )2 + (z + l1 )2

BcoilII (z) = z

(6)

where {c1 , c2 } are constants[1].

2.2

Harmonic oscillator

Before going into describing the two models used to express the frequency of a harmonic oscillating dipole in an inhomogeneous magnetic field in [1], it is wise to take a look at the force acting on a dipole in an external field. We known that the force dF on a small current element I dl is orthogonal to the external magnetic field Bext , according to the Lorentz force, and given by dF = I dl × Bext ,

(7)

ˆ and Bext = Br ρˆ + Bz zˆ, using cylindrical coordinates. By completing where in our case dl = dφφ the cross-product in eq.(7), where I is constant, one can integrate over the entire atomic current loop and thereby get the total force acting on the loop, by the external field. To carry out the integration one may express ρˆ in terms of φ and change to Cartesian coordinates, i.e. ρˆ = cos φˆ x + sin φˆ y. The complete integration follows below Z 2π Z 2π ˆ − Br ˆzdφ F = I Bz ρdφ 0 0 Z 2π Z 2π zˆdφ = I Bz (cos φˆ x + sin φˆ y)dφ −Br 0 {z } |0 =0

= −2πI Br zˆ, and as shown, we are left with F = Fz zˆ = −2πI Br zˆ,

(8)

from which one can conclude that the magnetic field from the solenoid, i.e. eq.(5) must be studied more closely, in order to express Br in terms of Bcoilz [4, 5], expressed by either eq.(5) or eq.(6). One method of performing this study, is to consider the magnetic flux Φr through the shaded areas4 , shown in figure 3, which can be expressed as Φr = πr2 (z)Bz . It can be shown that Φr (z) is independent of ∆z given that r(z) is the distance to a certain magnetic field line. By a first order Taylor expansion of Φr (z + ∆z), i.e. dΦr (z) d = Φr (z) + ∆z πr2 (z)Bcoilz (z) dz dz dr dB coilz = Φr (z) + ∆zπ 2r Bcoilz + r2 , dz dz

Φr (z + ∆z) = Φr (z) + ∆z

(9)

one can conclude that in order for Φr to be independent of ∆z, the expression within brackets must equal to zero. In other words Bcoilz 4

dr r dBcoilz + = 0, dz 2 dz

i.e. along the axis

4

(10)

Ume˚ a University Department of Physics

Oscillating Magnetic Dipole in an Inhomogeneous Magnetic Field - an instruction for the experimental lab

−1 which in combination with the fact that all field lines are parallel with B, i.e dr dz −1 = Bcoilr Bcoil z as seen in figure 3, gives r dBcoilz , (11) Bcoilr (r) = − 2 dz i.e. an expression of the relation between the different components of the magnetic field. Since we only are interested in the field lines in the interval r ∈ [0, ], we now have the possibility to express the force along the axis, acting on a current loop, as a function of Bz , by substituting eq.(11) into eq.(8), that is

Fz = −2πI Br () = Iπ2

dBz dBz = [I → nI ] = m , dz dz

(12)

and since a permanent magnet consists of a superposition of many small current loop, the same equation holds for the entire magnet, as m → mp . Now when we know the force acting on our magnet it is possible, by completing the derivative I in eq.(12), using the expression for Bcoil stated in eq.(5) and Newton’s 2nd law of motion (see z exercise 2b, in section 2.3), to derive the following expression for the frequency of a harmonic oscillator inside an inhomogeneous magnetic field as r 3µ0 IN mp , (13) ωI = 2Mtot R3 given that R z and Mtot is the total mass of the oscillating object. It is also possible to arrive at the same expression for ωI , shown in eq.(13), by comparing the potential energy of a harmonic oscillation, i.e. 0.5Mtot ωI2 z 2 , with that of the potential energy of a dipole in an external field, i.e. mp · B, under similar assumptions [1]. Br

I

B Bz

r(z) z

R

Figure 3: The magnetic field of a circular coil, of radius R carrying the current I, is illustrated. Also indicated are the circular areas of radius r(z), where r(z) is the distance to a certain field line. Since there are a lot of underlying assumptions for the theoretical model for ωI , some of which are more or less questionable, Bisquert, J, et al. also considers an additional model for the oscillation frequency, aiming to correct for some of the previously taken assumptions. The second

5

Ume˚ a University Department of Physics

Oscillating Magnetic Dipole in an Inhomogeneous Magnetic Field - an instruction for the experimental lab

expression they end up with is far from pretty; s µ0 N Imp K− ωII = ln + A− − A+ , 8L/2l1 l2 Mtot K+ p 2 R± = R ± l2 , R+ + R+ + L2± p , K = ± 2 L± = L/2 ± l1 , + L2± R− + R− p p 2 2 2 R − R + L + L2± − 4Rl2 R R+ − + ± − p , A± = L2± p 2 2 2 2 + L2± R− R− + L2± R+ R+ + L2± + R+ + L2± + R−

(14)

but after direct, but long, calculations it is possible to show that eq.(14)→eq.(13) in the limit {L, l1 , l2 } → {0, 0, 0} [1].

2.3

Preparatory exercises

During this lab you will study two different theoretical models for the frequency of a harmonic oscillating dipole in an external inhomogeneous magnetic field. To prepare you for this task it is important to grasp how these two models differ, both in their underlying assumptions and derivation. Hence you should complete the following exercises: Exercise 1: Magnetic moment a) Would you say that eq.(3) when m → mp overestimates or underestimates the magnetic field strength, if you compare with eq.(4), assuming the later is more accurate? Make sure to motivate your answer. Hint: Given {mp , L, a}, how will the field strength according to eq.(3) differ from that of eq.(4) along the z axis? b) Derive eq.(5), i.e the field strength of a solenoid along its central axis, using Biot-Savart law, assuming a infinitely thin coil of radius R and N number of turns carrying the current I. c) In [1] they describe a method to check the correctness of eq.(6) (i.e. eq.(13) in [1]). Describe how they perform this check. Note: No calculation needed, only describe how they perform this check. Exercise 2: Axial force and harmonic oscillations a) With eq.(12) as a starting point, describe, both in words and figures, how the direction of the ˆ p = ±mp zˆ and Bcoilz are directed in respect to each other. force depends on the how m b) Complete the derivation of ωI using eq.(12) and eq.(5) as well as Fz = Mtot d2 z/dt2 , given that R z and Mtot is the total mass of the oscillating object. Hint: You will have to solve a differential equation on the form d2 z/dt2 + Cz = 0 c) What assumptions are made when deriving ωI and in what way do these assumption differ from those used when deriving ωII ? d) In the experiment described in [1] they use a parallelepipedal permanent magnet, whilst you will be using a cylindrical. Would you, based on the underlying assumption and derivation described in [1], say that the same expression for ωII , i.e. eq.(14), holds for a cylinder? Make sure to motivate your answer.

6

Ume˚ a University Department of Physics

3

Oscillating Magnetic Dipole in an Inhomogeneous Magnetic Field - an instruction for the experimental lab

Experimental procedure

3.1

Experimental setup

The experimental setup basically consists of: - An air track, with glider, to which a solenoid (R = 0.2 m and N = 154), is attached. - A current source and and ampere-meter, used to drive and measure I. - A cylindrical permanent magnet, with a dipole moment mp that is to be determined. - A Hall probe, used to measure magnetic fields. - A stopwatch, used to time the oscillation frequencies. In addition to the above stated equipment you will also have the possibility to use rulers, calipers and balances, in order to carry out the experiment. A simplified schematic sketch of the fundamental parts of the experimental setup is illustrated in figure 4, indicating both the air track, glider, magnetic dipole and solenoid.

y z x

Figure 4: A simple schematic sketch over the experimental setup, illustrating the air track (triangular) on which the glider (gray) may move close to frictionless. On top of the glider is a cylindrical permanent magnet (red) centered and attached. This ”dipole” is able to oscillate in the inhomogeneous magnetic field caused by the solenoid (circular), powered by a current source (not illustrated in the figure), surrounding the center of the air track.

3.2

Experimental exercises

Exercise 1: Determine the magnetic moment of your permanent magnet by measuring the magnetic field strength along its symmetry axis. In order to compare the accuracy of the two theoretical models for the angular frequency ω it is necessary to determine the magnetic moment of the permanent magnet used. One method to get an experimental estimation of mp is to use a Hall probe to measure the magnetic field caused by an assumed homogeneously magnetized dipole of finite size, at a given distance zM P from the probe, see figure 2. By alternating this distance along a given axis (in our case zˆ), experimental values for for the magnetic field at a given distance, i.e. Bexp (zM P ), can be expressed. The magnetic moment may then be estimated as the slope of a linear fit between the experimentally measured Bexp (zM P ) and the theoretically deduced m−1 p Bcal (zM P ). The procedure for carrying out this experimental exercise follows. Step 1: Place the magnetic dipole within the measurement stand and measure the field strength along its axis of symmetry. This measurement should be carried out for at least 10 distances, e.g. zM P = {2, 4, 6, . . . , 20} cm or zM P = {8.0, 8.5, 9.0, . . . , 15.0} cm. 7

Ume˚ a University Department of Physics

Oscillating Magnetic Dipole in an Inhomogeneous Magnetic Field - an instruction for the experimental lab

Step 2: Measure the Earth’s magnetic field strength Bearth and make sure to compensate for it, i.e. remove it from your Bexp (zM P ) data. One way of measuring an estimate of Bearth is to measure Bexp (zM P ) from both end-sides of your magnet, for a couple of ZM P , from which it is possible to compute an estimate for Bearth . Step 3: Use eq.(4) to express Bcal (zM P ) and thereafter estimate mp , as the slope of a linear fit between Bexp (zM P ) and m−1 p Bcal (zM P ). You will need to measure the magnets dimensions, i.e. L and a. The same kind of plot is given in Fig.4 in [1]. Exercise 2: Determine the limiting oscillation amplitude, such that the approximation R z does not affect the oscillation frequency significantly. In order to determine the oscillation frequency within the limitations of the underlying assumptions of the theoretical frequency models one must investigate at which oscillation amplitude zamp , i.e. distance from equilibrium, the approximation R z breaks. The reason for this exercise is determine which amplitude, that is start position, to use in the upcoming exercise. The procedure for this experimental exercise is stated below. Step 1: Mount and center your permanent magnet on the glider. Step 2: Make sure that the air track is level. Step 3: Turn on the air, so that the glider may move freely along the air track. Step 4: Set Icoil = 5.00 A and place the center the glider 1 cm from its equilibrium point. Step 5: When released, time a suitable number of oscillations, e.g. 10 oscillations, and determine the oscillation frequency ωexp (zini ). Hint: Try to decide if it is easier to time the oscillations at a end-side or at the midpoint of the oscillating movement. Step 6: Repeat Step 5 for increasing amplitudes, e.g. increase zinin+1 = zinin + 2 cm, until a clear pattern is visible, tentatively until 15 cm. Step 7: Use your measurement data to determine a suitable zamp to use in exercise 3. Make sure to motivate your decision. Exercise 3: Determine the magnetic moment of the dipole, by measuring the oscillation frequency in an inhomogeneous magnetic field. Assuming that eq.(13) and eq.(14) holds, one should be able to determine mp , by experimentally measure the oscillation frequency ωexp , over a range of currents. The procedure for carrying out this experimental exercise follows below. Step 1: Make sure that your magnet still is mounted and centered on the glider. Step 2: Set the current Icoil = 1.00 A and let the glider oscillate from the experimentally determined suitable starting position from Exercise 2. Time a suitable number of oscillations, e.g. 10 oscillations, and determine ωexp (Icoil )|zini =zamp , for Icoil = 1.00 : 0.50 : 5.00 A. Step 3: Measure Mtot and the rectangular dimensions of the solenoid, i.e {l1 , l2 }. Step 4: Use the same method as in Exercise 1 to determine mpexp using both eq.(13) and eq.(14) 2 and compare with the measured value in exercise 1. In other words plot ωexp towards both −1 2 −1 2 mp ωI and mp ωII and determine mp as the linear slope. 8

Ume˚ a University Department of Physics

4

Oscillating Magnetic Dipole in an Inhomogeneous Magnetic Field - an instruction for the experimental lab

Examination

Since you have been given the opportunity to practice your skills in scientific reading, it would be wrong not to give you the chance to practice your scientific writing as well. therefore the examination for this experimental lab will be a complete report, written in English and following the structure suggested in section 4.1. To facilitate your writing you may consider both the preparatory and experimental exercises as a guide, as well as the questions and comments given in section 4.2. The report should be well written, with your own words, using proper language and notations, so make sure to proofread your writing. The deadline for handing in the report to your opponents is no later than 2015-03-22 at 11.59 p.m.When you have corrected your report, with respect to the comments from your opponent, the final report, together with the additional document containing comments and corrections, should be sent in to either: [email protected] or [email protected], no later than 2015-03-27 at 11.59 p.m.

4.1

Report structure

Below follows a suggested report structure, which you should look upon as a recipe, i.e. you are free to adapt, add, or change to your liking, but the outcome should be at least as well-structured as the suggestion. Cover page - As always when handing in work, or publishing, it is important to know by whom the work has been carried out, what they have done, when and why, as well as contact information to the authors. Therefore it is essential that your cover page contains a suitable title, your names and contact information, which course you are taking and its credits, as well as the date of submission. It is also customary to state at which department and university the course is given. Abstract - Describe, in a short and concise way, what you have done, why, what results you have achieved and any conclusion drawn. The abstract may (preferably) be included on the cover page, or alternately at the beginning of your report. Introduction - The background to the experiment and the work carried out. Introduce your readers to what you have done. Theory - Describe the theoretical models, calculations and assumption used to carry out the experiment and computations. It is strongly recommended to include figures for clarity. Experimental procedure - Explain, using your own words and clear figures, the experimental setup (which components have you been using), method, and the measurement techniques, used throughout your experiment. You should also include the computational steps carried out in order to achieve the results presented later on. Do not forget to specify magnitudes of distances, currents, voltages etc. Based on this section, one should be able to repeat your experimental work and, ideally, end up with the same result. Results - What results have you achieved from the experimental work carried out. Make sure to always include error estimation when presenting measurement data and/or results. Discussion - Discuss your results, their accuracy, experimental errors and possible improvements in your work. The discussion may preferably end with a few sentences of final conclusion. References - Make sure to state all references when writing your report. 9

Ume˚ a University Department of Physics

Oscillating Magnetic Dipole in an Inhomogeneous Magnetic Field - an instruction for the experimental lab

Appendix - If you have large data sets, of e.g. experimental data, to present or harsh computations, e.g. Matlab-scrips, you may attach them in an appendix at the very end. This is to facilitate reading your report, but do not forget that each included table, graph, script must be referred to in the ”actual” report.

4.2

Things to think about

Throughout this instruction you have been given a lot of information considering various approaches for calculating and measuring magnetic dipole moments and frequencies of an oscillating dipole in an inhomogeneous magnetic field. It may therefore be hard to know what to put into your report, which is the reason for this section. Below follows a list of obvious and non-obvious things that might be worth to consider while writing your report. - The preparatory exercises do not have to be included as such, e.g. full derivations might be unnecessary. However they might be a good indicator for what to introduce, explain and discuss in your report. - Are every figure and table referred to in the main text? Do they also have a captions, which described the contents of the table or figure? - Are the results presented in a nice and clear way? - Make sure to clearly state what underlying assumptions there are for each and every model. How do they differ? - Do the more advanced models bring anything new to the table or do they only make computations unnecessarily troublesome? - Can you give physical interpretation to any of the more advanced models, and if so, do they make sense? - Would you say that measuring oscillations of a permanent magnet is a good way of estimating its magnetic moment? - How accurate are your results? Is any model more accurate than the other? - Which are the error sources of the experimental work carried out and how could one minimize these?

References [1] Bisquer, J, Hurtado, E, Maf´e, S, Pina, J (1990). Oscillations of a dipole in a magnetic field: An experiment. American association of Physics teachers, Vol. 58 No. 9. [2] Bisquer, J Hurtado, E, Maf´e, S, Pina, J (1991). Erratum: ”Oscillations of a dipole in a magnetic field: An experiment”. American association of Physics teachers, Vol. 59 No.6. [3] ”Wikipedia, Magnetic dipole. 2015 01 10. http://en.wikipedia.org/wiki/Magnetic dipole ¨ [4] Nordling, C , Osterman, J. (2006). Physics Handbook 8th . New Jersey, USA, Studentlitteratur. [5] Griffiths, D, Soroka, M, Throop, W. (2003). Introduction to electrodynamics 3th . Lund, Sweden, Pearson Education.

10