.!"#$%#&'&()#*+(,- Science Supplementary Material

SMT359 Electromagnetism

SMT359 AYRF

Are you ready for SMT359?

Contents

Page1 Introduction 12 Suggested prior study 23 Key concepts in physics 24 Mathematical skills 35 Other skills 46 Suggested further reading and preparatory work 57 References 58 Answers to self-assessment questions 6

If, after working through these notes, you are still unsure about whether SMT359 is the rightcourse for you, we advise you to seek further help and advice either from a Regional Advisor orfrom a Science Staff Tutor at your Regional Centre.

1 Introduction

If you are intending to study SMT359, you should make sure that you have the necessary background knowledgeand skills to be able to enjoy the course fully and to give yourself the best possible chance of completing itsuccessfully.

Read through these notes carefully and work through the questions. This is a useful exercise for all prospectivestudents of the course, even for those who have already studied other Open University science and mathematicscourses and who have completed the recommended prior study courses for SMT359 (see Section 2 below).Working through these diagnostic notes will serve as a reminder of some of the knowledge and skills which youwill be assumed to have, either from OU Level 2 science and mathematics courses or from other prior study.

If you find that you can answer nearly all the questions in these notes, within about three hours and with onlyoccasional reference to prerequisite material, it is likely that you are well prepared to start SMT359. If you havedifficulties with some questions, or take much longer than three hours, this will indicate that you have gaps inyour knowledge or that you need to improve your mathematical fluency.

Section 6 gives advice on specific remedial actions you can take. If you have substantial difficulties with five ormore questions, you probably have a considerable amount of catching up to do, and should seriously assesswhether SMT359 is the right course to attempt at this stage of your studies. Our experience is that studentswith insufficient preparation find Level 3 physics courses difficult, and often drop out.

Copyright c© 2005 The Open University SUP 85988 51.1 PT/TeX

2 Suggested prior study

SMT359 is a Level 3 course which makes intellectual demands appropriate to the third year of a degree. Thelaws of electromagnetism, expressed in their most powerful and general form, involve the use of advancedmathematical techniques. Most of these techniques are introduced in the prerequisite mathematics course,MST209 Mathematical methods and models, (or its predecessor MST207 Mathematical methods, models andmodelling). It is therefore strongly recommended that you have a good pass in MST209 (or MST207) and thatyou are familiar with the topics of complex numbers, vectors, vector calculus, partial differentiation, differentialequations, multiple integrals and line integrals, all of which are taught or used in that course.

SMT359 also assumes that you have previously studied physics, especially mechanics and electromagnetism, atan introductory level. A good pass in S207 The Physical World (or its predecessor S271 Discovering Physics) istherefore recommended.

3 Key concepts in physics

SMT359 presents electromagnetism as a coherent body of knowledge, developed from basic principles explainedfully within the course. Nevertheless, previous study of electromagnetism at the level of S207 is extremelyhelpful. In particular, a basic understanding of electrostatics, electrostatic potentials, electric and magneticfields, field lines and equipotentials, the Lorentz force law, circuit analysis, electromagnetic induction and waveswill provide a secure base for your studies.

The course also assumes a basic knowledge of mechanics, including Newton’s laws, uniform acceleration andcircular motion, work and energy, the force–potential energy relationship, the conservation laws of momentumand angular momentum, and properties of waves.

The following self-assessment questions test your understanding of some basic concepts in mechanics andelectromagnetism. The answers may be found in Section 8 of this booklet.

MechanicsQ1 (Newton’s second law; uniform acceleration) Aconstant force of magnitude 2 × 10−5 N acts on aparticle of mass 5× 10−6 kg.

(a) Determine the magnitude of the acceleration ofthe particle.

(b) If the particle is at rest at time t = 0, how farwill it have moved by time t = 1 s?

Q2 (momentum and kinetic energy) Determine

(a) the magnitude of the momentum, and(b) the kinetic energy

of a body of mass 3 mg moving at a speed of 5 cm s−1.

Q3 (force and the gradient of potential energy) Abody moves along the x-axis in a region where itspotential energy function is given by V (x) = Cx2,where C is a constant. Determine the force acting onthe body when it is at x = 2m, given thatC = 5Jm−2.

Q4 (centripetal acceleration, angular momentum,torque) A body of mass 10−2 kg moves at a constantspeed of 2m s−1 along a circular path of radius 0.5 m.

(a) What are the magnitude and direction of theacceleration?

(b) Specify the force acting on the body.

(c) What is the torque of the force about the centreof the circular path?

(d) What is the magnitude of the angularmomentum about the centre?

Electromagnetism and waves

The elementary charge is e = 1.60× 10−19 CThe electron mass is m = 9.11× 10−31 kg

(4πε0)−1 = 8.99× 109 Nm2 C−2

The speed of light is c = 3.00× 108 ms−1

The electronvolt is 1 eV = 1.60× 10−19 J

Q5 (electric current and charge) What electriccurrent is produced by a uniform beam of electronsmoving from left to right if 1020 electrons pass a givenpoint per second?

Q6 (Lorentz force) An electron moves with a speedof 1.0 km s−1 in a region where there is a uniformelectric field of magnitude 3.0 kV m−1 and a uniformmagnetic field of magnitude 2.0× 10−3 T. Determine

(a) the magnitude of the electric force,

(b) the magnitude of the magnetic force, given thatthe electron is travelling perpendicular to the magneticfield direction.

Q7 (the electronvolt) Determine the kinetic energyin joules, and also in electronvolts, of

(a) an electron moving at a speed of 1000 km s−1,

(b) a proton that has been accelerated from restthrough an electric potential difference of −5000 V.

2

Q8 (Coulomb force law and electrostatic potentialenergy) Determine (a) the magnitude of the electricforce between two protons, and (b) their electrostaticpotential energy, when they are separated by 1.0 ×10−11 m.

Q9 (sinusoidal waves, wavelength, period,frequency) Consider the wave described byx = (0.2m) cos(2.0z − 3.0t), where x and z aremeasured in metres and t in seconds. For this wavewrite down: (a) the amplitude, (b) the wavelength,(c) the angular frequency, (d) the period (e) the speedand direction of travel.

4 Mathematical skills

Mathematics is a vital tool in physics — it provides the language in which ideas are expressed and givesmethods that allow exact conclusions to be reached. You will therefore need to be fluent with algebraicmanipulation, vectors, differentiation and integration. In addition, electromagnetism makes use of some specialmathematical techniques including vector calculus, multiple and line integrals and differential equations. Allthese topics are covered in the prerequisite mathematics course MST209 Mathematical methods and models (orits predecessor MST207) or are developed as part of SMT359.

The following questions test your understanding of some mathematical techniques that it is assumed you will beable to use at the outset of your studies of SMT359.

Vectors

Q10 (geometric vectors; unit vectors; componentsand magnitude of a vector) Given the vector r =3i − 2j + k, where i, j, k are Cartesian unit vectors,determine

(a) the value of ry, the y-component of r,

(b) the magnitude |r| of vector r,

(c) a unit vector r̂ parallel to r.

Q11 (geometric vectors) Evaluate the expression,(r− r′)f(r)/|r− r′|3, where f(r) = |r|2 and r = i + 2jand r′ = i− j + 4k.

Q12 (scaling and adding vectors; dot product; crossproduct) Given the vectors a = 3i− k and b = i− j,determine

(a) −a,

(b) a− 2b,

(c) a . b,

(d) a × b,

(e) b × a.

Algebra, complex numbers andcoordinates

Q13 (trigonometry)

Given that sin 2x = 2 sin x cosx,cos 2x = cos2 x− sin2 x andsin2 x + cos2 x = 1, show that(sin 2θ)2 + 4 sin4 θ = 2(1− cos 2θ).

Q14 (complex numbers, modulus and polar form)Given z = 3− 2i (where i =

√−1) write down:

(a) the imaginary part of z,

(b) the modulus |z| of z,

(c) z2.

(d) Express z in polar form.

Q15 (complex numbers, Euler’s formula) Giveny = eikx, where k and x are real, determine

(a) the real part of y,

(b) |y|2.Q16 (cylindrical coordinates) Give the cylindricalcoordinates (r, φ, z) of the following points specifiedby Cartesian coordinates (x, y, z):

(a) (0, 1, 0)

(b) (1, 1, 1).

Q17 (spherical coordinates) A potential energyfunction is given by

V (x, y, z) = 12 (Ax2 + By2 + Cz2).

Express this function in spherical coordinates.For what values of B and C does the force dependonly on the distance from the origin?

Q18 (three-by-three determinants) Evaluate thedeterminant,∣∣∣∣∣∣

a b cx 0 −xy z y

∣∣∣∣∣∣Q19 (summation symbol and factorial notation)Evaluate

(a)3∑

j=1

j

(b)4∑

n=1

(n2 + 1)

(c) 5!

3

Calculus

Q20 (differentiation of functions of a single variable:products, quotients and composite functions)Differentiate the following

(a) y = x sinx,

(b) f(x) = (tan x)/x,

(c) u = eiωt, where i =√−1 and ω is constant,

(d) y = loge

(1 + x2

).

Q21 (properties of linear differential equations)

(a) Find the values of k for which cos kx and sin kxare solutions of the differential equation

d2y

dx2+ 4y = 0.

(b) Write down an arbitrary linear combination ofthe solutions found in part (a). What property ofthe differential equation guarantees that your linearcombination is a solution?

Q22 (partial differentiation) If f(x, y) = cx/(x2 +y2), where c is a constant, determine ∂f/∂x, ∂f/∂yand ∂2f/∂x∂y.

Q23 (integration) Evaluate∫ b

a

C

rdr, where C is a

constant.

Q24 (integration by substitution) Given that∫dx

a2 + x2=

1a

tan−1 x

a, determine

∫ 2

0

4 dy

1 + 9y2.

Q25 (grad, div and curl)

(a) If f = x2 + 2y + 3yz, evaluate grad f .

(b) If A = x2i + (2 + y)j + xyk, evaluate divA andcurlA.

Q26 (multiple integrals) Evaluate∫ y=2

y=0

∫ x=1

x=0

x2y dx dy.

Q27 (volume integrals) Evaluate the volume

integral I =∫

B

U0r2 dV using spherical coordinates,

where U0 is a constant and B is a sphere of radius Rcentred on the origin.

Q28 (vectors and line integrals) Evaluate the line

integral∫

C

E . dl, where E = E(r)eθ with eθ being a

unit vector in plane polar coordinates in the directionof increasing θ, and C is a circular contour of radiusR centred at the origin and which is traversed in thedirection of increasing θ.

5 Other skills

You should have the following skills:

Study skills The ability to:

• organize time for study, and pace study appropriately;

• read effectively and identify relevant information from scientific texts and accounts;

• seek help when it is required.

Writing skills The ability to:

• write coherently;

• give succinct and complete definitions;

• write a scientific account with appropriate diagrams and equations.

Problem-solving and modelling skills The ability to:

• recognize the physical principles and equations that apply in described situations;

• translate a problem described in words into a form suitable for mathematical analysis;

• recognize information supplied implicitly or explicitly;

• draw appropriate diagrams;

• check answers and interpret a mathematical solution in physical terms.

ICT skills The ability to:

• use applications and simulation software;

• access information on the Web;

• communicate using email and conferencing software.

4

6 Suggested further reading and preparatory work

If the SAQ questions show that you need to learn more about certain topics or improve certain skills, thefollowing sources are recommended.

Key concepts in physics

The Open University Level 2 course S207 The Physical World is published as a series of seven books by theInstitute of Physics, and is strongly recommended. Four of the books in this course are especially relevant forSMT359.

For introductory mechanics

Describing motion by R. Lambourne and A. Durrant

Predicting motion by R. Lambourne

For introductory electromagnetism

Static fields and potentials by J. Manners

Dynamic fields and waves by A. Norton

Many other physics textbooks describe the key physics concepts needed for SMT359. We recommend:

Fundamentals of physics by D. Halliday, R. Resnick and J. Walker

Physics by H. Ohanian

Physics for Scientists and Engineers by P. Tipler

Mathematical skills

The Open University Level 2 course MST209 is strongly recommended. The following Units are especiallyrelevant for SMT359.

Units 2 and 3 on differential equationsUnit 4 on vectorsUnit 12 on partial differentiationUnit 24 on vector calculus, line integralsUnit 25 on multiple integrals

If you need to refresh your knowledge of differentiation and integration, MST121 Block C and MS221 Block Care recommended. Complex numbers are covered in Unit D1 of MS221. Two extensive mathematical texts at anappropriate level to prepare for SMT359 are Basic Mathematics for the Physical Sciences and FurtherMathematics for the Physical Sciences, both by R. Lambourne and M. Tinker.

Other skills

Comprehensive guidance and advice on studying science courses may be found in The Sciences Good StudyGuide. Basic skills are covered in Chapters 1, 2 and 6, and writing skills in Chapter 9.

Specific skills needed for SMT359 are also addressed in the Revision and Consolidation chapters of S207. Inparticular, Chapter 6 of Predicting motion and Chapter 5 of Dynamic fields and waves develop problem-solvingskills. Chapter 5 ofClassical physics of matter also develops the skills needed to give concise and full definitions.

7 References

Describing motion, by R. Lambourne and A. Durrant, Institute of Physics Publishing 2000 (ISBN 0 7503 0715 3)

Predicting motion, by R. Lambourne, Institute of Physics Publishing 2000 (ISBN 0 7503 0716 1)

Classical physics of matter, by J. Bolton, Institute of Physics Publishing 2000 (ISBN 0 7503 0717X)

Static fields and potentials. by J Manners, Institute of Physics Publishing 2000 (ISBN 0 7503 0718 8)

Dynamic fields and waves, by A. Norton, Institute of Physics Publishing 2000 (ISBN 0 7503 0719 6)

Fundamentals of physics, by D. Halliday, R. Resnick and J. Walker, John Wiley and Sons 2004 (ISBN 0 471465097)

5

Physics, by H. Ohanian, W. W. Norton & Company 2002 (ISBN 0 3939 4896 X)

Physics for Scientists and Engineers, by P. Tipler, Worth Publishers 1998 (ISBN 1 5725 9673 2)

Basic Mathematics for the Physical Sciences, by R Lambourne and M. Tinker, John Wiley and Sons 2000(ISBN 0 4718 5207 4).

Further Mathematics for the Physical Sciences, by M. Tinker and R. Lambourne, John Wiley and Sons 2000(ISBN 0 4718 6723 3).

The Sciences Good Study Guide, by A. Northedge. J. Thomas, A. Lane and A. Peasgood, Open University 1997(ISBN 0 7492 3411 3)

8 Answers to self-assessment questions

Mechanics

Q1(a) Since F = ma, then

a =(2× 10−5 N

)/(5× 10−6 kg

)= 4Nkg−1 = 4ms−2.

(b) Use s = ut + 12at2 with u = 0.Then

s =12× 4m s−2 × (1 s)2 = 2m.

Q2(a) Momentum p = mv, so

p = mυ = 3× 10−6 kg × 5× 10−2 ms−1

= 1.5× 10−7 kg m s−1.

(b) Kinetic energy = 12mv2

=12× 3× 10−6 kg × (

5× 10−2 ms−1)2

= 3.75× 10−9 J ≈ 4× 10−9 J

to one significant figure.

Q3 Use Fx = −dV/dx = −2Cx. At x = 2 m andwith C = 5 Jm−2, this is Fx = −2× 5 J m−2 × 2 m =−20 N. Note: in three dimensions, the correspondingformula is

F = −grad V = −∇V

= −i ∂V/∂x− j ∂V/∂y − k ∂V/∂z.

Q4(a) The centripetal acceleration is of magnitudev2/r =

(2m s−1

)2/(0.5m) = 8m s−2. Its direction is

towards the centre of the circular path.

(b) F = ma. Therefore |F| = 10−2 kg ×8 m s−2 =8× 10−2 N. Its direction is towards the centre.

(c) Torque is Γ = r × F, that is |Γ| = |r||F| sin θ,where θ is the angle between r and F. But r and Fare antiparallel (i.e. θ = 180◦). Therefore Γ = 0.

(d) Angular momentum L = r × p. Therefore|L| = |r||p| sin θ. For circular motion, r and p areperpendicular (i.e. θ = 90◦). Therefore

|L| = |r||p| = mrv

= 10−2 kg × 0.5m× 2m s−1 = 10−2 kg m2 s−1

.

Electromagnetism and waves

Q5 Magnitude of electric current = number ofparticles per unit time × magnitude of electric chargeof particle

= 1020 s−1 × 1.60× 10−19 C = 1.60× 101 Cs−1

= 16A.

Because the charge of an electron is −e, the currentflows from right to left.

Q6 Lorentz force F = qE + qv × B.

(a) Magnitude of electric force is

qE = 1.60× 10−19 C × 3.0× 103 V m−1

= 4.8× 10−16 N.

(b) Magnitude of magnetic force is qv ×B

= qvB sin θ = qvB

= 1.60× 10−19 C × 1.0 × 103 ms−1 × 2.0× 10−3 T= 3.2× 10−19 N.

Q7(a) Kinetic energy = 12mv2

=12× 9.11× 10−31 kg × (

1.0 × 106 ms−1)2

= 4.56× 10−19 J =4.56× 10−19 J

1.60× 10−19 J eV−1

= 2.8 eV.

(b) Kinetic energy acquired = potential energy lost

= −qV = −1.60× 10−19 C× (−5000V)

= 8.00× 10−16 J =8.00× 10−16 J

1.60× 10−19J eV−1

= 5.0× 103 eV.

Q8(a) Electrostatic or Coulomb force is

F =1

4πε0

q1q2

r2

= 8.99× 109 N m2 C−2 ×(1.60× 10−19 C

)2

(1.0× 10−11 m)2

= 2.3× 10−6 N.

6

(b) Potential energy is

V =1

4πε0

q1q2

r

= 8.99× 109 Nm2 C−2 ×(1.60× 10−19 C

)2

1.0× 10−11 m= 2.3× 10−17 J.

Q9(a) Amplitude = 0.2 m.

(b) Wavelength = 2π/2.0 m = 3.1 m.

(c) Angular frequency = 3.0 rad s−1.

(d) Period = 2π/3.0 s = 2.1 s.

(e) Wave moves in +ve z-direction with speed3.0/2.0 m s−1 = 1.5 m s−1.

Vectors

Q10(a) ry = −2.

(b) |r|2 = 32 + (−2)2 + 12. Therefore |r| = 141/2.

(c) r̂ = 14−1/2r.

Q11 f(r) = |r|2 = 1 + 22 = 5.r− r′ = 3j− 4k. Therefore

|r− r′| =√

32 + 42 = 5

and(r− r′)f(r)|r− r′|3 =

(3j− 4k)5125

= 0.12j− 0.16k

.

Q12(a) −a = −3i + k.

(b) a− 2b = (3i− k)− 2(i− j) = i + 2j− k.

(c) a . b = axbx + ayby + azbz

= 3× 1 + 0× (−1) + (−1)× 0 = 3.

(d) a × b =(aybz − azby) i + (azbx − axbz) j + (axby − aybx)k= (0× 0− (−1)× (−1))i + ((−1)× 1− 3× 0)j+ (3× (−1)− 0× 1)k= −i− j− 3k.

(e) b × a = −(a × b) = i + j + 3k.

Algebra, complex numbers andcoordinates

Q13 The left-hand side of the required relationshipcan be written as follows:

(sin 2θ)2 + 4 sin4 θ = (2 sin θ cos θ)2 + 4 sin4 θ

= 4 sin2 θ(cos2 θ + sin2 θ

)= 4 sin2 θ.

Now cos 2x = cos2 x− sin2 x = 1− 2 sin2 x, so

4 sin2 θ = 4(

1− cos 2θ

2

)= 2(1− cos 2θ),

which is equal to the right-hand side of the requiredrelationship.

Q14(a) Im z = −2.

(b) |z| = (z∗z)1/2 = ((3 + 2i)(3− 2i))1/2

= (9 + 6i− 6i + 4)1/2 =√

13.

(c) z2 = (3− 2i)2 = 9− 12i− 4 = 5− 12i.

(d) Using the form reiθ, r = |z| = √13, and

θ = tan−1(Im z/Re z) = tan−1(−2/3) = −34◦.

Q15(a) y = cos kx + i sin kx. ThereforeRe y = cos kx.

(b) |y|2 = y∗y = e−ikx × eikx = 1.

Q16 Use r =√

x2 + y2, φ = cos−1(x/r) for y ≥ 0,and z′ = z.

(a) (0, 1, 0) → (1, π/2, 0).

(b) (1, 1, 1) → (√

2, π/4, 1).

Q17 Using x = r sin θ cosφ, y = r sin θ sinφ and z =r cos θ, thenV (r, θ, φ)= 1

2 (Ar2 sin2 θ cos2 φ + Br2 sin2 θ sin2 φ + Cr2 cos2 θ)= 1

2r2(A sin2 θ cos2 φ + B sin2 θ sin2 φ + C cos2 θ).The force depends on r only when C = B = A givingV (r) = Ar2/2.(Note: Strictly speaking V (r, θ, φ) and V (x, y, z) aredifferent functions and so different symbols should beused. However, you will find this abuse of notation inmany physics texts, including SMT359.)

Q18 The determinant is given by

det = a(0 + xz)− b(xy + xy) + c(xz − 0)= xza− 2xyb + xzc.

Q19(a) 1 + 2 + 3 = 6.

(b) (12 + 1) + (22 + 1) + (32 + 1) + (42 + 1) = 34.

(c) 5! = 5× 4× 3× 2× 1 = 120.

7

Calculus

Q20(a) dy/dx = 1× sinx + x cos x.

(b) f ′(x) =(x sec2x− tan x

)/x2.

(c) du/dt = iωeiωt = iωu.

(d) dy/dx = 2x/(1 + x2

).

Q21(a) For y = cos kx, dy/dx = (−k) sin kx,d2y/dx2 = −k2 cos kx. Substituting into the equationgives k2 = 4. Therefore k = ±2. Similarly for sin kx,we find k = ±2.

(b) The solutions in part (a) are y = cos 2x andy = ± sin 2x. An arbitrary linear combination ofthis is y = α cos 2x + β sin 2x, where α and β arearbitrary constants. The fact that the equation islinear guarantees that this is a solution.

Q22 The required partial derivatives are

∂f

∂x=

c

x2 + y2− 2cx2

(x2 + y2)2.

∂f

∂y=

−2cxy

(x2 + y2)2.

∂2f

∂x∂y=

∂2f

∂y∂x

=−2cy

(x2 + y2)2+

8cx2y

(x2 + y2)3.

Q23∫ b

a

C/r dr = C[loge r

]b

a= C loge(b/a).

Q24 Make the substitutions a = 1, x2 = 9y2, anddx = 3dy. Then∫ 2

0

4 dy

1 + 9y2=

43×

∫ 6

0

dx

a2 + x2

=43×

[1a

tan−1 x

a

]x=6

x=0

=43

tan−1 6.

Q25 (a)

grad f =∂f

∂xi +

∂f

∂yj +

∂f

∂zk

= 2xi + (2 + 3z)j + 3yk.

(b)

div A =∂Ax

∂x+

∂Ay

∂y+

∂Az

∂z= 2x + 1 + 0= 2x + 1.

curlA =(

∂Az

∂y− ∂Ay

∂z

)i +

(∂Ax

∂z− ∂Az

∂x

)j

+(

∂Ay

∂x− ∂Ax

∂y

)k

= (x− 0)i + (0− y)j + (0− 0)k= xi− yj.

Q26∫ y=2

y=0

∫ x=1

x=0

x2y dx dy =∫ y=2

y=0

y[x3/3

]10

dy

=∫ y=2

y=0

(y/3) dy =[y2/6

]20

= 2/3.

Q27∫B

U0r2 dV =

U0

∫ r=R

r=0

∫ θ=π

θ=0

∫ φ=2π

φ=0

r2r2 sin θdφdθdr

= U0

[∫ R

0

r4 dr

][∫ π

0

sin θ dθ

] [∫ 2π

0

dφ

]

= U0

[r5

5

]R

0

[− cos θ]π

0

[φ

]2π

0= U0 × R5

5× 2× 2π

= 4πU0R5

5.

Q28 For a circular contour, of radius R, dl = R dθ,i.e. dl = R dθeθ, and θ is integrated from 0 to 2π. Then

E . dl = E(R)R dθ and∫ 2π

0

E(R)R dθ = 2πE(R)R.

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