2014-2015 Magnetostatics Cours

8/15/2019 2014-2015 Magnetostatics Cours

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Chapter 4

Magnetostatics

IntroductionAs previously stated, in the static regime (or steady state), where the sources {ρ,

−→J } do not

vary in time, the coupling between the electric and magnetic eld vanishes. The magnetic eld,which concerns us here, is only created by constant currents. This is the area of Magnetostatics .

Chapter 1 contains, in principle, all laws and properties necessary to solve any problem inelectromagnetism, so that, very much like what we have done in chapter 2, we will rst briey goover the relations valid in magnetostatics.

4.1 Maxwell’s equations applied to Magnetostatics and con-sequences

4.1.1 Magnetic force

The force exerted by a magnetostatic eld −→B (M, t ) on a point charge q placed at point M at

time t and having a velocity −→v (M, t ) is given by:

−→F = q −→v (M, t ) ∧

−→B (M, t ) (4.1)

Property : The magnetic force does not produce any work since δW = −→F . −→dl = q −→v ∧−→B .−→dl =0. From the work-energy theorem, one can infer that the magnetic force does not change the kineticenergy of a particle. In particular, a magnetic eld cannot set a particle in motion .

A nice visualization of this force can be obtained in so-called bubble chambers 1 A bubblechamber consists in a liquid that is kept, in a metastable phase, at a temperature above its boilingpoint through an increased pressure. When a particle goes through the chamber, it may collide withthe electrons of the atoms forming the liquid. The energy given by the particule to the electronslocally heats up the liquid and hence bubbles form on the particle trajectory. The bubbles aretherefore the signature of a particle going through the chamber. When the particle is charged, anapplied magnetic eld allows to distinguish between positively and negatively charges species (seegure 4.1) through the direction of curling trajectories.

1. Bubble chambers were invented by Donald Glaser in 1952. For more information on this technique and itsuse, see http://cerncourier.com/cws/article/cern/29120 .

91


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92 CHAPTER 4. Magnetostatics

Figure 4.1: Trajectories of an electron and a positron created by an incoming γ particle visualizedin a bubble chamber. The γ particle hits the electron of a hydrogen atom. The atomic electron isejected while a pair electron/positron is created. The two latter particles curl in opposite directionsthanks to the applied magnetic eld. Taken from http://www.alternativephysics.org/book/MatterEnergy2.htm .

⋆ Comment on magnetic monopoles : An electric eld is created by an ensemble of charges. Eachone of these can be dened as a“positive”or a “negative” charge. In other words, one can deneelectric monopoles . However, a magnetic eld is created by a current - whether macroscopicor local as we will see in the next chapter -. In other words, in standard materials, if you cut amagnet into two parts, you will always obtain both a North pole and a South pole 2 . Indeed, therehas, so far, been no consensus on the existence or not of magnetic monopoles , althoughtheir existence has been theoretically predicted by Paul Dirac in 1931 who claimed that they couldexplain charge quantization 3 . Experimentally, numerous claims have been made for the discoveryof these monopoles. To date, they have all been disproved.... except the latest one by David Halland Michael Ray in 2014 4 (at least not yet !).

4.1.2 Corollary : Laplace force

Property #1: The elemental magnetic force exerted by a magnetostatic eld −→B on aportion of linear conductor

−→dl through which ows a current I is given by the Laplace force:

−→dF = I

−→dl ∧

−→B (4.2)

2. As can be seen from the equation div−→B = 0, there is no “magnetic charges”.

3. See for example the discussion in J.D. Jackson, Classical Electrodynamics , section 6.11.4. They have observed synthetic magnetic monopoles in Bose-Einstein condensates. See Ray et al., Nature , 505 ,

657, 2014, or https://www.amherst.edu/aboutamherst/news/faculty/node/532493 .


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4.1. MAXWELL’S EQUATIONS APPLIED TO MAGNETOSTATICS AND CONSEQUENCES 93

♦ Proof :

−→dF = i n i q i

−→vi ∧

−→B dV

= −→

J ∧−→B dV

= −→

J dSdl ∧−→B

= JdS −→dl ∧

−→B

−→dF = I

−→dl ∧

−→B

Property #2: From the above demonstration, one can also extract the force exerted by amagnetostatic eld

−→B on a conducting surface

−→dS through which ows a surface current density

−→J s : −→

dF =−→J s ∧

−→B dS

♣ Application to the denition of the Ampère: 1 Amp is the intensityof the currents I 1 and I 2 owing through two innite wires, separatedby a distance of 1 m and exerting a force on one another of 2.10 − 7 Nper unit length.

♦ Proof : We will see in section 4.2.2 that the magnetic eld createdby the rst innite wire at a distance d is

−→B 1 = µ 0 I 12πd

−→u y , so that the forceexerted by the rst wire on the second one is :

−→df 1→ 2 = I 2

−→dl ∧

µ0I 12πd

−→uy

= µ0I 1I 2

2πd dl−→u z ∧−→uy

−→df 1→ 2 = −

µ0I 1I 2dl2πd

−→u x

This denition is equivalent as dening µ0 = 4π. 10− 7 u.S.I.

4.1.3 Typical values for magnetic elds

The table below gives the orders of magnitude for a few magnetic elds 5 :

Residual magnetic eld within an excellently shielded box 10− 14

TEarth’s magnetic eld at its surface 2-4× 10− 5 TMagnetic eld of a usual magnet 10 − 2 TMagnetic eld generated by a large electromagnet 2 TMagnetic eld generated by a large superconducting magnet 10-50 TMagnetic eld generated by a neutron star 10 8 T

5. The international unit for the magnetic eld is the tesla (T). However the gauss is also commonly used : 10 4

G = 1 T.


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4.1.4 Maxwell’s equations in magnetostatics

The decoupling between the electric and magnetic elds reduces the number of equations to betaken into account. The Maxwell’s equations of magnetostatics are :

Conservation of magnetic ux : div−→B =

−→∇ .

−→B = 0 (4.3)

Maxwell-Ampère : −−→

curl −→B =

−→∇ ∧

−→B = µ0

−→J (4.4)

4.1.4.a Conservation of the magnetic ux

Denition : The magnetic ux ΦB of a magnetic eld −→B through a surface ( S ) is :

ΦB = ‹ −→B . −→dS (4.5)Φ

B is in unit of webers (Wb) 6 .

As we have seen on numerous occasions, we have :

div−→B = 0 (4.6)

which implies that −→B is a vector which conserves its ux 7 .

4.1.4.b Ampère’s law

The integral form of Maxwell-Ampère’s equation in the steady-state regime gives Ampère’slaw :

˛ (C )−→B .

−→dl = µ0I (S ) (4.7)

where (C ) is a closed contour delimiting a surface ( S ), I (S ) is the current encircled by ( C ).

4.1.4.c Discontinuity equations at interfaces

The discontinuity equation for the magnetic eld is :−→B 2 −

−→B 1 = µ0

−→J s ∧−→n 1→ 2 (4.8)

where −→B i is the magnetic eld in the medium i,

−→J s is the surface current density at the interface

between the two media, and −→n 1→ 2 is the unit vector, normal to the interface, directed from medium(1) to medium (2).

6. 1 Wb = 1 T.m 2 .7. See section 2.1.3.c for the consequences of this equation.


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4.1. MAXWELL’S EQUATIONS APPLIED TO MAGNETOSTATICS AND CONSEQUENCES 95

4.1.5 Vector potential and Biot-Savart law

4.1.5.a Vector potential and potential propagation

In chapter 1, we have seen that the magnetic eld derives from a vector potential :−→B =

−−→curl

−→A (4.9)

In section 1.3.4, we have derived the general equation for the propagation of the vector potential :

−→△

−→A −

1c2

∂ 2−→A

∂t 2 −

−−→grad div

−→A +

1c2

∂φ∂t

= − µ0−→J

In magnetostatics, this equation, within the Coulomb or Lorenz gauge 8 , this equation amounts toa Poisson equation: −→

△−→A = − µ0

−→J (4.10)

which solution is :−→A (M ) =

µ04π ˚ −→J (P )P M dV (4.11)

♦ Proof : In electrostatics, one had a Poisson equation for the electrostatic potential: ∆ φ = − ρε 0which solution was expressed as φ(M ) = ˝

ρ4πε 0 P M dV . By analogy, one derives the solution of

the Poisson equation for the vector potential.

4.1.5.b Biot-Savart Law

Hence, the magnetic eld created by the current density−→J can be expressed under the form of

the Biot-Savart law 9 :−→B (M ) =

µ0

4π ˚

−→J (P ) ∧

−−→P M

P M 3 dV (4.12)

♦ Proof :

−→A =

Ax = µ 04π ˝ J x (P )

P M dV

Ay = µ 04π ˝ J y (P )

P M dV

Az = µ 04π ˝ J z (P )

P M dV

−−→curl

−→A =

∂ ∂x

∂ ∂y

∂ ∂z

∧

Ax

Ay

Az

For the component along z :

B z = ∂A y

∂x −

∂A x∂y

= ∂

∂xµ04π ˚ J y (P )P M dV − ∂ ∂y µ04π ˚ J x (P )P M dV

= µ0

4π ˚ J y (P ) ∂ ∂x 1P M − J x (P ) ∂ ∂y 1P M dV 8. In statics, both gauges are equivalent: div

−→A + 1c 2

∂φ∂t = div

−→A = 0.

9. This law was derived by Jean-Baptiste Biot and Félix Savart in 1820.


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and if

−−→P M =

x

y

z

P M = x 2 + y2 + z2∂ ∂x

1P M

= − 1

P M 2∂P M

∂x = −

1P M 2

xP M

so that :

B z = µ0

4π ˚

J x (P ) y

P M 3 − J y (P )

xP M 3

dV

B z = µ0

4π ˚ −→J ∧−−→P M P M 3

.−→u z dV

4.1.5.c Biot-Savart law for a wire

In the case of a wire through which ows a constant current I , one can express the magneticeld created by I :

−→B (M ) =

µ0I 4π ˛

−→dl ∧

−−→PM

P M 3 (4.13)

This is the Biot-Savart law .

♦ Proof :

−→B (M ) =

µ04π ˚

−→J (P ) ∧

−−→P M

P M 3 dV

−→B (M ) =

µ04π ˚

−→J (P ) ∧

−−→P M

P M 3 Sdl

−→B (M ) =

µ04π ˚ J (P )S

−→dl ∧

−−→P M

P M 3

where S is the section of the wire.

4.1.6 Field lines and symmetries

4.1.6.a Field lines

Denition : Magnetic eld lines are the lines that are, at all points in space, tangent tothe magnetic eld

−→B . They can be determined through the relation :

−→B (M ) ∧

−→dl =

−→0 (4.14)

where −→dl is an innitesimal vector along the eld line, centered around point M .


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4.2. EXAMPLES OF USUAL CHARGE DISTRIBUTIONS 97

4.1.6.b Properties

Property # 1 : The vector potential and the magnetic eld reproduce all symmetries and invariances of the current distribution −→J (P ) that creates this potential and this eld.

Property # 2 : The eld lines for the magnetic eld are closed (possibly at innity) and enlace the current sources.

−→B is an axial vector, whereas

−→A is a polar vector . They then obey

the symmetry properties described in section 1.2.1.♦ Proof : The eld lines must be closed, otherwise ‚

−→B .

−→dS ¬0 which is not possible, due to the

conservation of ux. The eld lines must embrace the current sources : ¸ −→B .

−→dl = µ0I due to

Maxwell-Ampère’s equation.

4.1.7 Energetics in magnetostatic

4.1.7.a Magnetostatic energy

Denition : The volume density of magnetostatic energy is given by :

u = B 2

2µ0(4.15)

This is a particular case of equation 1.24.

As a consequence, the total electrostatic energy U in a volume V can be written as :

U = ˚ (V )

u dV

U = 12 ˚ (V ) B 2µ0 dV ≥ 0 (4.16)

This energy is always positive. It implies that the presence of a eld adds energy to the system.

4.2 Examples of usual charge distributions

4.2.1 Example #1 : Circular loop

Let us consider a circular loop of radius R and axis Oz through which ows a current I . Wewill determine the eld on the axis of this circular loop.


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Figure 4.2: Circular loop.

The problem is of cylindrical symmetry. We will therefore use the cylindrical coordinates. Sincethe current distribution

−→J (r, θ, z ), and the magnetic eld conserves the symmetries of the current

distribution, −→B (r, θ, z ). For a point M on the Oz axis, M belongs to all ( −→u r , −→u z ) antisymmetry

planes. On the Oz axis, the magnetic eld therefore is along the Oz direction. Using Biot-Savartlaw :

−→B =

µ0I 4π ˛

−→dl ∧

−−→P M

P M 3

= µ0I

4π ˆ 2π

ϕ=0

Rdϕ −→uθ ∧ (− R −→u r + z−→u z )

(r 2 + z2)32

But we know that the eld will be along Oz . The contribution of −→uθ ∧−→u z will therefore cancel outin the integration. Hence :

−→B =

µ0I 4π ˆ

2π

ϕ=0

Rdϕ −→u θ ∧ (− R −→u r )

(r 2 + z2)32

= µ0I

4π ˆ 2π

ϕ=0

R 2dϕ−→u z(R 2 + z2)

32

= 2πµ 0IR 2

4π (R 2 + z2)32

−→u z

−→B =

µ0I

2R sin3 α −→u z

4.2.2 Example #2 : Innite wire

Let us consider an innite wire through which ows a current I . We will determine the magneticeld created by this wire.

Figure 4.3: Innite wire.


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4.2. EXAMPLES OF USUAL CHARGE DISTRIBUTIONS 99


−→J (r, θ, z), and the magnetic eld conserves the symmetries of the current

distribution, −→B (r, θ, z). All points M in space belong to a ( −→u r , −→u z ) symmetry plane. The magnetic

eld therefore is along −→uθ . Then, by taking the Ampère contour drawn on the above gure :

˛ −→B . −→dl = µ0I B (r )2πr = µ0I

−→B =

µ0I 2πr

−→uθ

Given the symmetries and invariances, the vector potential can be written as : −→A = A (r, θ, z)−→u z .

Then :

˛ −→A.−→dl = ¨ −→B . −→dS Az (r 1)h − Az (r 2)h = ¨ µ0I 2πr −→uθ dz dr −→uθAz (r 1)h − Az (r 2)h =

µ0Ih2π ˆ

r 2

r = r 1

1r

dr

−→A (r ) =

−→A (r 0) −

µ0I 2π

ln rr 0

−→u z

4.2.3 Example #3 : Innite solenoid

Let us consider an innite solenoid of radius R and axis Oz , having n turns per unit length, andthrough which ows a current I . We will determine the eld created by this innite solenoid 10 .

Figure 4.4: Innite solenoid.


−→J (r, θ, z), and the magnetic eld conserves the symmetries of the current

10. The case of a nite solenoid will be seen in the exercise sheet.


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distribution, −→B (r, θ, z). All points M in space belong to a ( −→u r , −→u θ ) symmetry plane (because the

solenoid is innite). The magnetic eld therefore is along the Oz direction. An Ampère contour

which does not include any current shows that the eld is homogeneous outside the solenoid. Since−→B (r → + ∞ ) = −→0 , the eld is null everywhere outside the solenoid. An Ampère contour includinga current gives :

˛ −→B . −→dl = µ0I (S )B (r 1)h − B (r 2)h = µ0nhI

B (r 1) = µ0nI −→B = µ0nI −→u z

Noticing that all points M

in space belong to a (−→u

r, −→u

z ) antisymmetry plane and given thepreviously established symmetries and invariances, the vector potential can be written as : −→A =A(r, θ, z)−→uθ . Then, along a circle of axis Oz encompassing the solenoid :

˛ −→A.−→dl = ¨ −→B . −→dS Aθ 2πr = µ0nI × πR 2 for r > R

−→A (r ) =

µ0nIR 2

2r−→u θ for r > R

Aθ 2πr = µ0nI × πr 2 for r < R−→A (r ) =

µ0nIr2

−→uθ for r < R

4.3 Magnetic dipole

The notion of magnetic dipole is important. Indeed, we will see in the next chapter that it isat the basis of the microscopic description of magnetic materials.

4.3.1 Denition

Denition : A magnetic dipole is a localized distribution of current loops, of nite spatial extension δ , centered around point A.This distribution can then be modeled by a closed loop of electric currentfor which one can dene a magnetic moment

−→M (in A.m2) :

−→M = I ¨

(S )

−→dS (4.17)

⋆ A few comments :– In the case of a plane current loop of normal −→n :

−→M = I S −→n (4.18)


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4.3. MAGNETIC DIPOLE 101

– The dipole approximation can here also be used (see section 2.4.1). It consists in consid-ering the effect of a magnetic dipole at point M such that r = AM ≫ δ . In other words, one

neglects the spatial extension δ of the current distribution around point A and replaces it bya dipole −→M placed in A.– This model can account for Earth’s magnetic eld (see additional reading section 4.6).– This model can account for orbital magnetic moments . Indeed, a simple semi-classical

model describes the electronic motion in an atom as a circular movement due to the attractiveCoulomb force exerted by the nucleus. The period T of such a motion is very short, so thatit can be associated to an effective current I ≃ − eT . Then :

M = IS −→n = −eS T

−→n

Since the motion derives from a central force : S T = L2m . Hence :

M = −eL2m

−→n = −

e2m

−→L = γ

−→L

where γ = − e2m is the gyromagnetic ratio.

4.3.2 Magnetic eld and potential created by an magnetic dipole

4.3.2.a Vector potential created by an magnetic dipole

Figure 4.5: Schematic representation of a dipole.

In the dipole approximation , the vector potential created at point M by an magnetic dipole−→M

placed at point A such that −−→AM = r −→u , with −→u an unit vector, is 11 :

−→A (M ) =

µ04π

−→M ∧ −→u

r 2 (4.19)

4.3.2.b Magnetic scalar potential created by an magnetic dipole

In the regions of space where there are no sources (−→J =

−→0 ), one has −→rot

−→B =

−→0 . One can

therefore dene a magnetic scalar potential Φm such that :−→B = −

−−→grad Φ m (4.20)

11. The proof of this expression is troublesome so that we will not derive it here. However, it can be found inJ.D. Jackson, Classical Electrodynamics , section 5.6.


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The magnetic eld is then said to derive from the magnetic scalar potential.In the case of a magnetic dipole, in the dipole approximation :

Φm (M ) = µ04π

−→M .−→rr 3

(4.21)

4.3.2.c Magnetic eld created by an magnetic dipole

In the dipole approximation , the magnetic eld created at point M by an electrostatic dipole−→M placed at point A such that

−−→AM = r −→u , with −→u an unit vector, is 12 :

−→B (M ) =

µ04π

3−→M .−→u .−→u −

−→M

r 3 (4.22)

4.3.2.d Field lines for an magnetic dipole

One can note that, within the dipole approximation , the magnetic eld created by a magneticdipole (equation 4.22) has the same mathematical structure than the one giving the electric eldcreated by an electrostatic dipole (equation 2.30). Thus, the eld lines for a magnetic dipole canbe directly derive. They are schematized gure 4.6.

Figure 4.6: Schematic representation of the eld lines created by a magnetic dipole. Taken from

http://en.wikipedia.org/wiki/Magnetic_dipole . The dipole is here modeled by a closed cur-rent loop.

4.3.3 Mechanical action of an external eld on a magnetic dipole

4.3.3.a Magnetic dipole in an external magnetic eld

In the dipole approximation , an magnetic dipole−→M , of center A , placed in an external magnetic

eld −→B is submitted to forces such that 13 :

12. The proof of this expression is also troublesome so that we will not derive it here. However, it can be foundin J.D. Jackson, Classical Electrodynamics , section 5.6.

13. The proof of this expression is also troublesome so that we will not derive it here. However, it can be foundin J.D. Jackson, Classical Electrodynamics , section 5.7.


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4.4. METHODS IN MAGNETOSTATICS 103

– the resultant force is : −→F (A) =

−→M .

−−→grad A

−→B (A) (4.23)

– the angular momentum is : −→Γ (A) =

−→M∧

−→B (A) (4.24)

4.3.3.b Application in the case of a rigid magnetic dipole

If the considered dipole is rigid, the current loop is not deformable. Then the norm of themagnetic moment

−→M is time-independent. Then the resulting force is :

−→F (A) = −

−−→grad A (U m ) (4.25)

U m = −−→M .

−→B (A) (4.26)

The force exerted on the dipole therefore derives from the potential energy U m . The dipoletends to align with the external eld −→B (A).

There again, you can notice the resemblance between these expressions and the ones seen forthe resultant force, angular momentum and potential energy in the case of an electrostatic dipole(section 2.4.3.a).

4.4 Methods in magnetostatics

There are numerous solving methods in magnetostatics which we will not review here 14 . Forthe purpose of this course, you will mainly be using one of the following methods :

– Directly compute the magnetic eld using symmetries, invariances and :

−→B (M ) =

µ04π ˚

−→J (P ) ∧

−−→P M

P M 3 dV

– Directly compute the vector potential using symmetries, invariances and :

−→A (M ) =

µ04π ˚

−→J (P )P M

dV

– Determine the magnetic eld using symmetries, invariances and Ampère’s theorem :

˛ −→B . −→dl = µ0I int

Moreover, as stated in section 2.5.2, there are numerous numerical methods available to computemore complex problems.

4.5 General eld analogies

In the course of these four chapters, we have seen similar mathematical expressions for differentquantities : div

−→B = 0 ; ∆ φ = 0 ; φ1 − φ2 = ´

−→E .

−→dl ; ... You may also have noted that you

had encountered the same expressions in different occasions outside electromagnetism : whetherin electricity, in thermal conduction, gravitation, or particle diffusion, you have the same kind of

14. An introduction to those can be found in chapters 5 of J.D. Jackson, Classical Electrodynamics and in chapters5 to 8 of E. Weber, Electromagnetic Theory .


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equations. The way to solve them is therefore similar and one can establish analogies betweenthe different quantities at play. Weber has written a very nice chapter on this 15 and below is

an excerpt giving the correspondance between different problems dealing with scalar potentialelds.ext dddpdage

4.6 Additional reading

The following article 16 , describes how a dynamo eld can simulate the Earth’s magnetic eld,including the switching between two directions.

15. E. Weber, Electromagnetic Theory , chapter 3.16. Berhanu et al., ArXiv:physics/0701076.


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16/17


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2014-2015 Magnetostatics Cours