PHY 042: Electricity and Magnetism
Magnetostatics
Prof. Hugo Beauchemin
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Introduction
 We did a complete survey of electrostatics concepts and tools that
proved to be useful to make predictions in a large variety of
experiments and understand very different observable phenomena
 Laws and concepts derived so far are ONLY applicable to statics
situations
Do the concepts of electrostatics also apply to situations where
charges are in motion and if yes under which conditions???
 To answer: perform basic electric experiments with moving charges
Empirical studies of systems of charges in steady motion will make
us discovering very different phenomena. Completely new laws and
concepts with no apparent connection with the electrostatics will be
introduced to understand these new phenomena.
 Charge in steady motion will bring magnetostatics
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Electric current
 Need to define the physical meaning of “charges in motion”
before discussing the phenomena that can be produced by this.
 “Electric current” is the name given to a flow of moving charges
in a macroscopic perspective
 Study forces on systems baring non-zero electric currents
 Defined as the variation in the number of charges contain in a
system over a given period of time (moving charges  velocity)

Units: 1 Coulomb/Second = 1 Ampere
 Charges can vary locally:
Different charge variations at different regions of the system
 Charge is not quantized, so variation can be as small as we want

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Current density
 If the charge distribution is continuous, the electric current must be
defined in terms of a charge density and not a number of charges
 Current: how much the charge density will vary in a small region
over time

Assuming a linear charge density (current is in a 1D wire):
Rate of charges passing through a point P in the linear conductor
 From this relationship between current and linear charge density:

Current is a flux, i.e. a flow of charges through a given boundary

That flux depends on the velocity and density of the charge carriers

The velocity v is a vector so the current has a direction!
 With this definition, I is rather the 1D version of a current density
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Volume current density
 In experimental set-up or technical devices:
Charges flow in a constrained geometry such as wire
 Infinitely thin wires are an idealization, materials have a volume

 Wire approximation can be used when field point r >> radius of wire
 To remove the confusion between the two definitions of I (current
and current density), consider the volume current density:
Rate of charges flowing through a small surface element
normal to the direction of flow direction
 The flow does not need to be normal to the surface it is crossing,
but we know how to define a flux of a vector field through a
surface:
I is the current, J is the current density
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Surface current density
 Charge carriers won’t necessarily move at the same speed, so we
should really talk about the average velocity of the charge carriers

Remember Maxwell’s speed distribution and equipartition theorem
dlT
 Charges can also move on a surface
E.g.: Non-zero tangential electric field
on a conductor before equilibrium
 Charges passing through a transverse line element

K
 Will often be interested in the amount of current in an element of
line ds or in a confined space assuming we know its density
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Equation of continuity
 The amount of electric charges in a full system is not changed by
the electromagnetic phenomena that occurs in it.
 The electric charge is conserved
 That was trivial in electrostatics as charges were not altered by
electrostatics phenomena
 Conservation of charges however impose crucial constraints on
electric currents
A differential equation
completing Maxwell’s
equations
 This is a precise mathematical statement about local charge
conservation (conserved at all points, not just globally in V)
 Fundamental law of Nature (global gauge symmetry)
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Back to magnetostatics
 Now that we quantified the motion of charges with the concept of
current, we can comeback to our original question:
What is the interaction between two charges that are in
motion, i.e. how the configuration of moving charges is
affected by the presence of other moving charges?
 Almost the exact same question as what was asked when
electrostatics was introduced  we will apply similar approach to
answer:
1.
Start from the observation of some phenomena involving mutual
influence of moving charges
2.
Build empirical laws describing the effect
3.
Generalize it to a wider set of experiments and phenomena, ideally in
a coherent way with what was obtained in electrostatics
 i.e. ideally in terms of field equations
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Oersted’s discovery
 Started from an “accidental” observation:
Oersted discovered (1819) that electric current can also
exert an effect (a force) on magnetic compass needle
• Public demonstration of a
connection between electric
current, heat and light
• Already knew that putting a
compass perpendicular to a
wire yields nothing
• Oersted idea: wouldn’t a
magnetic force radiate away
from wire like light does?
 From Oersted observation, Ampere decided to test if currents
were exerting force on each others and found that it was the case
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Ampere’s experiment
 Ampere performed a set of systematic empirical studies of the
force exerted by a current on another to understand what were the
factors, both qualitative and quantitative, characterizing this force

This is very similar to what Coulomb did
 Ampere designed many experimental setups to understand the
phenomena and successfully establish a mathematical law
adequate to his experimental observations and measurements
X
Y
E
F
P
Q
C
A
D
B
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Figure 3: Reconstruction of Ampère’s balance for demonstrating the forces between
Empirical Ampere’s law
C
I
From his systematic studies:
I’
C’
O
Force exerted by full circuit
C on full circuit C’ is given
by:
m0 is the permeability of free space = 4p x 10-7 N/A2
• An exact value, not an empirical value like e0
• Fixes current unit and scale
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What does it tell us? (I)
 Is this law reducible to the same law of electricity found by
Coulomb, but applied to moving charges?
 NO!!! The force is NOT due to the attraction/repulsion between
charge carriers:

The full circuits C and C’ are neutral

A test charge at rest is not attracted by any of the circuits

Reversing the direction of one current change the direction of the
force without changing the distribution of charges

Each circuits exert a force on a magnet bar
 This features a completely DIFFERENT set of phenomena:
 Moving charges generate MAGNETISM
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What does it tell us? (II)
 Fundamental elements about magnetism that can be extracted
from this empirical quantitative law:

The force is not along the direction of motion of charges but
perpendicular to it

The force between two current elements is radial
 Like Coulomb’s law
 Null, if current elements are perpendicular

The force behave as 1/r2
 Again, as Coulomb’s law

Superposition principle applies

Third law of Newton applies
 The relationship between current elements is very different
than between charge elements in Coulomb’s law, and
constant values are different too, but other features are 13
essentially the same…
Conditions of applicability
 As for Coulomb’s law, the experimental setup (or equivalent setups)
used to derive the law imposes conditions of applicability to the law:

Currents involved in the experiments are steady currents, i.e
correspond to a continuous non-varying flow of charges
 Charge are not pilling anywhere
 Magnetotstatics
 Law applies to full circuits in mechanical equilibrium
 Point charges in motion are NOT electric circuits with steady
currents
 Can shape the circuits to study force between finite current elements
 Must be achievable from realistic circuit (using approximations)
 Another difficulty in quantifying E&M…
 This is a classical description (average over quantum
fluctuations)
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The Magnetic field
 To generalize the Coulomb’s law to electrostatics we introduced the
concept of electric field. Can we do it here too?
 Similarly as when we defined the electric field from Coulomb’s law:

gives the effects of the force due to C’ on C

Can factorize this law as a term that gives the effect of C’ on C regardless
of the value of I times a factor that depends on I.

Ampere’s law satisfies action-reaction relationship leaving us to think
that there is something transmitting the force from C’ to C
 Can extract from the Ampere’s empirical law a new concept:
The magnetic field B
 Generalization: not anymore limited to the force on a full circuit,
the B field will have an effect on any steady current elements

Equivalent as test charge with a similar asymmetry between I and I’
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Biot-Savart law
 The hope is to be able to describe B-fields by a set of diff. equations

Allow to find B-fields uniquely everywhere for a wider range of setups
 Start with B-fields generated by known current density distributions
at a field point
by factorizing the Ampere’s empirical law

Not formulated in terms of close loops because we can shape circuits
in a such a way that only the B-field due to a current element matters

Still constrained by other conditions of applicability of empirical law
The source
can be a line,
a surface or a
volume
current
density
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Magnetic forces
 The magnetic field created by a current element will exert
a force on other circuit elements

Allow to test and thus give an empirical meaning to B-field

This formulation is slightly different than what we got for
the Ampere’s empirical law from which it is extracted (slide
11), showing that a generalization have been integrated in
the law, from the introduction of the concept of B-field
 No more closed loop
Already more
general than
the empirical
Ampere’s law
of slide 11 even
if it seems to
express the same
thing
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Lorentz’ force
 If the (average) velocity of the charge carriers is constant through
the material distribution and the B-field is also constant:

Reasonable assumption for steady currents
 The charge distribution can also be affected by an electric field
 Can’t be applied (yet) to moving point particles

Ampere/Biot-Savart’s law is not applicable to point charges

The charge will radiate energy and the system won’t satisfy curl(E)=0

Nevertheless, we will eventually reach the conclusion that Lorentz’
force holds for any charges in motion
 We often ignore these non-steady state effects in semi-realistic problems
 E.g.: Cyclotron, cycloid motion, etc.
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Work and Energy
 Similarly as what was done in electrostatics, we would like to use the
definition of work and potential energy in mechanics together with
the concept of magnetic force from the Ampere’s law to define the
work of a magnetic force and the energy of a magnetic field
 But… the magnetic force is always perpendicular to the direction of
the flow of charge
 The magnetic force may alter the direction in which a charged
system moves, but cannot speed it up or slow it down
 A magnetic force does NO work on a current
 This doesn’t mean that there is no energy stored in a magnetic field.
It means that we will need to proceed differently than what we did in
electrostatics to define such magnetic energy

Need to have electrodynamics and Faraday’s law to define a procedure
to determine the energy stored in a magnetic field
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Field equations for B
 We would like to generalize the laws of magnetostatics to
experimental contexts or phenomena beyond the limitations of the
Ampere’s empirical law or Biot-Savart’s law
Still limited to steady current
 We will use magnetostatics to perform this generalization

 To do this, we would need to obtain the two field equations:

Helmholtz theorem: can find B everywhere if boundaries are known
and if we know the “?” in:
and
 We need to find the right-hand side of these equations
 Strategy:
Compute
and
in situations in which B is known from B-S
 Generalize the result to general geometries

 Similar to what we did to obtain Gauss’ law
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Curl of B (I)
 From the B field of an infinite wire, we have:
 It can be proved by computing the
circulation of B on a close path C
wrapping the wire.
 We can show that if the wire is outside
the path, the circulation of B is null on C

Note that a current not enclosed by
C does not contribute to circulation but
does contribute to B
 If a bundle of straight wires passes through the
path C, all the wires in it will contribute to the
circulation of the B-field around C
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Curl of B (II)
 For any path C around any set of wires, we have:
 The magnetic field is a non-conservative field
 This is only valid, for the magnetic field produced by a wire, but
can be generalized to any general steady current J
 The proof uses the assumption that J is a steady current
 It is again in the curl of the field that statics conditions appear
 This equation is called the Ampere’s law

Different physics content (more general) than the empirical
Ampere’s law introduced earlier, because it introduces LOCALITY

Equivalent to the Gauss’ law

Will be useful to find B when the system has strong symmetries
Divergence of B
 We can similarly compute the divergence of the magnetic field,
starting from the general formula of B produced by a general J
 B circulates, so B doesn’t diverge!

Might not be clearly manifest for a sum of B
such as in a solenoid since B is central

Solenoid are the equivalent of capacitors:
devices to store a magnetic field
 Physical meaning of divergenceless B-field:

B doesn’t go from one charge to another

It begins and end nowhere

There is no source point at the origin of B
 There are no analogue to the electric monopole for B

Dirac said that one single magnetic monopole would be sufficient to
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quantize the whole electric charge
Vector potential
 Similarly as for the electric field, the magnetic field cannot take
any form: it is a constrained field having to satisfy
Q: Does it implies that B can come from a potential?
A: Yes! Remember Helmholtz theorem:
Divergenceless fields are the curl of some vector potential:
 We can rewrite the Ampere’s law in terms of potential
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Is it a useful concept? (I)
 The concept of vector magnetic potential doesn’t seem to share
the great advantage brought by the scalar electric potential:

We don’t reduce the vector problem to a scalar problem

There are no straightforward interpretation geometrical
interpretation in terms of equipotential
 It is a measure of how much A is curling, not the gradient of A

There are no clear connection with potential energy since B does no
work

There are no simple empirical law formulated in terms of A
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Is it a useful concept? (II)
 If
we can have a Poisson equation:
 That has the same virtue as for the scalar potential in terms of
generalization of Biot-Savart to broader set of experiments, but

Need to have three times as many boundary conditions, so require
more information on the system that what V needed

Three equations to solve so problem more tedious
 There are advantages to introduce A but they are much milder than
the advantages of introducing V
 Can we simply just avoid this concept?
No! In fact, the Aharonov-Bohm effects reveals the effect of the
vector potential A, so it is needed to understand new phenomena
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Gauge invariance
 Is it reasonable to assume
?
 Let suppose that
 We can always find a function l(x,y,z) such that

This is the Poisson equation for the source
and this can be solved
such that we can always find a function l such that
 The magnetic field specifies the curl of A but says nothing about its
divergence, so we are free to choose this divergence as we want
 B is gauge invariant!

Gauge invariance plays a crucial role in particle physics
 Yes the vector potential is a very important concept

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Might just not be as practicable in magnetostatics as V in electrostatics
Boundary conditions
 Do we expect any discontinuities in B-field at boundaries of two
regions and if yes, what are the discontinuity conditions?

means that the tangential component is continuous, and
means that the normal component is not when s ≠ 0
 For B, we have the opposite:
and
 The normal component of B is continuous and the tangential one
is not when K ≠ 0
 And in terms of the potential
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