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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

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 density4

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:

5I is the current, J is the current density

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

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

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 6

dlT

K

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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

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)

A differential equationcompleting Maxwell’s

equations

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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

From Oersted observation, Ampere decided to test if currents were exerting force on each others and found that it was the case

• 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?

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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

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Empirical Ampere’s law

From his systematic studies:

Force exerted by full circuit C on full circuit C’ is given by:

C

C’

I

I’

O 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

essentially the same…

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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)

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 loopAlready more general than the empirical Ampere’s law

of slide 11 evenif it seems to

express the samething

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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.

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 field19

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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:

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

and

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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

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

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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 quantize the whole electric charge

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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

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 Might just not be as practicable in magnetostatics as V in

electrostatics

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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