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Maxwell’s Equations K. Craig 1

Maxwell’s Equations

• When James Clerk Maxwell worked out his theory of

electromagnetism in 1865, he ended up with 20 equations

that describe the behavior of electric and magnetic fields.

• Oliver Heaviside in Great Britain and Heinrich Hertz in

Germany combined and simplified these equations into four

equations in the two decades after Maxwell’s death in 1879.

• Today we call these four equations – Gauss’s Law for

Electric Fields, Gauss’s Law for Magnetic Fields, Faraday’s

Law, and the Ampere-Maxwell Law – Maxwell’s Equations.

• They are on everyone’s list of the most important equations

of all time, and perhaps, they are the least understood!


Gauss’s Law for Electric Fields

enc

S0

qˆE nda

Electric charge produces an electric field, and the flux of that field passing through any closed surface is proportional to the total charge contained within that surface. Induced electric fields are not included here.

Integral Formvalid over a surface

Electric Flux

Total charge:free & bound

ε0 = 8.85E-12 C/(V-m)

real or imaginary surface,any size or shape

net charge


• An electric field is a vector and is the electrical force per

unit charge exerted on a small positive-charged object q0.

• E has units of N/C, which is the same as V/m.

• Rules of Thumb

– Electric field lines must originate on positive charge and

terminate on negative charge.

– The net electric field at any point is the vector sum of all

electric fields at that point.

– Electric field lines can never cross.

– Electric field lines are always perpendicular to the

surface of a conductor in equilibrium.

• The permittivity ε of a material determines its response to

an applied electric field.

e

0

FE

q

r

0

relativepermittivity


Examples of Electric Fields

Strength is shown by either the arrow length or the spacing of the lines.


Electric Field Equations for Simple Objects


• Electric Flux through a closed surface

– Electric flux is a scalar quantity and has units of electric

field times area (V-m).

– How should you think of electric flux? Use field lines, which

are continuous in space, to represent the electric field (see

previous slide of electric field examples). The strength of

the electric field at any point is indicated by the spacing of

the field lines at that location, i.e., the electric field strength

is proportional to the density of field lines (the number of

field lines per square meter) in a plane perpendicular to the

field at the point under consideration. Remember that field

lines are actually continuous in space.

– Integrating that density over the entire surface gives the net

number of field lines penetrating the surface, which is the

Electric Flux ФE, with direction of penetration taken into

account.

ES

ˆE nda


• Enclosed Charge

– This is the total charge, both free and bound charge,

within the closed surface over which the flux is

determined.

– Any charge located outside the surface produces an

equal amount of inward (negative) flux and outward

(positive) flux, so the net contribution to the flux through

the surface must be zero.

– For groups of discrete charges enclosed by surfaces of

any shape, simply add the individual charges.

– Charge density may be specified in one, two, or three

dimensions: λ = linear charge density (C/m), σ = area

charge density (C/m2), ρ = volume charge density (C/m3)

enc iL S V

i

q q d da dV

qenc


• Permittivity of Free Space ε0

– The permittivity of a material determines its response to an

applied electric field. In nonconducting materials (called

insulators or dielectrics) charges do not move freely, but

may be slightly displaced from their equilibrium positions.

ε0 is not a constant, but is frequency dependent.

– Even though we use the permittivity of free space in

Gauss’s Law, the law applies to electric fields within

dielectrics as well as in free space (vacuum) provided that

you account for all of the enclosed charge, including

charges that are bound to the atoms of the material.

ε0 = 8.85E-12 C/(V-m)

Dielectric in External

Electric Field

Net field in dielectric is less than the

external field.polarized

r

0

relative

permittivity


Differential Formvalid at individual points

in space

0

E

The electric field produced by electric charge diverges from positive charge and converges upon negative charge.

Divergence of E Field:tendency of E field to

“flow” away from positivecharge

ˆ ˆ ˆi j kx y z

SV 0

1ˆE lim E nda

V

yx zEE E

Ex y z

scalar

Only places where the divergence is not zero are those locations where charge is present.


• Both flux and divergence deal with the “flow” of a vector

field, but with an important difference; flux is defined over

an area, while divergence applies to individual points.

• Points of positive divergence are sources (positive charge),

while points of negative divergence are sinks (negative

charge).

• The mathematical definition of divergence may be

understood by considering the flux through an infinitesimal

surface surrounding the point of interest. If you were to

form the ratio of the flux of a vector field through a surface

to the volume enclosed by that surface as the volume

shrinks toward zero, you would have the divergence of the

vector field:

SV 0

1ˆE lim E nda

V


Vector Fields with Various Values of Divergence

Positive Divergence: Points 1, 2, 3, and 5Negative Divergence: Point 4Zero Divergence: Points 6 and 7, as the flow lines are spreading out but also getting shorter at greater distance from the center. The spreading out compensates for the slowing down of the flow.Note: The key factor in determining the divergence at any point is whether the flux out of an infinitesimally small volume around the point is greater than, equal to, or less than the flux into the volume.


Gauss’s Law for Magnetic Fields

SˆB nda 0

The total magnetic flux passing through any closed surface is zero. The inward magnetic flux must be exactly balanced by the outward magnetic flux.

Integral Formvalid over a surface

Magnetic Flux

real or imaginary surface,any size or shape


• The key difference in the electric field and magnetic field

versions of Gauss’s Law arises because opposite electric

charges (positive and negative) may be isolated from one

another, while opposite magnetic poles (north and south)

always occur in pairs. There are no magnetic monopoles in

nature; thus the right side of Gauss’s Law for Magnetic

Fields is identically zero.

• Just as the electric field may be defined by considering the

electric force on a small test charge, the magnetic field may

be defined using the magnetic force experienced by a

moving charged particle, as shown by the Lorentz Equation:

• B is called the magnetic induction or magnetic flux density

with unit Tesla (T) or Wb/m2. (Wb = weber)

BF qv B BFB

q v sin( )

v particlevelocity w/r B


• Distinctions between Magnetic and Electric Fields

– Like the electric field, the magnetic field is directly proportional

to the magnetic force. But unlike the electric field, which is

parallel or antiparallel to the electric force, the direction of the

magnetic field is perpendicular to the magnetic force.

– Like the electric field, the magnetic field may be defined

through the force experienced by a small test charge, but

unlike the electric field, the speed and direction of the test

charge must be taken into consideration when relating

magnetic forces and fields.

– Because the magnetic force is perpendicular to the velocity at

every instant, the component of the force in the direction of the

displacement is zero, and the work done by the magnetic field

is therefore always zero.

– Whereas electrostatic fields are produced by electric charges,

magnetostatic fields are produced by electric currents.


Examples of Magnetic FieldsMagnetic fields may be represented using field lines whose density in a plane perpendicular to the line direction is proportional to the strength of the field. Don’t forget that magnetic fields, like electric fields, are continuous in space.


• Rules of Thumb

– Magnetic field lines do not originate and terminate on

charges; they form closed loops.

– The magnetic field lines that appear to originate on the

north pole and terminate on the south pole of a magnet

are actually continuous loops (within the magnet, the

field lines run between the poles).

– The net magnetic field at any point is the vector sum of

all magnetic fields present at that point.

– Magnetic field lines can never cross, since that would

indicate that the field points in two different directions at

the same location – if the fields from two or more

sources overlap at the same location, they add (as

vectors) to produce a single, total field at that point.


• All static magnetic fields are produced by moving electric

charge. The contribution to the magnetic field at a specified

point P from a small element of electric current is given by

the Biot-Savart Law:

• µ0 is the permeability of free space and I is the current.

• Magnetic Flux, like Electric Flux, is a scalar quantity. The

unit is webers (Wb = T-m2). Outward and inward magnetic

flux must be equal and opposite through any closed surface;

magnetic field lines always form complete loops.

0

2

ˆId rdB

4 r


Magnetic Field Equations for Simple Objects



B 0

Divergence of B Field

The divergence of the magnetic field at any point is zero. Since it is not possible to isolate magnetic poles, you can’t have a north pole without a south pole, and the “magnetic charge density” must be zero everywhere.

Vector fields with zero divergence are called “solenoidal” fields, and all magnetic fields are solenoidal.


Faraday’s Law

Integral Form

C S

dˆE d B nda

dt

Changing magnetic flux through a surface induces an emf in any boundary path of that surface, and a changing magnetic field induces a circulating electric field.

The negative sign tells us that the induced emf opposes the change in flux, i.e., it tends to maintain the existing flux. This is called Lenz’s Law.

E in this expression is the induced electric field at each segment of the path C measured in the reference frame in which that segment is stationary.

circulation of Earound path C


• Induced Electric Field

– Charge-based electric fields have fields that originate on

positive charge and terminate on negative charge, and

thus have non-zero divergence at those points.

– Induced electric fields produced by changing magnetic

fields have field lines that loop back on themselves, with

no points of origination or termination, and thus have

zero divergence.

Both types of electric field accelerate electric charges and are represented by field lines, but are different in structure.


• Electric Field Circulation

– Since the field lines of induced electric fields form closed loops,

these fields are capable of driving charged particles around

continuous circuits, i.e., generate an electric current. The

circulation of the electric field around a circuit is known as

electromotive force (emf). It is not actually a force, but rather a

force per unit charge integrated over a distance.

– So the circulation of the induced electric field around a path is

the work done by the electric field in moving a unit charge

around that path; it is the energy given to each coulomb of

charge as it moves around the circuit.

CE d Electric Field Circulation = electromotive force (emf)

C

C C

F dF WE d d

q q q


• Rate of Change of Flux

– This expression represents the rate of change of

magnetic flux through any surface. How can the

magnetic flux through a surface change with time?

• The magnitude of vector B might change.

• The angle between vector B and the surface normal

might change.

• The area of the surface S might change.

BS

d dˆB nda = Rate of Change of Flux

dt dt

Note that the magnitude of the induced emf depends only on how

fast the flux changes.


• Lenz’s Law

– There is a great deal of physics wrapped up in the minus sign:

Lenz’s Law. Heinrich Lenz had important insight concerning

the direction of the current induced by the changing magnetic

flux. Currents induced by changing magnetic flux always flow

in the direction so as to oppose the change in flux.

C S

dˆE d B nda

dt

Note that changing magnetic flux induces an electric field whether or not a conducting path exists in which current may flow. Lenz’s Law tells us the direction of the circulation of the induced electric field around a specified path even if no conduction current actually flows along that path.

If the magnetic flux through the circuit is increasing (decreasing), the induced current produces its own magnetic flux in the opposite direction to offset the increase (decrease).


BE

t

A circulating electric field is produced by a magnetic field that changes with time.Curl of

electric field

CS 0

1E lim E d

S

Curl of the electric field is a measure of the E field’s tendency to circulate about a point. The direction of the curl is the normal direction of the surface for which the circulation is a maximum.



• Curl of a Vector Field – measure of the field’s tendency to

circulate about a point.

– To physically understand the concept of curl, imagine

holding a tiny paddlewheel at each point in a flow. If the flow

would cause the paddlewheel to rotate, the center of the

wheel marks a point of nonzero curl. The direction of the

curl is along the axis of the paddlewheel, with the positive-

curl direction determined by the right-hand rule.

– Vector fields with zero curl at all points are called irrotational.

x y z

y yz x z x

ˆ ˆ î j kx y z

ˆ ˆ ˆ ˆ ˆ Ê i j k iE jE kEx y z

E EE E E Eˆ ˆ î j ky z z x x y

Each component indicates the tendency of the field to rotate in one of the coordinate planes. The overall direction represents the axis about which the rotation is greatest, with the sense of rotation given by the right-hand rule.


Vector Fields with Various Values of Curl

High-Curl Locations: Points 1, 2, 3, 4, and 5Zero-Curl Locations: Points 6 and 7


• Curl of the Electric Field

– Charge-based electric fields diverge away from points

of positive charge and converge toward points of

negative charge. Such fields cannot circulate back on

themselves. All electrostatic fields have no curl.

– Electric fields induced by changing magnetic fields are

very different. Whenever a changing magnetic field

exists, a circulating electric field is induced. Unlike

charge-based electric fields, induced fields have no

origination or termination points – they are continuous

and circulate back on themselves. Integrating

around any boundary path for the surface through

which is changing produces a nonzero result, which

means that induced electric fields have curl. The

faster changes, the larger the magnitude of the curl

of the induced electric field.

E d

B

B


Ampere-Maxwell Law

Integral Form

0 enc 0C S

dˆB d I E nda

dt

An electric current or a changing electric flux through a surface produces a circulating magnetic field around any path that bounds that surface.

Circulation of magnetic field around closed

path C

A magnetic field is produced along a path (real or imaginary) if any current is enclosed by the path or if the electric flux through any surface bounded by the path changes over time.


• Some Background

– Ampere’s Law relating a steady electric current to

a circulating magnetic field was well known when

Maxwell began his work in the 1850s.

– However, Ampere’s Law was known to apply only

to static situations involving steady currents.

Maxwell added another source term – a changing

electric flux – that extended the applicability of

Ampere’s Law to time-dependent conditions.

– It was the presence of this term in the equation,

now called the Ampere-Maxwell Law, that allowed

Maxwell to discern the electromagnetic nature of

light and to develop a comprehensive theory of

electromagnetism.


• Magnetic Field Circulation

– Shown is the magnetic field around a current-carrying

wire.

– The magnetic field strength decreases precisely as 1/r,

where r is the distance from the wire.

– As you circle around the wire, the direction of the

magnetic field is along your path, perpendicular to an

imaginary line from the wire to your location.

CB d


• Permeability of Free Space (vacuum) µ0 = 4π x 10-7 Vs/Am

– Just as the electric permittivity characterizes the

response of a dielectric to an applied electric field, the

magnetic permeability determines a material’s response

to an applied magnetic field.

– As in the case of electric permittivity in Gauss’s Law for

electric fields, the presence of this quantity does not

mean that the Ampere-Maxwell Law applies only to

sources and fields in a vacuum. This form of the

Ampere-Maxwell Law is general, so long as you consider

all currents, bound as well as free.

– Unlike the effect of dielectrics on electric fields, the effect

of magnetic substances on magnetic fields is that the

magnetic field is actually stronger than the applied field

within many magnetic materials.


• Shown is the effect of a magnetic core on the field inside a

solenoid.

• Relative permeability µr = µ/µ0

– Diamagnetic material: µr < 1 slightly, e.g., gold and silver

– Paramagnetic material: µr > 1 slightly, e.g., aluminum

– Ferromagnetic material: permeability depends on the

applied magnetic field, e.g., µr varies from several

hundred to several thousand.

Materials become magnetized when exposed to an external magnetic field, and the induced magnetic field is in the same direction as the applied field.


• Enclosed (Encircled) Electric Current

– What current do we include on the right side of the

Ampere-Maxwell Law? The enclosed current is the net

current (the direction of current relative to the direction of

integration must be considered) that penetrates the

membrane.

The enclosed current is exactly the same irrespective of the shape of the surface you choose, provided that the path of integration is a boundary edge of that surface.


• Rate of Change of Flux

– This term is the electric flux analog of the changing

magnetic flux term in Faraday’s Law.

– To demonstrate the need for the changing-flux term in

the Ampere-Maxwell Law, we can study the

inconsistency in Ampere’s law when applied to a

charging capacitor.

S

dÊ nda

dt

0 encC

B d I

E

S S S0 0 0

Q / A QÊ nda da da

0 dS

d dQÊ nda I displacement current

dt dt

No conduction current flows between capacitor plates.

σ is charge density on plates = Q/A.

Even though no charge actually flows.


0 0

EB J

t

Differential Form

A circulating magnetic field is produced by an electric current and by an electric field that changes with time.

curl of the magnetic field electric current density &

time rate of change of electric field

The curl of the magnetic field is nonzero exactly at the location through which an electric current is flowing, or at which an electric field is changing.

All magnetic fields circulate back on themselves and form continuous loops.


• Curl of the Magnetic Field

– All magnetic fields, whether produced by electric

currents or by changing electric fields, circulate back

upon themselves and form continuous loops.

– In addition, all fields that circulate back on themselves

must include at least one location about which the path

integral of the field is nonzero. For the magnetic field,

locations of nonzero curl are locations at which current is

flowing or an electric field is changing.

– Just because magnetic fields circulate, do not conclude

that the curl is nonzero everywhere in the field. A

common misconception is that the curl of a vector field is

nonzero wherever the field appears to curve.

– How can a curving field have zero curl?

B


– The answer lies in the amplitude as well as the

direction of the magnetic field.

Offsetting components of the curl of the magnetic field using the fluid-flow and small paddlewheel analogy

The key concept is that the magnetic field may be curved at many different locations, but only at points at which

the current is flowing (or the electric flux is changing) is the curl of the B vector nonzero.


• Electric Conduction Current Density

– This is defined as the vector current flowing

through a unit cross-sectional area perpendicular

to the direction of the current.

– The units of current density are amperes per

square meter (A/m2).

– The electric current density includes all currents,

including the bound current density in magnetic

materials.


• Displacement Current Density

– This term involves the rate of change of the

electric field with time. This has units of amperes

per square meter (A/m2), identical to those of

electric conduction current density, J.

– Does the displacement current density represent

an actual current? Not in the conventional sense!

But the displacement current density is a vector

quantity that has the same relationship to the

magnetic field as does the conduction current

density, J.

– The key concept is that a changing electric field

produces a changing magnetic field even when no

charges are present and no physical current flows.

0

E

t


Summary


One More Thing…

• Gradient

– The third differential operator (divergence and curl are the

other two operators) is the gradient.

– The gradient applies to scalar fields, i.e., a field that is

specified by only its magnitude at various locations.

– What does the gradient tell you about a scalar field? Two

important things: the magnitude of the gradient indicates how

quickly the field is changing over space, and the direction of

the gradient indicates the direction in that the field is

changing most quickly with distance.

– The result of the gradient operation is a vector, with both

magnitude and direction.

ˆ ˆ ˆi j kx y z

Note this identity: 0


Electrostatics & Magnetostatics

• The static form of Maxwell’s Equations are derived from the

complete forms by imposing the conditions that there are no

time varying E and B fields. Mathematically, these

conditions are represented by the requirements that:

• Maxwell’s Equations then are: E B0 0

t t

enc

S0

qˆE nda

SˆB nda 0

CE d 0

0 encC

B d I

0

E

B 0

E 0

0B J

Each equation is a function of either E or B; no equation contains both E and B. Electric and magnetic fields are uncoupled; electricity and magnetism are independent subjects.


Quasistatics

• Quasistatics reside between electrostatics and

magnetostatics (no time changes) and electrodynamics

(fully dynamic coupled behavior).

• The primary significance of quasistatics is that it establishes

the foundations of modern circuit theory.

• When electric field effects dominate magnetic field effects,

the laws of electroquasistatics result.

• When magnetic field effects dominate electric field effects,

the laws of magnetoquasistatics result.

• What are the quantitative criteria under which these laws

are applicable?


Electroquasistatics

• Consider an arbitrary system where the electric field effects

are so dominant that the magnetic field can be neglected.

• This assumption is equivalent to restricting consideration to

systems or elements having high impedance or,

equivalently, that behave as an open circuit.

• The mathematical approach is via a dimensional argument.

– L is taken as the characteristic length of the system

– is taken as a characteristic time such as the period of the

electromagnetic process.

BE

t

Faraday’s Law

E B

L

dimensional analysis leads to


• For magnetic effects to be negligible compared with the

electric field effects

• Now consider

E B

L

0 0

EB J

t

Ampere-Maxwell Law

In a high-impedance system, e.g., a capacitor, the current density term will be negligibly small compared with the displacement current because no charge can be transported across the open space. It is the field-induced current, rather than the flow of charge across the gap of a capacitor that is primarily responsible for the current through a capacitor.


0 0

B E

L

2

B 1 E

L c

2

0 0

1c

2

1 ELB

c

c = speed of light


• Substitution

• We know

2

1 ELB

c

E B

L

2 2

E EL

L c

L1

c

This is an important result; it is a quantitative criterion for the appropriate application of electroquasistatics. The characteristic time or period of the electromagnetic process must be sufficiently long to ensure that L

1c

1and

c f

L1

c

L1

Lf1

c

λ = wavelengthf = frequency

Alternative expressions to assess the electroquasistatic approximation:


enc

S0

qˆE nda

SˆB nda 0

CE d 0

0

E

B 0

E 0

0 enc 0C S

dˆB d I E nda

dt

0 0

EB J

t

Maxwell’s Equations for Electroquasistatics


Magnetoquasistatics

• Consider an arbitrary system where the magnetic field

effects are so dominant that the electric field can be

neglected.

• This assumption is equivalent to restricting consideration to

systems or elements having low impedance or,

equivalently, that behave as a short circuit.

• The mathematical approach is via a dimensional argument.

– L is taken as the characteristic length of the system

– is taken as a characteristic time such as the period of the

electromagnetic process.


• Consider the Ampere-Maxwell Equation

– It is necessary for the B field to be much more dominant

than the E field. This implies that the entire displacement

current must be negligible. This is possible only if it is

much smaller than both the current density term and the

magnetic field term. Alternatively, it means that both the J

and B terms need to be much larger than the

displacement current term.

– In order for the J term to be large, the system under

consideration must be of low impedance or behave like a

short circuit, e.g., a system consisting of a conducting

wire loop (an inductor) where the J effects will be large

compared with the E effects.

0 0

EB J

t


• Apply the dimensional argument to the Ampere-Maxwell

Law, requiring that the B field term be much larger than

the E field term. This results in the inequality

• From Faraday’s Law

• Substitution leads to

0 0

B E

L

BE

t

Faraday’s Law

E B

L


0 0 2

B BL

L

L1

c

2

0 0

1c

The condition that must be satisfied to ensure validity of magnetoquasistatics is the same as that for electroquasistatics.


enc

S0

qˆE nda

SˆB nda 0

0

E

B 0

BE

t

0 encC

B d I 0B J

C S

dˆE d B nda

dt

Maxwell’s Equations for Magnetoquasistatics


• In electroquasistatics, the change from the complete form of

Maxwell’s Equations is in Faraday’s law. Here the term

has been removed from the complete form of Faraday’s

Law. In magnetoquasistatics, the change is in Ampere’s

Law; the displacement current term has been removed.

• In quasistatics there is a decoupling of the electric and

magnetic fields, similar to statics; however, this decoupling

is partial but, unlike statics, it maintains a time variation of

the E and B fields.

• Electroquasistatics is dependent only on the time variations

in the E field. Magnetoquasistatics is dependent only on the

time variations in the B field.

• We now look at how energy in electroquasistatics and

magnetoquasistatics is stored via the E and B fields,

respectively.

B

t


Energy Storage

• Here we consider energy storage in electroquasistatics

and magnetoquasistatics. The results of this analysis

illustrate further the consequences of the decoupling of

the E and B fields in quasistatics.

• We will show that the electromagnetic energy is stored in

the electric field for systems characterized by the

electroquasistatics approximation while the

electromagnetic energy is stored in the magnetic field for

systems characterized by the magnetoquasistatic

approximation.


• Shown is an energy balance in volume V bounded by

closed surface S.

dS V VP dS wdV p dV

t

Rate of energy flow into the system

through surface S (Ein)

Energy accumulation rate in the system

volume V (Ea)

Energy dissipation rate in the system

volume V (Ed)

P is the rate of energy flow per unit area through the surface S into the volume V. w is the energy density within V and pd is the rate of dissipation of the energy density within V.

Ein = Ea + Ed


Energy Storage in

Electroquasistatics• In electroquasistatics, the Ampere-Maxwell Law is:

• Faraday’s Law is:

• Manipulation:

0 0

EB J

t

E 0

0 0

0 0 0

E B E EB E J

t


• Vector Simplification

• Result

• Recognize , integrate the entire

equation above over the volume V, and apply the

divergence theorem to the left side of the equation.

• In the divergence theorem, the vector A is replaced

by:

C A A C A C

0

0

1 1B E E E E J

t 2

B E E B

0

1B E


• Result:

• We see that:

• Energy accumulation is accomplished via the E field. In an

element or system characterized by the electroquasistatic

approximation (e.g., ideal capacitor), the energy is stored in

the electric field.

dS V VP dS wdV p dV

t

0S V V

0

1 1E B dS E E dV E J dV

t 2

0

0

1P E B rate of energy flux

1w E E electromagnetic energy density

2


Energy Storage in

Magnetoquasistatics

• In magnetoquasistatics, the Ampere-Maxwell Law is:

• Faraday’s Law is:

• Manipulation:

0B J

BE

t

0 0 0

E B B BB E E J

t


• Vector Simplification

• Result

• Recognize , integrate the entire equation

above over the volume V, and apply the divergence

theorem to the left side of the equation.

• In the divergence theorem, the vector A is replaced by:

C A A C A C

0 0

1 1 1E B B B E J

t 2

B E E B

0

1B E


• Result:

• We see that:

• Energy accumulation is accomplished via the B field. In

an element or system characterized by the

magnetoquasistatic approximation (e.g., ideal inductor),

the energy is stored in the magnetic field.

dS V VP dS wdV p dV

t

S V V0 0

1 1 1E B dS B B dV E J dV

t 2

0

0

1P E B rate of energy flux

1 1w B B electromagnetic energy density

2


Kirchhoff’s “Laws”

• The fundamental concepts of modern circuit theory can be

derived as a consequence of quasistatics. The formulation of

the equations of motion for an electrical network via circuit

theory requires the satisfaction of three steps:

– Kirchhoff’s Current “Law”

– Kirchhoff’s Voltage “Law”

– Constitutive relations for the individual elements

• One of the major consequences of quasistatics that is utilized

in circuit theory is that the electric and magnetic fields are

highly localized in the capacitive and inductive elements,

respectively, and negligible elsewhere. Fields exist only within

the capacitive and inductive elements, not at nodes.


Kirchhoff’s Current “Law”

• Quasistatic Ampere-Maxwell Law applied to a node

in an electrical network is , as the E and B

fields are negligible at a node.encI 0

r

k

k 1

i 0


Kirchhoff’s Voltage “Law”

• Kirchhoff’s Voltage “Law” can be derived from the

quasistatic form of Faraday’s Law. The B field is dominant

only within inductive elements, being negligible elsewhere,

and we select a hypothetical path around the network, as

shown.

CE d 0

C S

dˆE d B nda

dt

Electroquasistatics Magnetoquasistatics

b

ae E d voltage at point a w/r to point b

C

n

k

k 1

E d 0

e 0

maxwell’s equations · maxwell’s equations k. craig 1 ... faraday’s law, and the...

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