Chapter 3: Static Fields Index (I)

Chapter 3: Static Fields
Group 4
Joaquín Bernal Méndez
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Chapter 3: Static Fields
1
Index (I)

Maxwell's Equations for Static Fields

Electrostatics


Solution for the Electrostatic Field
Application of Gauss's Law for Calculating
Electrostatic Fields

Electrostatic Energy

Multipole Expansion

The Electric Dipole
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Index (II)

Magnetic Field of Steady Currents

Solution for the Magnetostatic Field

Biot-Savart Law
Application of Ampere's Law for calculating
Magnetostatic Fields

Magnetostatic Energy

Multipole Expansion


The Magnetic Dipole
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Static Case


Maxwell's equations:
We will assume stationary charges and steady
currents
Independent
of the time
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Independent
of the time
Chapter 3: Static Fields
4
Static Case

We get two separate set of equations:
Electrostatic
Magnetostatic


We have two separate problems (the electric fields is
no longer a source for the magnetic field and
viceversa)
Each pair of equations can be solved by means of the
Helmholtz's Theorem: electrostatic field and
magnetostatic field
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Electrostatic
Helmholtz's Theorem:
Electrostatic:
Solution:
Solution:
With:
With:
Electrostatic potential
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Electrostatic Field

Electrostatic field as a function of ρ:
By using:
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Example: point charge
Given
located at
,
exerts a force on which is:
Coulomb's Law
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Difference of Potential

In principle, the electrostatic potential can be
considered an useful mathematical tool for calculating
the electrostatic field


The potential formulation reduces a vector problem
down to a scalar one
It is also possible to calculate potential differences
given the electrostatic field:
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Physical Meaning of the
Electrostatic Potential

Consider a region where there exits an electrostatic field

Let q a point charge that we carry from infinity to a point



We will do it very slowly (quasi-static process)
: Force due to the electric field
: Force that we have to exert to move the point charge

We : Work done by the electrostatic field

W : Work that we will have to do
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Physical Meaning of the
Electrostatic Potential

Energy balance:
The electrostatic potential at a point is
equal to the work per unit charge it
takes to carry a point charge from
infinity to that point in a quasi-static
process
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Potential Energy



It can be defined for conservative forces
The potential energy in the interaction between an
electric field and a point charge is the work done to
carry the point charge from infinity to its current
location in a quasi-static process:
Electrostatic force is a conservative force and
therefore it can be derived from a potential U:
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M
iss
in
g
Electrostatic Equations
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Poisson's Equation
Poisson's Equation

In a region where there is no charge (ρ=0) :
Laplace's Equation
V is an harmonic scalar field
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Electrostatic Equations
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Calculating the Electrostatic Field
from ρ
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Calculating the Electrostatic Field
from ρ
A direct calculation of the electric field uses to be
cumbersome (vector integral)

Usually it is easier to calculate V as an intermediate step

Warning: this can not be done for idealized charge
distributions that extend to infinity, since V blows up

Examples: infinite wire of charge, infinite plane of charge...
 The reason of this is that Helmholtz's theorem can not be
applied for sources of infinite extent
For this case the electric field must be directly calculated or
even better...


We can exploit the symmetry of the problem to calculate the
electrostatic field by means of Gauss's Law

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Gauss's Law



Gauss's law in integral form:
Total charge
enclosed within
the surface
It is always true but it is not always useful
For highly symmetric distributions of charge it
allows us to calculate the electric field in a very
easy way, avoiding a direct integration

There are three kinds of symmetry which make it
possible this application of Gauss's law: spherical,
cylindrical and plane
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Calculating Electric Fields by
Ussing Gauss's Law

●
Example: point charge located at the origin
●
Spherical symmetry: electric field:
●
Symmetry arguments:
Gaussian surface = centered sphere with radius r
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Sphere with Uniform Surface
Charge Density



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Spherical symmetry: the same as
point charge
Gaussian surface = centered sphere
of radius r :
Case r<R: there is no charge inside Sr
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Sphere with Uniform Surface
Charge Density

Case r>R: all the charge is enclosed
by the gaussian surface
The electric field is zero inside the charged
sphere, whereas outside the sphere the field
coincides with the field that would create a point
charge Q located at the center of the sphere
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Sphere with Uniform Volume
Charge Density



Spherical symmetry:
Guassian surface = centered sphere
of radius r:
Case r<R:
The electric field is
proportional to r
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Sphere with Uniform Volume
Charge Density



Case r>R: all the charge is enclosed
by the gaussian surface
The electric field outside the sphere coincides with the field that
would create a point charge Q located at the center of the
sphere
In this case, since there is no ρS , the electric field is
continuous at the surface of the sphere
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Infinitely Long Straight Wire with
Uniform Line Charge Density


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Symmetry arguments: cylindric symmetry
Gaussian surface = coaxial cylinder of radius r
and height L
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Infinite Plane with a Uniform
Surface Charge

Plane symmetry:
Moreover
Odd function

Gaussian surface=cylinder with arbitrary cross section S
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Infinite Plane with a Uniform
Surface Charge


Therefore:
This field meets the boundary condition (discontinuous
field):
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Index (I)

Maxwell's Equations for Static Fields

Electrostatics


Solution for the Electrostatic Field
Application of Gauss's Law for Calculating
Electrostatic Fields

Electrostatic Energy

Multipole Expansion

The Electric Dipole
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Electrostatic Energy



In general, energy stored by an electric field can be written
as:
In electrostatics we can obtain an alternative expression
in terms of the potential and the charge density:
Therefore:
Over the region where the sources are
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Electrostatic Energy

¿How can it be inferred that the first term vanishes?
Divergence Theorem


SR can be infinitely enlarged: sphere enclosing all space
When
even faster:
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but the integrand goes down
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Electrostatic Energy




In electrostatics, energy stored in the distribution:
It is the work required to assemble the charge distribution ρ
carrying the charges from infinity by means of a quasistatic process
It is also the amount of work that you'd get back out if you
dismantled the system
For surface charge densities:
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Example

Energy of a uniformly charged sphere of radius R
Method 1:
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Example

Energy of a uniformly charged sphere of radius R
Method 2:
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Example

Energy of a uniformly charged sphere of radius R

By taking
we get the energy of a point charge
The point charge is a idealized model that
implies an infinite stored energy
(An infinite work is needed to carry every dq to the same point)
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Energy of a Discrete Charge
Distribution
It includes the energy
that can be attributed to
the assembling of the
point charges themselves
Potential due to the i-th
charge evaluated on the
i-th charge is infinite
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Energy of a Discrete Charge
Distribution

An useful expression can be obtained
by dropping the troublesome term:
V* stands for the potential at the location of each charge due to
all the other charges

In this case UE represents the work done to bring together all the
point charges, but it does not include the energy needed for
creating each point charge, which is infinite

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Index (I)

Maxwell's Equations for Static Fields

Electrostatics


Solution for the Electrostatic Field
Application of Gauss's Law for Calculating
Electrostatic Fields

Electrostatic Energy

Multipole Expansion

The Electric Dipole
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Multipole Expansion


To calculate the electrostatic field and potential of an
arbitrary charge distribution you have to carry out an
integration that can be complicated :
Multipole expansion: the idea is to analyze the field
created by the charge distribution for large distances
between the source and the field point

The problem can be simplified

A new useful concept arises: electric dipole
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Multipole Expansion
If the field point is far away from
the source point:
We get the potential of a point charge (as
expected). But let's make a more accurate
approximation...
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Multipole Expansion
When
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Multipole Expansion
Dipole term
●
Monopole term: potential of a point charge
●
Q is the net charge of the distribution:
●
This is the dominant term for large distances as long as Q≠0
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Dipole Term
Dipole term:
● Dominant term in the potential when Q=0
● It falls off as 1/r2
● It does not have radial symmetry
Dipole moment of the
charge distribution
We define:
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The Electric Dipole



It is the simpler charge distribution that does not show a
monopole term in its potential
It will be very useful for modeling the behavior of a
dielectric in an electric field
Two charges of equal magnitude but opposite sign
separated by a distance d :
Dipole moment of the dipole
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The Potential of a Dipole

This is not exact: there may be higher multipole contributions

This is exact in the limit case (pure dipole):

Equipotential surfaces:
(dashed line)
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Electric Field of a Dipole

The electric field of the pure dipole can
be obtained from the potential:

Result:

It falls off like

Axially symmetric electric field
Recommended link:
http://www.falstad.com/vector3de/
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Interaction Energy with an
External Electric Field

Interaction energy:
points to
the middle of
the dipole
The energy of interaction reaches a minimun when the
electric field is parallel to the dipole moment
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Force on a Dipole

Force on a dipole:
with
is independent of
Electrostatics

An uniform electric field does not exert a net force
on a dipole
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Force on a Dipole

Uniform external electric field:
The dipole tends to rotate around its center

For a non-uniform electric field there exists a net force:
The dipole tends to rotate and is pushed toward
the region where the electric field is more intense
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Torque on a Dipole


The torque is zero when the electric field is parallel to the
dipole and is maximun when they are perpendicular
The dipole tends to line up with the electric field
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Index (I)

Maxwell's Equations for Static Fields

Electrostatics


Solution for the Electrostatic Field
Application of Gauss's Law for Calculating
Electrostatic Fields

Electrostatic Energy

Multipole Expansion

The Electric Dipole
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Index (II)

Magnetic Field of Steady Currents

Solution for the Magnetostatic Field

Biot-Savart Law
Application of Ampere's Law for calculating
Magnetostatic Fields

Magnetostatic Energy

Multipole Expansion


The Magnetic Dipole
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Steady State Case

Maxwell's equations for stationary charges and steady
currents:
Electrostatics
Magnetostatics

Magnetostatics: current density is a solenoidal field

Closed field lines for the current density

The net charge does not change with time:
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Mathematical
condition for
steady currents
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Solution for Magnetostatics
Helmholtz's Theorem:
Magnetostatics:
Solution:
Solution:
With:
With:
Magnetic vector potential
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Magnetic Field
Magnetic field as a funtion of

:
By using:
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Line Currents

Current on a wire: practical interest

Our formulas can be particularized by using:

Therefore:
In magnetostatics the magnetic field is often
calculated directly, without using the
potential vector as an intermediate step
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Example: Magnetic Field of a
Straight Segment of Wire
( Cylindrical coord. )
Change:
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Example: Magnetic Field of a
Straight Segment of Wire
Infinite straight wire:
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Biot-Savart Law

Let two loops carry steady currents:

Loop 2 creates a magnetic field
that exerts a force on loop 1:

But we already now that:

By combining this two expressions:
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Force between two parallel wires


The force is attractive if the currents are parallel and
repulsive if the currents are antiparallel
The force is proportional to the currents and inversely
proportional to the distance between the two wires
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Index (II)

Magnetic Field of Steady Currents

Solution for the Magnetostatic Field

Biot-Savart Law
Application of Ampere's Law for calculating
Magnetostatic Fields

Magnetostatic Energy

Multipole Expansion


The Magnetic Dipole
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Ampere's Law
I
°

Where:



~ ¢ d~r = ¹0 I(°)
B
Is the net current crossing any surface enclosed by
The sign of the current is given by the right hand rule
The Ampere's law allow us to find the magnetic field due to
a steady current distribution with an axial symmetry:

A careful choice of the amperian loop, , will enable us
to pull the magnetic field outside the integral sign
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Axial Symmetry

Possible axial symmetries

Toroidal field:

Poloidal field:
(cylindrical coordinates)

Example:
magnetic field
of an infinite
straight wire
Example:
magnetic field
of a circular
loop
Theorem:
If
is poloidal
is toroidal
If
is toroidal
is poloidal
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Application of Ampere's Law

Magnetic field of a infinite straight wire:

Axial symmetry:
Cross product

This also can be seen in this way:
Poloidal
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Application of Ampere's Law

Amperian loop: circumference centered
at the axis of symmetry (wire):
with:
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Application of Ampere's Law
Magnetic field of the infinite straight wire:




The field lines are
circumferences centered at the
wire
Direction of the field: right
hand rule
The magnitude of the field
decreases as the inverse of the
distance to the wire
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Surface Current on Infinite
Cylinder

Symmetry arguments:
Poloidal

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Amperian loop: circumference of radius r
centered at the z axis

Case r<R:

Case r>R:
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Volume Current on Infinite
Cylinder

Symmetry:
Poloidal


Toroidal
Amperian loop: circumference of radius r
centered at the z axis
Solution (left as an exercise):
The magnetic field inside is proportional to
the distance to the axis whereas outside
the field coincides with that of an infinite
straight wire carrying a current I
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Toroidal Coil

N closely wound turns of a wire carrying a current I on a
toroid
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Toroidal Coil

Poloidal

Toroidal
Amp. loop: circumference of radius r centered at the z axis

Outside the coil:

Inside the coil:
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Magnetic Field Lines Due to the
Toroid Indicated by Iron Filings
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Infinite Solenoid

We can consider that we have an uniform
surface current:
where n =turns per unit length

Symmetry argument:
Toroidal

Poloidal
An additional argument can be employed to
simplify the magnetic field ...
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Infinite Solenoid


From Maxwell's equations we have:
We apply this equation on a cylinder of height d
and radius r (see figure):
since


Therefore:

Hence,
and we are left with:
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Infinite Solenoid



Let's apply Ampere's Law by using a
rectangle (see figure) as amperian loop
Since the magnetic field only has z
component the horizontal segments do not
contribute to the integral:
Then:
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Infinite Solenoid


Amperes law gives us the difference between
the inner field and the outer field
This difference is independent of ri and re



Therefore the magnetic field must be
uniform in both regions (inside and outside)
But we know that he magnetic field goes to
zero for large distances
Therefore we get to the following solution:
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Index (II)

Magnetic Field of Steady Currents

Solution for the Magnetostatic Field

Biot-Savart Law
Application of Ampere's Law for calculating
Magnetostatic Fields

Magnetostatic Energy

Multipole Expansion


The Magnetic Dipole
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Magnetostatic Energy



In general, the energy stored in the magnetic field is:
In magnetostatics we can obtain an alternative
expression which is a function of the vector potential and
the current density:
Therefore :
Over the region where the sources are
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Energy of n loops of current

For line currents:
We define:
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Magnetic flux through the
loop i
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Example: Energy Stored in a
Toroidal Coil

Toroidal coil of N turns and rectangular cross section:

Method 1:
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Example: Energy Stored in a
Toroidal Coil

Toroidal coil of N turns and rectangular cross section:

Method 2:
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Index (II)

Magnetic Field of Steady Currents

Solution for the Magnetostatic Field

Biot-Savart Law
Application of Ampere's Law for calculating
Magnetostatic Fields

Magnetostatic Energy

Multipole Expansion


The Magnetic Dipole
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Multipole Expansion

Vector potential for the loop:

For large distances we have found that:

Then:
The dipole term is the first non-zero term of the expansion
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Magnetic Dipole Moment

The dipole term can be written as (not proven here):

Magnetic dipole moment of the loop:
Any surface bounded by the loop

For a plane loop (common case):
Magnitude: area of the plane surface bounded by the
loop
Direction: It is a vector perpendicular to the plane loop
in the direction given by the right hand rule
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Magnetic Field Due to a Magnetic
Dipole

Magnetic field of a magnetic dipole:



Identical in structure to the electric field of an electric dipole
This expression is highly accurate as long as the field point
is far away from the current loop.
This expression is exact for a pure dipole:

Infinitesimally small current loop whose current tends to
infinity in such a way that m=IS is constant
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Torques and Forces on Magnetic
Dipoles
An external magnetic field can exert a force and a
torque on a magnetic dipole

There exists a strong analogy with the formulas of torque
and force of an external electric field on an electric dipole


Interaction Energy:

Force on the magnetic dipole:

Torque on the magnetic dipole:
From this formulas we conclude that the magnetic dipole
will tend to:

Rotate to align parallel to the external magnetic field

Be pushed toward the region where the magnetic field is
stronger

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Summary Chapter 3 (I)


For stationary charges and steady currents the Maxwell's
equation's get uncoupled and the electric and magnetic
problems can be studied separately
Helmholtz's theorem provides a solution for the electrostatic
problem

The electrostatic field is obtained from the gradient of a scalar
field: electrostatic potential



The electrostatic potential has a physical meaning
Usually, the integral representation of the electrostatic
potential can be evaluated in a simpler manner than that of
the electric field
Gauss's law allow us to calculate the electric field due to
highly symmetric charge distributions, avoiding the integration
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Summary Chapter 3 (II)

We have found a specific formula for the electrostatic energy
associated to an electrostatic field


Coincides with the work required to carry all the charges which
are creating the electrostatic field from infinity to its present
location
The multipole expansion gives us an approximation of the
electric field “far way” form the charge distribution

When the net charge of the distribution is zero the field is
dominated by the dipole term. The electric dipole is the
simpler example of such a distribution of charges.

We will see that the electric dipole model is useful to explain
the response of insulators to external electric fields.
 An external electric field will in general exert a force
and a torque on a dipole
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Summary Chapter 3 (III)

The Maxwell's equations for stationary currents can be solved
thanks to the Helmholtz's theorem: magnetostatic




The magnetostatic field is obtained as an integral involving the
volume current density
This integral expression can be easily particularized to account
for line currents, which have a high practical interest
Ampere's Law, when applied to highly symmetric distributions
of currents, enables us to calculate the magnetostatic field in a
very straightforward way (avoiding integration)
There is an specific formula for calculating the energy
associated to the magnetostatic field of a distribution of
currents.
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Summary Chapter 3 (IV)

Multipole expansion gives us an approximation of the magnetic
field “far away” from a loop of current



The dominant term of this expansion is the dipole term
The magnetic field associated to the dipole term can be written
as a function of the magnetic dipole moment of the loop, which
depends on the area of the loop and on the current carried by the
loop
A “small” loop can be considered a magnetic dipole: the
magnetic field due to it coincides with that associated to the
dipole term of the multipole expansion

An external magnetic field will, in general, exert a force and
a torque on a magnetic dipole.
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