Diapositivas Jhon Aguilar

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ELECTROMAGNETISMO





Atomo



BREVE HISTORIA

Uno de sus hitos iniciales puede situarse hacia el año 600 a. C. cuando el

!il"so!o #rie#o Tales de $ileto o%ser&" 'ue !rotando una &arilla de ()%ar

con una piel o con lana se o%ten*an pe'ueñas car#as +e!ecto tri%oel,ctrico-

'ue atra*an pe'ueños o%etos / !rotando )ucho tie)po pod*a causar la

aparici"n de una chispa. Cerca de la anti#ua ciudad #rie#a de $a#nesia se

encontra%an las deno)inadas piedras de Magnesia 'ue inclu*an

)a#netita. os anti#uos #rie#os o%ser&aron 'ue los tro1os de este )aterial

se atra*an entre s* / ta)%i,n a pe'ueños o%etos de hierro. as pala%ras

magneto +e'ui&alente en español a i)(n- / )a#netis)o deri&an de ese

top"ni)o. a electricidad e&olucion" hist"rica)ente desde la si)ple

percepci"n del !en")eno a su trata)iento cient*!ico 'ue no se har*a

siste)(tico hasta el si#lo 2VIII. El tel,#ra!o el,ctrico +Sa)uel $orse

3455 precedido por auss / 7e%er 3488- puede considerarse co)o la pri)era #ran aplicaci"n en el ca)po de las teleco)unicaciones pero no

ser( en la pri)era re&oluci"n industrial sino a partir del cuarto !inal del

si#lo 2I2 cuando las aplicaciones econ")icas de la electricidad la

con&ertir(n en una de las !uer1as )otrices de la se#unda re&oluci"n

industrial.



a ener#*a el,ctrica es esencial para la sociedad de la in!or)aci"n de la

tercera re&oluci"n industrial 'ue se &iene produciendo desde la se#unda

)itad del si#lo 22 +transistor tele&isi"n co)putaci"n ro%"tica

internet...-.

3600 + 7illia) il%ert9 adopt" el t,r)ino de electricidad )ediante

e:peri)entos de !rotaci"n de distintos )ateriales.

Si#lo 2VIII + Stephen ra/9 detallo las caracter*sticas necesarias para la

conducti&idad el,ctrica.

3;55 + Charles <ran=ois de Cisterna/ du <a/9 creo lo conceptos de car#as

positi&as / car#as ne#ati&as.

Si#lo 2I2 + Hans Christian Oersted9 planteo la hip"tesis de 'ue los

!en")enos )a#n,ticos / el,ctricos estu&iesen relacionados.

A)pere9 de)ostr" la relaci"n de Oersted entre 3488 / 3486.

348; + Oh)9 !or)ul" la de Oh) en la 'ue relaciona%a tensi"n corriente /

resistencia.



$ichael <arada/9 descu%ri" la inducci"n electro)a#n,tica / el concepto

de l*neas de ca)po lo 'ue le per)iti" en 3483 crear el pri)er )otor

el,ctrico. Ta)%i,n consi#ui" de)ostrar 'ue la car#a el,ctrica en un

conductor se acu)ula en la super!icie e:terior de ,ste

independiente)ente de lo 'ue ha/a en su interior. +!oto-

34>3+ Her)ann Von Hel)holt19 de)ostr" 'ue la electricidad era una

ener#*a / 'ue co)o tal cu)pl*a la le/ de conser&aci"n.

346> + ?a)es Cler@ $a:ell9 esta%lece las lla)adas Ecuaciones de

$a:ell 'ue de)ostraron / detallaron la relaci"n )ate)(tica entre

ca)pos el,ctricos / )a#n,ticos los cuales se#n de)ostr" ten*an la

)is)a naturale1a 'ue la lu19 naturale1a de onda a las cuales se deno)in"

ondas electro)a#n,ticas.

3443 + Edison9 Tho)as Al&a Edison in&ent" la %o)%illa / tras otros dos

años descu%ri" el e!ecto ter)oi"nico o e!ecto Edison.

3444 + Di@ola Tesla9 crea el pri)er #enerador de corriente alterna /

posterior)ente un )otor 'ue !unciona%a con ella.



(a) A negatively charged

rubber rod suspended by athread is attractedto a positively charged glassrod.

(b) A negatively chargedrubber rod is repelled by



Carga eléctrica

La carga eléctrica es la propiedad de la materia que

señalamos como causa de la interaccin electromagnética!

Se dice que un cuerpo est" cargado positi#amente

cuando tiene un de$ecto de electrones! Se dice que un

cuerpo est" cargado negati#amente cuando tiene un

e%ceso de electrones!

&or tanto tam'ién podemos de$inir la carga eléctrica como

el e%ceso o de$ecto de electrones que posee un cuerpo

respecto al estado neutro!

La unidad de carga en el sistema internacional es el

Coulom' (C)! Se usan tam'ién el microcoulom' (* +C ,

*-./C)0 el nanocoulom' (*nC , *-.1C) o el picocoulom'

(*pC , *-.*2C)!



Induccin de Carga So're un Aislante



When materials behave in this way, they are said to beelectrifed, or to have become electrically charged.

In a series of simple experiments, it was found that there are

two kinds of electric charges, which were given the namespositive and negative by Benjamin Franklin (1706–1790).

n the basis of these observations, we conclude thatcharges of the same sign repel one another and charges with

opposite signs attract one another.

Another important aspect of electricity that arises fromexperimental observations is that electric charge is alwaysconserved in an isolated system



Electrical conductors are materials in which some of the

electrons are free electrons! that are not bound to atoms and

can move relatively freely through the material" electrical

insulators are materials in which all electrons are bound to

atoms and cannot move freely through the material.

emiconductors are a third class of materials, and theirelectrical properties are somewhere between those of

insulators and those of conductors

#onductores$ hierro, plata, oro, aluminio, bronce, cobre%

Aislantes$ vidrio, pl&stico, cer&mica, madera seca, caucho%

'emiconductores$ silicio, germanio%





rom #oulombs experiments, we can generali*e the following propertiesof

the electric force between two stationary charged particles. +he electric

force.

is inversely proportional to the suare of the separation r between

the particles and directed along the line -oining them

is proportional to the product of the charges q and q’ on the two

particles

is attractive if the charges are of opposite sign and repulsive if the

charges have the same sign

is a conservative force.







+wo point charges separated by adistance r exert a force on each other that isgiven by #oulombs law. +he force

exerted by on is eual in magnitudeand opposite in direction to the force

exerted by on

+hen$

(a)When the charges are of thesame sign, the force is repulsive.

(b) When the charges are ofopposite signs, the force isattractive







t3e electric $ield #ector at a point in space

is de$ined as t3e electric $orce acting on

a positi#e test c3arge q placed at that point

divided by the test charge:

E is t3e $ield produced '4 some c3arge orc3arge distri'ution separate from t3e test

c3arge0 it is not t3e $ield produced '4 t3e test

c3arge itsel$!

T3e presence o$ t3e test c3arge is not

necessar4 $or t3e $ield to e%ist! T3e test c3arge

ser#es as a detector of the electric $ield!



qo´





A test c3arge at point P is a distance r

from a point c3arge q.

(a)If q is positive, t3en t3e $orce on t3e test

c3arge is directed a5a4 $rom q.

(a) For the positi#e source c3arge0 t3e

electric

$ield at P points radially outward $rom q.

(c) If q is negative, then the $orce on t3e testc3arge is directed to5ard q.

(d) For the negative source c3arge0 t3e

electric $ield at P points radiall4 in5ard

to5ard q.























We have dened the electric eld mathematically through

+he electric eld vector E is tangent to the electric eld line ateach point. +he line has a direction, indicated by an arrowhead,that is the same as that of the electric eld vector.

+he number of lines per unit area through a surfaceperpendicular to the lines is proportional to the magnitude ofthe electric eld in that region. +hus, the eld lines are closetogether where the electric eld is strong and far apart wherethe eld is weak.



+he magnitude of the eld is greater on surface Athan on surface /.

+he electric eld lines for a point charge. (a) or a positive point charge, thelines are directed radially outward. (b) or a negative point charge, the lines aredirected radially inward. 0ote that the gures show only those eld lines that liein the plane of the page. (c) +he dark areas are small pieces of thread

suspended in oil, which align with the electric eld produced by a small chargedconductor at the center



+he lines must begin on a positive charge andterminate on a negative charge.

In the case of an excess of one type of charge, somelines will begin or end innitely far away.

+he number of lines drawn leaving a positive chargeor approaching a negative charge, is proportional to

the magnitude of the charge.

0o two eld lines can cross.



















If 1 is uniform in both magnitude and

direction$ +he line density is proportional to the

magnitude of the electric eld. +herefore,the total number of lines penetrating thesurface is proportional to the product EA.

1l 2u-o el3ctrico es la medida deln4mero de l5neas de campo ueatraviesan cierta supercie.





0ote that the number of lines that cross this area A is equal tothe number that cross the area A’, which is a pro-ection of areaA onto a plane oriented perpendicular to the eld. We

conclude that the 2ux through A is

6aximum value 1A when the surface is perpendicular to theeld (678)" the 2ux is *ero when the surface is parallel to

the eld (6

798)

7E.!



#onsider a general surface divided up into a largenumber of small elements, each of area

where we have used the denition of the scalar product$



:sing the symbol to represent an integral over a closed

surface, we can write the net 2ux through a closedsurace as

1l 2u-o puede ser positivo (;), negativo (<)

o cero (8).





Esta'lece la relacin entre el $lu7o eléctrico a tra#és de una

super$icie cerrada (Gaussiana) 4 la carga neta encerrada en

su interior

q

r

dA

E

pero0

We nd that the net 2ux

through the gaussiansurface is

8r

kq E =

8> r A π =

0ε

q=Φ



ε

intq

dA E c =⋅=Φ

∫ =a =ey de >auss establece ue el 2u-o el3ctrico

neto atrav3s de cualuier supercie >aussianacerrada es igual a la carga neta ue se encuentradentro de ella, dividida por ?@

q





#onditions$

+he value of the electric eld can be argued bysymmetry to be constant over the surface.

+he dot product in 1uation, can be expressed as asimple algebraic product E . dA because E and dA are parallel.

+he dot product in 1uation is *ero because 1 and d Aare perpendicular

+he eld can be argued to be *ero over the surface.









E , - (r 8 a)



#ontinued







;;

;;

;;

;;

;;

;;

;;

;;

;;

;;

<<<<<<<

<<<<<<<

<<<<<

<<<<<<<

<<<<<<

Separacin92m: sigma,;nc<m=Calcule E en %,*0>m 4 %,?m!

R*<9 2@22/N<c R2<9 -



No 3a4 mo#imiento neto de la carga dentro del conductor

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

E

*! El campo eléctrico es cero en cualquier punto en el interior

del conductor



2! Cualquier e%ceso de carga en un conductor aislado0 de'e residir

enteramente so're su super$icie

Super$icie Gaussiana

Conductor Aislado

B! El campo Eléctrico precisamente $uera del conductor es

perpendicular a la super$icie del mismo 4 tiene magnitud de B?@

E7e! 2;.









dD , Fds , qCEds



+he potential diDerence between two points Aand B in an electric eld is dened as the change in

potential energy of the system when a test charge is movedbetween the points A and /, divided by the test charge

+he work done by an external agentin moving a charge q through anelectric eld at constant velocity is$

+he electric potential at any point in an electric eld is$

EFG7H#

Fi$erencia de &otencial



E

d

A

FesplaHamiento de una partcula cargada0

desde A 3asta 0 en presencia de un campo

eléctrico uni$orme E





We can calculate the change in the potential energy of thechargeJeld system from 1uations

We conclude that a system consisting of a positive chargeand an electric eld loses electric potential energy when thecharge moves in the direction of the eld. A systemconsisting of a negative charge and an electric eld gains

electric potential energy when the charge moves in thedirection of the eld.0ow consider the more general case, in this case, euationis$

E· s = E s cosθ = E d,

con s cosθ = d



Conclusión:

VB-VA = VC-VA, luego VB = VC .

Todos los puntos en un plano perpendicular al

campo eléctrico uniforme, están al mismo

potencial. El nombre de superficie equipotencialse da a cualquier superficie que contiene una

distribución continua de puntos que tienen el

mismo Potencial.

+he change in potential energy of thechargeJeld system is









+o nd the electric potential at a point

located a distance r rom the charge,we

begin with the general expression for

potential diDerence$

where is a unit vectordirected

from the charge toward thepoint.

+he uantity can be expressed as



T3en t3e electric potential created '4 a point

c3arge at an4 distance r $rom t3e c3arge is

ds , ds cosr K J

dr , ds cosJ

Considere)os el potencial VA = 0 para un r

)u/ #rande +A en el in!inito- V α 1/rA .

∑=i

i

r

qk V





1s la energ5a potencial del sistema dedos part5culas separadas por una

distancia r!L

1s tambi3n el traba-o necesario oreuerido para traer la carga L ,desde el innito, hasta una distanciar!L de !

r *2

q*

q2

F!

38

83

38

r qqk

V qU

=

=









+he electric potential dV at some point due to the charge element dqis

where r is the distance rom thecharge element to point . +o obtainthe total potential at point , we

integrate Equation



Anillo cargado uni$ormemente





ESFERA AISLANTE CARGADA UNIFORMEMENTE





















Condensador de Placas Paralelas

















COMINACION EN &ARALELO





COMINACION EN SERIE















E7emplo!

A parallel plate capacitor is c3arged 5it3 a 'atter4 to a c3arge o as



A parallel.plate capacitor is c3arged 5it3 a 'atter4 to a c3arge o, as

shown in Figureb(a). !he battery is then remo#ed0 and a sla' o$

material t3at 3as a dielectric constant " is inserted 'et5een t3e

plates0 as s3o5n in igure (')! ind t3e energ4 stored in t3e capacitor'e$ore and a$ter t3e dielectric is inserted!















+he current is the rate at which charge 2ows through this

surface. If M! is the

amount o charge that passes through this area in a timeinterval ʌt, the average

current "av is equal to the charge that passes through A perunit time

If the rate at which charge 2ows varies intime, then the current varies in time" wedene the instantaneous current " as thedi#erential limit o average current

+he 'I unit of current is the ampere (A)$



At v A xV d ol ⋅∆=⋅∆=





A current density H and an electric eld 1are established in a conductor whenevera potential diDerence is maintained

across the conductor. In some materials,the current density is proportional to theelectric eld$

+herefore, we can express the magnitude of the currentdensity in the wire as



density in the wire as

/ecause

+he uantity is called the resistance o theconductor. $e can defne the resistance as the ratio ofthe potential diDerence across a conductor to thecurrent in the conductor$

+he inverse of conductivity











metales





otencia entre#ada a la resistencia.

Ener#*a disipada por la resistencia







"irect #urrent #ircuits



Electromoti$e Force



The e)! o! a %atter/ is the )a:i)u) possi%le &olta#e that the %atter/can pro&ide %eteen its ter)inals.

Do i)a#ine )o&in# throu#h the %atter/

!ro) a to b and )easurin# the electric

potential at &arious locations. As e pass!ro) the ne#ati&e ter)inal to the positi&e

ter)inal the potential increases %/ an

a)ount . Hoe&er as e )o&e throu#h

the resistance r the potential decreases %/

an a)ount Ir here I is the current in the

circuit. Thus the ter)inal &olta#e o! the %atter/ ab V V V −=∆



Los circuitos que se analiHar"n ser"n aquellos que se encuentran en

estado estacionario0 o sea que en ellos las corrientes son constantes

en magnitud 4 direccin!

Kna corriente que es constante en direccin es llamada corriente

directa (FC)!

+he resistor represents a load on the battery because the battery must supply energy tooperate the device. +he potential diDerence across the load resistance is

#ombining this expression with 1uation



+his euation shows that the current in this simple circuitdepends on both the load resistance % external to the batteryand the internal resistance r. If % is much greater than r, as itis in many real<world circuits, we can neglect r. If we multiply1uation by the current &, we obtain

/ecause power , +he total power output of the battery is delivered to theexternal load resistance in the amount and to theinternal resistance in the amount

ε I R I 8

EJEMPLO



% i i i d



%esistors in eries and

'arallel

or a series combination of two resistors, the currents are thesame in both resistors because the amount of charge that

passes through %& must also pass through %' in the same timeinterval.

+he potential diDerence applied across the series combinationof resistors will divide between the resistors$

+he potential diDerence across the battery is also applied tothe eui$alent resistance



+his relationship indicates that the euivalent resistance of a

series connection of resistors is the numerical sum of the

individual resistances and is always greater than any

individual resistance.

the eui$alent resistance



%esistors in 'arallel



eq

Where %eq is an equivalent single resistancewhich will have the same e#ect on the circuit as



which will have the same e#ect on the circuit asthe two resistors in parallel" rom this result

or









NirchhoDs rst rule is a statement of conservation of electric charge.

If we apply this rule to the -unction shown in gure we obtain$



NirchhoDs second rule follows from the law ofconservation of energy. When the charge returns to thestarting point, the chargeJcircuit system must have thesame total energy as it had before the charge was moved.





9

9



%# #ircuits #harging a#harging a

#a-acitor#a-acitor



#a-acitor#a-acitor

+he value of the maximum charge on the plates depends on the

voltage of the battery. nce the maximum charge is reached, thecurrent in the circuit is *ero because the potential diDerenceacross the capacitor matches that supplied by the battery.

where q() is the potential di#erence across the capacitor and "% isthe potential di#erence across the resistor. *or the capacitor,notice that we are traveling in the direction rom the positive plate



notice that we are traveling in the direction rom the positive plateto the negative plate" this represents a decrease in potential.

we nd that the initial current in the circuit is a maximumand is eual to

At this time, the potential diDerence from the batteryterminals appears entirely across the resistor. =ater, when thecapacitor is charged to its maximum value !, charges cease to

2ow, the current in the circuit is *ero, and the potentialdiDerence from the battery terminals appears entirely acrossthe capacitor

#orrientem&xima

#argam&xima

+his current is eual to the time rate of change of the chargeon thecapacitor plates$



+o nd an expression for q, we solve this separabledi#erential equation. $e frst combine the terms on theright<hand side$

0ow we multiply by dt and divide by q to obtain

Integrating this expression, using the fact that

atweobtain





"ischarging a#a-acitor



0ow consider the circuit shown in igure which consists of acapacitor carrying an initial charge ! , a resistor, and a switch.$hen the switch is open, .

" the switch is closed at the capacitor begins to discharge

through the resistor. At some time t during the discharge, thecurrent in the circuit is " and the charge on the capacitor is q. +hus,

we eliminate from euation to obtain the appropriate loopeuation for the circuit

When we substitute into this e+pression, itbecomes



Integrating this expression, using the fact that

OiDerentiating this expression with respect to time gives theinstantaneous current as a function of time$



E!/'E











$hen the particle’s velocity vector maes anyangle with the magnetic feld the magnetic



angle with the magnetic feld, the magnetic

orce acts in a direction perpendicular to both v andB- that is, is perpendicular to the plane ormed byv and B .

he magnetic orce e+erted on a positive charge

is in the direction opposite the direction o themagnetic orce e+erted on a negative charge movingin the same direction.

he magnitude o the magnetic orce e+erted on

the moving particle is proportional to sin whereis the angle the particle’s velocity vector maes

with the direction o B. $e can summari/e theseobservations by writing the magnetic orce in theorm



0agnitude o the magnetic orce on a charged particle moving in a magnetic feld



is the smaller angle between v and B. *rom thise+pression, we see that FB is /ero when v is parallelor antiparallel to B 1 23 or &4356 and ma+imumwhen v is perpendicular to B .

Important differences: electric and magnetic forces:



he electric orce acts along the direction o theelectric feld, whereas the magnetic orce acts

perpendicular to the magnetic feld.

he electric orce acts on a charged particle

regardless o whether the particle is moving,whereas the magnetic orce acts on a charged particle only when the particle is in motion.

he electric orce does wor in displacing a charged particle, whereas the magnetic orce associated with asteady magnetic feld does no wor when a particle is

displaced because the orce is perpendicular to thedisplacement.

$e see that the 7" unit o magnetic feld is the 8ewton percoulomb9meter per second, which is called the tesla 16:

An electron in a television picture tube moves

EXAMPLE:



toward the ront o the tube with a speed o 4.3 +

&31e+p;6 m(s along the + a+is 1*ig. '<.=6.7urrounding the nec o the tube are coils o wirethat create a magnetic feld o magnitude 3.3'= ,directed at an angle o ;35 to the + a+is and lyingin the +y plane. )alculate the magnetic orce on

and acceleration o the electron. *ind a vectore+pression or the magnetic orce on the electron





*or ease in visuali/ation, part o the horseshoemagnet in part 1a6 is removed to show the endace o the south pole in parts 1b6, 1c6, and 1d6 themagnetic feld is directed into the page and coversthe region within the shaded squares. $hen thecurrent in the wire is /ero, the wire remainsvertical, as shown in *igure 1b6 >owever, when thewire carries a current directed upward, as shownin *igure 1c6, the wire de?ects to the let. " we

reverse the current as shown in *igure 1d6 the

A



q #d

L

A

+he magnetic force exerted on a charge q moving witha drit velocity

iso fnd the total orce acting on the wire, we multiply

the orce exerted on one charge by the number ofcharges in the segment. /ecause the volume of thesegment is the number o charges in the segmentis, where n is the number of charges per unitvolume.

$e can write this e+pression in a moreconvenient orm by noting that, rom Equation



convenient orm by noting that, rom Equationthe current in the wire is

where L is a vector that points in the directiono the current I and has a magnitude equal tothe length L o the segment. 8ote that thise+pression applies only to a straight segment owire in a uniorm magnetic feld.





Case 1. A curved wire carries a current I and islocated in a uniorm magnetic feld B. Because thefeld is uniorm, we can tae B outside the integralin Equation

ds represents the vector sum o all the lengthelements rom a to b. *rom the law o vectoraddition, the sum equals the vector L, directedrom a to b. hereore, reduces to

L’

b

a

B

I

@ *rom this we conclude that the magnetic orceon a curved current9carrying wire in a uniormmagnetic feld is equal to that on a straight wire



magnetic feld is equal to that on a straight wireconnecting the end points and carrying the samecurrent

Case 2 . An arbitrarily shaped closed loop carrying a current " is

placed in a uniorm magnetic feld. $e can again e+press the mag9



placed in a uniorm magnetic feld. $e can again e+press the mag

netic orce acting on the loop in the orm o Equation but this time

we must tae the vector sum o the length elements ds over theen9

tire loop:

dsB

I

0= B F

*rom this we conclude that themagnetic orce on an arbitraryclosed loop carrying a current, placed in a e+tern uniormmagnetic feld, is equal to /ero.











Ejem-lo





Fireccin de A 4 en consecuencia de µ

Energia &otencial de un Sistema de un Fipolo

Magnético en un Campo Magnético





+he particle moves in a circle because the magneticforce / is perpendicular to v and /



+he angular speed of the particle

+he period of the motion (the time interval the particlereuires to complete one revolution) is eual to the

circumference of the circle divided by the linear speedof the particle$



Ejem-lo



Selector de elocidades





he /ass -ectrometer

A mass spectrometer separates ions according to their mass<to<charge ratio. In one version of this device, known as theBainbridge mass spectrometer, a beam o ions frst passesthrough a velocity selector and then enters a second

uniform magnetic eld /8 that has the same direction asthe magnetic eld in the selector that has the samedirection as the magnetic eld in the selector





Cuando un conductor que lle#a una corriente se coloca en presencia



q p

de un campo magnético0 se genera un #olta7e

en la direccinperpendicular tanto a la corriente como al campo magnético!

Los electrones #an a la iHquierda 4 e%perimentan $uerHa 3acia arri'a!

Los protones #an a la derec3a 4 e%perimentan $uerHa 3acia arri'a!

nqA

IBd

V H =



=ey de /iot<'avart.

Si un alam're conduce una corriente constante I0 el campo magnético

dB en un punto P de'ido a un elemento ds (el cual tiene la direccin de

la corriente) tiene las siguientes propiedades9

0 Fr Ids

dB ×

= µ



8

> r

dB =

π

Le4 de iot.Sa#art!

AmT ⋅×= −;0

30>π µ

∫ ×=8

0 F

> r r ds I B

π

µ

&ermea'ilidad del espacio li're!

Campo total creado en algnpunto por una corriente

$inita!



Ejem-lo





Ejem-lo







area



Recuerde9

uerHa que e%perimenta un conductor

que lle#a una corriente I0 en presencia

de un campo magnético e%terno!

Campo magnético generado por un

conductor que lle#a una corriente I 0 en

&unto e%terno a una distancia a de el!

uerHa so're el alam're * de longitud l.



Magnitud de la $uerHa *!

ReemplaHando en *!

uerHa por unidad de longitud entre dos alam'res

conductores paralelos!











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Solenoide ideal! Campo

e%terno cero0 e interno

uni$orme













EXPEIME!"O #E FAA#A$





E7emplo



use ,2P





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Diapositivas Jhon Aguilar