Republic of the Philippines
ISABELA STATE UNIVERSITY
Cabagan Campus
COLLEGE OF COMPUTING STUDIES, INFORMATION AND
COMMUNICATION TECHNOLOGY
Magnetism
Learning outcomes:
•
•
•
State the principles of magnetic energy and density
Learn about the Ampere’s law and Lenz’s law
Solve problems about Electricity and magnetism
Learning Content
In this module introduces Magnetism in relation to electricity and current flow including Magnetic
interaction, Magnetic forces, Ampere’s law, Lenz’s law, Magnetic energy and Magnetic energy
density a that links Electricity and Magnetism
Introduction
Magnetism, phenomenon associated with magnetic fields, which arise from
the motion of electric charges. This motion can take many forms. It can be an electric
current in a conductor or charged particles moving through space, or it can be the motion of
an electron in an atomic orbital. Magnetism is also associated with elementary particles, such
as the electron, that have a property called spin
The most common source of magnetic fields is the electric current loop. It may be an electric
current in a circular conductor or the motion of an orbiting electron in an atom.
Ampère’slaw
• The magnetic field in space around an electric current is proportional to the
electric current which serves as its source, just as the electric field in space is
proportional to the charge which serves as its source. Ampère’s law states that for
any closed loop path, the sum of the length elements times the magnetic field in
the direction of the length element is equal to the permeability times the electric
current enclosed in the loop.
• Oersted’s 1819 discovery about deflected compass needles demonstrates that a
current- carrying conductor produces a magnetic field. Several compass needles
are placed in a horizontal plane near a long vertical wire. When no current is
present in the wire, all the needles point in the same direction (that of the Earth’s
magnetic field), as expected.
• When the wire carries a strong, steady current, the needles all deflect in a
direction tangent to the circle. These observations demonstrate that the direction
of the magnetic field produced by the current in the wire is consistent with the
right-hand rule.
• When the current is reversed, the needles also reverse. Because the compass
⃗ , we conclude that the lines of 𝐵
⃗ form circles
needles point in the direction of 𝐵
around the wire, as discussed in the preceding section. By symmetry, the magnitude
of B is the same everywhere on a circular path centered on the wire and lying in a
plane perpendicular to the wire. By varying the current intensity and distance a
from the wire, we find that B is proportional to the current intensity and inversely
proportional to the distance from the wire, as described by the following equation
⃗ . d 𝑠 for a small length element ds on the circular path
• Now let us evaluate the dot product 𝐵
defined by the compass needles and sum the products for all elements over the closed circular
⃗ are parallel at each point, so 𝐵
⃗ . d 𝑠 = B ds.
path. Along this path, the vectors d 𝑠 and 𝐵
Furthermore, the magnitude B is constant on this circle and is given by equation. Therefore,
⃗ . d 𝑠over the closed path, which is equivalent to the line integral of
the sum of the products 𝐵
⃗ . d 𝑠 , is
𝐵
Where is the circumference of the circular path. Although this result was calculated for the
special case of a circular path surrounding a wire, it holds for a closed path of any shape
surrounding a current that exists in an unbroken circuit.
• As a result, the general case, known as Ampère’s law, can be also stated as follows:
⃗ .d 𝑠around any closed path equals μ0I, where I is the total continuous
The line integral of𝐵
current passing through any surface bounded by the closed path.
• Ampère’s law describes the creation of magnetic fields by all continuous current
configurations,but atour mathematical level it is useful only for calculating the
magnetic field of current configurations having a high degree ofsymmetry.
(a) When no current is present in the
wire, all compass needles point in
the same direction (toward the
Earth’s north pole).
(b) When the wire carries a strong
current, the compass needles deflect
in a direction tangent to the circle,
which is the direction of the
magnetic field created by the
current.
Lenz’s law
• Faraday’s law of induction indicates that the induced Emf and the change in flux
have opposite algebraic signs. This has a very real physical interpretation that has
come to be known as Lenz’slaw:
• The polarity of the induced Emf is such that it tends to produce a current that creates a
magnetic flux to oppose the change in magnetic flux through the area enclosed by the
currentloop.
• That is, the induced current tends to keep the original magnetic flux through the
circuit from changing. This law is actually a consequence of the lawof
conservation of energy.
• Wenowconcentrateonthenegativesignintheequationt
hat expresses Faraday's law. The direction of the flow
of
induced
currentinaloopisaccuratelypredictedbywhatisknowna
sLenz'slaw(orLenz'srule).
Depicting Lenz’s Law
• To understand Lenz’s law, we consider an example as shown in Figure 65. In the
figure we show a bar magnet approaching a loop. The induced current flows in the
direction
indicated
becausethis
current
generatesaninducedmagneticfieldthathasthefieldlinespointingfrom
lefttoright.Theloopisthenequivalenttoamagnetwhosenorthpolefacesthecorrespondi
ng
northpoleofthebarmagnetthatisapproachingtheloop.Theloopthenrepelstheapproach
ing magnet and thus opposes the change in the original magnetic flux that generated
the induced current.
Example:
1. A coil consists of 200turns of wire having a total resistance of 2.0Ω. Each
turn is a square of side 18cm, and a uniform magnetic field directed
perpendicular to the plane of the coil is turned on. If the field changes
linearly from 0 to 0.50T in 0.80s,
(a) Whatisthemagnitudeoftheinducedemfinthecoilwhilethefieldischanging?and
(b) What is the magnitude (intensity) of the induced current in the coil while
the field is changing?
(Ans. (a) |ε| = 4.1 V; (b) I = |ε|/R = 2.05 A)
2. A circular coil of wire with 350 turns and a radius of 7.5 cm is placed horizontally
on a table. A uniform magnetic field pointing directly up is slowly turned on, such
that the strength of the magnetic field can be expressed as a function of time as: B(t)
= 0.02(T /s2 ) × t 2 . What is the total EMF in the coil as a function of time? In which
direction does the current flow?
Answer: EMF = (-N) * (pi * r ^ 2 ) * (d/dt B) = -350 * pi * (0.075 m)^2 * 2 * 0.020 T * t = - .25 t
(Tm^2/ s^2) = -.25 t V/s Clockwise - looking from the top
Magnetic energy
Energy stored in a magneticfield
Rlcircuit
• ConsideraseriesRLcircuit.Whenthe switch S is
closed, the current immediately starts to increase. The
induced emf (or back emf) in the inductor is large, as
the current is changing rapidly. As time goes on, the
current increases more slowly, and the potential
difference across the inductordecreases.
• Ittakesenergytoestablishacurrentinaninductor;t
A series RL circuit. As the current
intensity increases toward its
maximum value, an emf that
opposes the increasing current is
induced in the inductor.
his
energyiscarriedbythemagneticfieldinsidetheindu
ctor.
• Consideringtheemfneededtoestablishaparticular
current and the power involved, wefind:
• As the current intensity through the coil increases, the magnetic field of the
coil
also
increasesandelectricalenergyisstoredinthecoilasamagneticfield.Themagneticenergy
UB stored in the coil is givenby
•
Incapacitorswefoundthatenergyisstoredintheelectricfieldbetweentheirplates.In
inductors, energy is similarly stored, only now in the magnetic field. Just as
with capacitors, where the electric field is created by a charge on the capacitor
and electric energy is stored inside the capacitors, we now have a magnetic
fieldcreatedwhenthereisacurrentthroughthe
inductor.Thus,justaswiththecapacitors,themagneticenergyisstoredinsidetheind
uctor.
•
Again, although we introduce the magnetic field energy when talking about
energy
in
inductors,itisagenericconcept–
wheneveramagneticfieldiscreated,ittakesenergytodoso, and that energy is
stored in the fielditself.
• The SI unit of magnetic energy is the joule(J).
Magnetic energy density
• Forsimplicity,consideranidealsolenoidwhoseinductanceisgivenby
L = μo(N2 /l)A = μon2Al
• The magnetic field inside a solenoid is given by
B = μonI. As a result I= B/μon
• SubstitutingtheexpressionsforLandforIintoequationleadsto
•
Because Al = V is the volume of the solenoid, the energy stored per unit volume in the
magnetic field or the magnetic energy density, uB = UB/V, inside the inductor is
•
Although this expression was derived for the special case of a solenoid, it is valid for
any region of space in which a magnetic field exists regardless of its source. From
equation above, we see that magnetic energy density is proportional to the square of
the square of the field magnitude.
• The SI unit of magnetic energy density is the joule per cubic meter(J/m3).
Example: The earth’s magnetic field in a certain region has the magnitude 6.0 x 10-5 T.
Find the magnetic energy density in this region.
(Ans. 1.4 x 10-3 J/m3)
Linking electricity and magnetism
•
Thereisastrongconnectionbetweenelectricityandmagnetism.Withelectricity
,thereare
positiveandnegativecharges.Withmagnetism,therearenorthandsouthpoles.Simi
larto charges,likemagneticpolesrepeleachother,whileunlikepolesattract.
•
Animportantdifferencebetweenelectricityandmagnetismisthatinelectricityitispo
ssibleto have individual positive and negative charges. In magnetism, north and
south poles are always found in pairs. Single magnetic poles, known as magnetic
monopoles, have been proposed theoretically, but a magnetic monopole has never
beenobserved.
•
Inthesamewaythatelectricchargescreateelectricfieldsaroundthem,northandsout
hpoles will set up magnetic fields around them. Again, there is a difference. While
electric field lines begin on positive charges and end on negative charges, magnetic
field
lines
are
closed
loops,
extendingfromthesouthpoletothenorthpoleandbackagain(or,equivalently,fromthen
orth pole to the south pole and back again). With a typical bar magnet, for example,
the
field
goes
fromthenorthpoletothesouthpoleoutsidethemagnet,andbackfromsouthtonorthinsid
ethe magnet.
•
Electric fields come from charges. So do magnetic fields, but from moving
charges, or currents, which are simply a whole bunch of moving charges. In a
permanent
magnet,
the
magneticfieldcomesfromthemotionoftheelectronsinsidethematerial,or,morepreci
sely,
fromsomethingcalledtheelectronspin.TheelectronspinisabitliketheEarthspinningo
nits axis.
•
The magnetic field is a vector; the same way the electric field is. The electric
field at a particular point is in the direction of the force a positive charge would
experience
if
it
were
placedatthatpoint.Themagneticfieldatapointisinthedirectionoftheforceanorthpoleo
fa magnet would experience if it were placed there. In other words, the north pole
of
a
compass
pointsinthedirectionofthemagneticfieldthatexertsaforceonthecompass.
•
ThesymbolformagneticfieldinductionormagneticfluxdensityistheletterB.TheSI
unitis the tesla(T).
•
One of various manifestations of the linking between electricity and
magnetism
is
electromagneticinduction.Thisinvolvesgeneratingavoltage(aninduced
electromotiveforce)bychangingthemagneticfieldthatpassesthroughacoilofwir
e.
•
In other words, electromagnetism is a two-way link between electricity and
magnetism.
An
electriccurrentcreatesamagneticfield,andamagneticfield,whenitchanges,createsavo
ltage. The discovery of this link led to the invention of transformer, electric motor,
and generator. It also explained what light is and led to the invention ofradio.
Self- Assessment:
1. Find the forces exerted by S poles of magnets given below.
2. Find resultant magnetic field at point O, produced by I1, I2 and I3.
3. A, B and C wires are given below. Find the magnetic field of A, B and C at points X and
Y.
4. Solenoid having number of loops N and surface area A is shown in picture given below. If
we change the position of solenoid as shown in the picture below, find the equation used for
finding induced emf of solenoid.
5. Draw the directions of magnetic field lines at point A, B, C and D in the picture given
below.
References:
Published Works:
▪ Halliday, David; Resnick, Robert; Walker, Jearl. Fundamentals of Physics 7th
ed. John Wiley & Sons, Inc.
▪ Feynman, Richard; Leighton, Robert; Sands, Matthew. Feynman Lectures on
Physics. Addison-Wesley Publishing Company.
▪ Serway, Raymond; Faughn, Jerry. College Physics 7th ed. Thompson,
Brooks/Cole.
▪ Sears, Francis; Zemansky Mark; Young, Hugh. College Physics 7th ed.
Addison- Wesley Publishing Company.
▪ Beiser, Arthur. Physics 5th ed. Addison-Wesley Publishing Company.
▪ Jones, Edwin; Childers, Richard. Contemporary College Physics 7th ed.
Addison- Wesley Publishing Company.
▪ Alonso, Marcelo; Finn, Edward. Physics 7th ed. Addison-Wesley Publishing
Company.
▪ Michels, Walter; Correll, Malcom; Patterson, A. L. Foundations of Physics
7th ed. Addison-Wesley Publishing Company.
▪ Hecht, Eugene. Optics 2th ed. Addison-Wesley Publishing Company.
▪ Eisberg, R. M. Modern Physics, John Wiley & Sons, Inc.
URL:
▪ http://ocw.mit.edu/OcwWeb/Physics/8-02TSpring2005/LectureNotes/index.html
▪ http://physics.bu.edu/~duffy/PY106/Charge.html
▪ http://science.jrank.org/pages/1729/Conservation-Laws-Conservationelectric-charge.html
▪ http://web.pdx.edu/~bseipel/ch31.pdf
▪ http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magcur.html#c1
▪
▪
▪
https://courses.lumenlearning.com/physics/chapter/18-4-electric-field-concept-of-a-fieldrevisited/#:~:text=The%20electric%20field%20E%20is%20defined%20to%20be%20E%3DF
,is%20the%20distance%20from%20Q.
https://physexams.com
https://www.csun.edu/~rd436460/100B/lectures/chapter19-1-3.pdf