Energy Stored in Magnetic Field
ÎJust
like electric fields, magnetic fields store energy
uE =
1 ε E2
2 0
2
Electric field
B
uB =
2μ0
ÎLet’s
Magnetic field
see how this works
PHY2049: Chapter 30
47
Energy of an Inductor
Î
How much energy is stored in an inductor
when a current is flowing through it?
Î
Start with loop rule
ε = iR + L di
dt
Î
Multiply by I to get power equation
ε
Î
Î
Î
di
I = i R + Li
dt
2
Identify PL, the rate at which energy
is being stored in the inductor
Identify energy stored in inductor
Similar to capacitor: U C
PL =
dU L
di
= Li
dt
dt
1 2
U L = ∫ Lidi = Li
2
q2
=
2C
48
Energy in Magnetic Field (2)
ÎApply
UL =
ÎUse
l
to solenoid (constant B field)
1
2
Li 2 =
1
2
(
)
μ 0 n 2 lA i 2
r
formula for B field: B = μ 0 ni
N turns
2
B
UL =
lA
2μ0
UL
ÎCalculate energy density: u B =
V
B2
uB =
2μ0
ÎThis
B field
V = Al
u E = 12 ε 0 E 2
E field
is generally true even if B is not constant
PHY2049: Chapter 30
49
Energy Calculation Examples
ÎCalculate
uB for earth field, B = 5 x 10-5 T
(
−5
)
2
5 × 10
B2
−3
3
uB =
=
10
J/m
2 μ 0 2 × 4π × 10 − 7
ÎCalculate
uB for neutron star, B = 108 T
( )
8
2
10
B2
21
3
=
×
uB =
4
10
J/m
2 μ 0 2 × 4π × 10 − 7
ÎCalculate
uB for magnetar, B = 1011 T
(
11
)
2
10
B2
27
3
=
×
uB =
4
10
J/m
2 μ 0 2 × 4π × 10 −7
Equivalent mass density
uB
10
3
ρ = 2 4 × 10 kg/m
From E = mc2
c
PHY2049: Chapter 30
50
Web Sites
ÎOriginal
magnetar discovery
‹ http://science.nasa.gov/newhome/headlines/ast20may98_1.htm
‹ http://www.firstscience.com/site/articles/solarflares.asp
ÎMore
recent magnetar discovery (Feb. 2005)
‹ http://www.physorg.com/news3112.html
ÎOnline
articles on magnetars
‹ http://solomon.as.utexas.edu/~duncan/magnetar.html
‹ http://www.space.com/scienceastronomy/magnetar_formation_05
0201.html
‹ http://apod.gsfc.nasa.gov/apod/ap041126.html
ÎArticles
on neutron stars (second one has videos_
‹ http://www.astro.umd.edu/~miller/nstar.html
‹ http://antwrp.gsfc.nasa.gov/htmltest/rjn_bht.html
PHY2049: Chapter 30
51
Gigajoule Magnet at CERN
ÎCMS
experiment at CERN
‹ p-p
collisions at world’s highest energy in 2007
‹ Hope to discover new particles, find the origin of mass and new
fundamental forces
Compact Muon Solenoid
PHY2049: Chapter 30
52
Human
PHY2049: Chapter 30
53
CMS Experiment Magnet
ÎLarge
central solenoid magnet to study particle production
‹B
= 4T, R = 3.15 m, L = 12.5 m
‹ UB = 2.6 x 109 J = 2.6 gigajoules!!
UB
(
)
B2
42
2
lA =
3.15
=
π
×
(12.5 )
−
7
2μ0
2 × 4π × 10
http://www.spacedaily.com/news/energy-tech-04b.html
PHY2049: Chapter 30
54
CMS Articles and Pictures
ÎHome
page
‹ http://cmsinfo.cern.ch/Welcome.html/
ÎPictures
of detector
‹ http://cmsinfo.cern.ch/Welcome.html/CMSdetectorInfo/CMSdetect
orInfo.html
‹ http://cmsinfo.cern.ch/Welcome.html/CMSmedia/CMSmedia.html
ÎInteresting
article on solenoid, with pictures
‹ http://cmsinfo.cern.ch/Welcome.html/CMSdocuments/MagnetBroc
hure/MagnetBrochure.pdf
ÎOther
documents & pictures about CMS
‹ http://cmsinfo.cern.ch/Welcome.html/CMSdocuments/CMSdocume
nts.html
PHY2049: Chapter 30
55
Transformers
ÎPurpose:
change alternating (AC) voltage to a bigger (or
smaller) value
Input AC voltage
in the primary
produces a flux
Changing flux
in secondary
induces emf
dΦB
Vp = N p
dt
dΦB
Vs = N s
dt
Ns
Vs = V p
Np
PHY2049: Chapter 30
56
Transformers
ÎNothing
comes for free, however!
‹ Increase
in voltage comes at the cost of current.
‹ Output power cannot exceed input power!
‹ power in = power out
i pVp = isVs
is V p N p
=
=
i p Vs N s
PHY2049: Chapter 30
57
Transformers: Sample Problem
ÎA
transformer has 330 primary turns and 1240 secondary
turns. The input voltage is 120 V and the output current
is 15.0 A. What is the output voltage and input current?
Ns
⎛ 1240 ⎞
Vs = V p
= 120 ⎜
⎟ = 451V
Np
⎝ 330 ⎠
i pV p = isVs
“Step-up”
transformer
Vs
⎛ 451 ⎞
i p = is
= 15 ⎜
⎟ = 56.4 A
Vp
⎝ 120 ⎠
PHY2049: Chapter 30
58
Transformers
¾ This is how first experiment by
Faraday was done
¾ He only got a deflection of the
galvanometer when the switch
is opened or closed
¾ Steady current does not make
induced emf.
PHY2049: Chapter 30
59
Applications
Microphone
Tape recorder
PHY2049: Chapter 30
60
ConcepTest: Power lines
ÎAt
large distances, the resistance of power lines becomes
significant. To transmit maximum power, is it better to
transmit (high V, low i) or (high i, low V)?
‹ (1)
high V, low i
‹ (2) low V, high i
‹ (3) makes no difference
Power loss is i2R
PHY2049: Chapter 30
61
Electric Power Transmission
i2R: 20x smaller current ⇒ 400x smaller power loss
PHY2049: Chapter 30
62