Energy Stored in Magnetic Field
ÎJust
like electric fields, magnetic fields store energy
uE =
1 ε E2
2 0
2
Electric field energy density
B
uB =
2μ0
ÎLet’s
Magnetic field energy density
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
di
ε i = i R + Li
dt
2
Power produced = dissipated + stored
Î
Let PL = power stored in inductor
Î
Identify energy stored in inductor
Î
q2
=
2C
Similar to capacitor:
UC
dU L
di
PL =
= Li
dt
dt
1 2
U L = ∫ Lidi = Li
2
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 on Neutron Stars, Magnetars
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
Quiz on Energy
ÎEach
circuit has identical values of V, R and L. After the
switch has been closed for a long time, which circuit has
the largest energy stored in the inductor L?
‹ (1)
‹ (2)
‹ (3)
‹ (4)
‹ (5)
1
2
3
1
2
only
only
only
&2
&3
V
Answer is modified from what was given in class.
Answers 2 and 3 were accepted
R
R
R
L
V
R
L
V
R
L
R
PHY2049: Chapter 30
52
Gigajoule Magnet at CERN
ÎCMS
experiment at CERN
‹ p-p
collisions at world’s highest energy in 2009
‹ Hope to discover new particles, find the origin of mass and new
fundamental forces
Compact Muon Solenoid
PHY2049: Chapter 30
53
Human
PHY2049: Chapter 30
54
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
55
Articles on CMS and LHC
ÎLarge
Hadron Collider at CERN
‹ http://public.web.cern.ch/public/en/LHC/LHC-en.html
ÎHome
page and picture of CMS experiment
‹ http://cms.cern.ch/
ÎWikipedia
articles
‹ http://en.wikipedia.org/wiki/Compact_Muon_Solenoid
‹ http://en.wikipedia.org/wiki/Large_Hadron_Collider
PHY2049: Chapter 30
56