Energy Stored in Magnetic Field

ÎJust like electric fields, magnetic fields store energy

1 2 Electric field energy density uEE = 2 ε0 B2 uB = Magnetic field energy density 2 0 μ

ÎLet’s 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 di ε =+iR L dt

Î Multiply by i to get power equation di ε iiRLi=+2 Power produced = dissipated + stored dt dUL di Î Let P = power stored in inductor P ==Li L L dt dt 1 2 Î Identify energy stored in inductor ULidiLi== L ∫ 2 q2 Î Similar to capacitor: U = C 2C 48 Energy in Magnetic Field (2)

ÎApply to solenoid (constant B field) l

11222 ULinlAiL ==(μ0 ) 22 r

ÎUse formula for B field: B = μ0ni N turns B 2 UlAL = 2μ0 U L ÎCalculate energy density: uVAl= = B V 2 B 1 2 uB = B fielduEE = 2 ε0 E field 2μ0

ÎThis is generally true even if B is not constant

PHY2049: Chapter 30 49 Energy Calculation Examples

-5 ÎCalculate uB for earth field, B = 5 x 10 T 2 −5 B 2 ()510× u == 10−33 J/m B −7 2μ0 24×× 10 8 ÎCalculate uB for neutronπ star, B = 10 T 2 8 B 2 (10 ) u == 410J/m ×21 3 B −7 2μ0 24×× 10 11 ÎCalculate uB for magnetar,π B = 10 T 2 11 B 2 (10 ) u == 410J/m ×27 3 B −7 2μ0 24×× 10 uB 10π 3 Equivalent mass density ρ =× 410kg/m 2 c2 From E = mc 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) 1 only (2) 2 only Answer is modified from what was given in class. Answers 2 and 3 were accepted (3) 3 only (4) 1 & 2 (5) 2 & 3 R R R

V L V R L V R L

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 9 UB = 2.6 x 10 J = 2.6 gigajoules!!

μ B 224 UlA==π π ×3.152 12.5 B −7 ()() 2 0 24×× 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