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Unformatted text preview: Induction in Stationary Circuit
Switch closed (or opened)
Current induced in coil B (directions as shown) Steady state current in coil A
No current induced in coil B A B PHY2049: Chapter 30 33 Inductance
Inductance in a coil of wire defined by Can also be written From Faraday’s law E = N d di =L dt dt
B This is a more useful way to understand inductance Calculate emf generated in coil from rate of change of current Inductors play an important role in circuits when current is changing! PHY2049: Chapter 30 34 Self  Inductance
Consider a single isolated coil:
Current (red) starts to flow clockwise due to the battery But the buildup of current leads to changing flux in loop Induced emf (green) opposes the change This is a selfinduced emf (also called “back” emf) di =L E= N dt dt
B d 12V
L is the selfinductance units = “Henry (H)”
PHY2049: Chapter 30 induced emf 35 Inductance of Solenoid
Total flux (length l) di di E= N = μ0 n Al = L dt dt dt
B 2
To make large inductance: Lots of windings Big area Long d PHY2049: Chapter 30 36 LR Circuits
Inductance and resistor in series with battery of EMF V Start with no initial current in circuit
Close switch at t = 0 Current is initially 0 (initial increase causes voltage drop across inductor) Find i(t)
Resistor: V = Ri Inductor: V = L di/dt Apply loop rule V R L PHY2049: Chapter 30 37 Analysis of LR Circuit
Differential equation is General solution:
(Check and see!) K = V/R (necessary to make i = 0 at t = 0) Rise from 0 with time constant =L/R Final current (maximum) PHY2049: Chapter 30 38 Current vs Time in RL Circuit (Initially Zero Current in Inductor) i t = imax 1 e () ( tR / L ) t in units of L/R
PHY2049: Chapter 30 39 LR Circuits (2)
Switch off battery: Find i(t) if current starts at i0 di 0= L dt Ri di R + i=0 dt L
Exponential fall to 0 with time constant =L/R Initial current (maximum) PHY2049: Chapter 30 40 Current vs Time in RL Circuit (For Initial Current imax in Inductor) t in units of L/R
PHY2049: Chapter 30 41 Exponential Behavior
= L/R is the “characteristic time” of any RL circuit
Only t / is meaningful t=
Current falls to 1/e = 37% of maximum value Current rises to 63% of maximum value t =2
Current falls to 1/e2 = 13.5% of maximum value Current rises to 86.5% of maximum value t =3
Current falls to 1/e3 = 5% of maximum value Current rises to 95% of maximum value t =5
Current falls to 1/e5 = 0.7% of maximum value Current rises to 99.3% of maximum value
PHY2049: Chapter 30 42 ConcepTest: Generators and Motors
A current begins to flow in a wire loop placed in a magnetic field as shown. What does the loop do?
(a) moves to the right (b) moves up (c) rotates around horizontal axis (d) rotates around vertical axis (e) moves out of the page B This is how a motor works !!
PHY2049: Chapter 30 43 Electric Motors
Current is supplied from an external source of emf (battery or power supply) Forces act to rotate the wire loop A motor is essentially a generator operated in reverse! PHY2049: Chapter 30 44 Motor
Forces act to rotate the loop clockwise. When loop is vertical, current switches sign and the forces reverse, in order to keep the loop in rotation. This is why alternating current is normally used for motors. PHY2049: Chapter 30 45 Motors Generators Electrical mechanical energy Mechanical electrical energy
46 PHY2049: Chapter 30 Energy Stored in Magnetic Field
Just like electric fields, magnetic fields store energy
Electric field energy density Magnetic field energy density Let’s see how this works PHY2049: Chapter 30 47 Energy of an Inductor
How much energy is stored in an inductor when a current is flowing through it? Start with loop rule di E = iR + L dt
Multiply by i to get power equation Let PL = power stored in inductor Identify energy stored in inductor Similar to capacitor:
48 di Ei = iR + Li dt Power produced = dissipated + stored Energy in Magnetic Field (2)
Apply to solenoid (constant B field)
UL =
1 2 l
r N turns Li 2 = 1 2 ( μ0 n2 lA i 2 ) Use formula for B field: Calculate energy density:
B field E field This is true even if B is not constant!
PHY2049: Chapter 30 49 Energy Calculation Examples
Calculate uB for earth field, B = 5 x 105 T Calculate uB for neutron star, B = 108 T Calculate uB for magnetar, B = 1011 T Equivalent mass density From E = mc2
PHY2049: Chapter 30 50 Web Sites on Neutron Stars, Magnetars
Original magnetar discovery
http://science.nasa.gov/newhome/headlines/ast20may98_1.htm http://www.firstscience.com/site/articles/solarflares.asp More recent magnetar discovery (Feb. 2005)
http://www.physorg.com/news3112.html Online articles on magnetars
http://solomon.as.utexas.edu/~duncan/magnetar.html http://www.space.com/scienceastronomy/ magnetar_formation_050201.html http://apod.gsfc.nasa.gov/apod/ap041126.html Articles 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
Each 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 This is a little tricky. After a long time, the current is steady and thus there is no emf across inductor. Final inductor current is V/R for #2 and #3. R R R V L R V R L V R L PHY2049: Chapter 30 52 Gigajoule Magnet at CERN
CMS experiment at CERN
pp 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
Large 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!! B2 42 uB = = 2 μ0 2 4 10 7 = 6.4 106 J/m 3 B2 42 UB = lA = 2 μ0 24 10 7 ( 3.152 12.5 = 2.6 109 J )( ) http://www.spacedaily.com/news/energytech04b.html
PHY2049: Chapter 30 55 Articles on CMS and LHC
Large Hadron Collider at CERN
http://public.web.cern.ch/public/en/LHC/LHCen.html Home page and picture of CMS experiment
http://cms.cern.ch/ Wikipedia articles
http://en.wikipedia.org/wiki/Compact_Muon_Solenoid http://en.wikipedia.org/wiki/Large_Hadron_Collider PHY2049: Chapter 30 56 ...
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 Spring '08
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 Physics, Current, Inductance

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