Compare to principle of superposition applied to determine the electric field inside and outside an infinite parallel plate capacitor ELEC 3105 BASIC EM AND POWER ENGINEERING Force and torque on Magnetic Dipole Magnetic dipole = product of current in loop with surface area of loop DEFINITION OF MAGNETIC DIPOLE I = current in loop A “S” = surface area of the loop 𝑛 = unit vector normal to loop surface - RHR Magnetic moment 𝑚 vector in the direction of 𝑛 𝑚 = 𝐼𝐴 = 𝐼"𝑆" Units of Am2 ẑ FORCE ON A MAGNETIC DIPOLE Consider a circular ring of current I placed at the end of a solenoid as shown in the figure. The current in the solenoid produces a magnetic field in which the current loop is placed into. By postulates 1 and 2 of magnetic fields, the current ring will be subjected to a magnetic force. out of page into page B I ẑ FORCE ON A MAGNETIC DIPOLE F Circular ring F out out F down F out F B F down I Cancel in pairs around the ring out F down F down Will add in same direction on ring giving a net force. ẑ FORCE ON A MAGNETIC DIPOLE F Circular ring F out out F 2r down F down F B F down I Will add in same direction on ring giving a net force. down Using postulate 1: Gives: We need for find Br dF I Bd Fdown 2rIBr B B z B r F down FORCE ON A MAGNETIC DIPOLE z Gaussian cylinder B z We will relate Br to z z Total magnetic flux through Gaussian cylindrical surface must be zero. As many magnetic field lines that enter the surface, leave the surface. No magnetic charges or monopoles. B 0 Another important property of B Recall B 0 everywhere No net magnetic flux through any closed surface. Closed surface S B da 0 S Using divergence theorem B dv vol 0 vol 0 11 3-D view FORCE ON A MAGNETIC DIPOLE z Flux through top: r B z z 2 z top z Flux through side: 2rzB r side Flux through bottom: r B z 2 z bottom side top bottom 0 3-D view FORCE ON A MAGNETIC DIPOLE side top z 0 bottom z 2rzB r B z z r B z 0 2 r 2 z z r B z z B z B 2 z z z r r B B 2 z z r We can now use this in our force on current ring expression 3-D view ẑ FORCE ON A MAGNETIC DIPOLE F Circular ring B F out out F down 2r I F down We have found Br r B F 2rIB B z down r 2 z r B B z F down r B 2rI 2 z B r z F down ẑ FORCE ON A MAGNETIC DIPOLE Circular ring F F out out 2r F I F down down F r B 2rI 2 z z down F down F down B B A I z z F down B r I z 2 z B m z B B z z B F m z Force pulls dipole into region z z of stronger magnetic field B r F down FORCE ON A MAGNETIC DIPOLE z In general Fx m Fy m Fz m Bx x B y y Bz z Solenoid with axis along x Solenoid with axis along y Solenoid with axis along z Form suggests some sort of dot product with the del operator 3-D view FORCE ON A MAGNETIC DIPOLE z In general F m B F m B F m B x x y y z z F m B 3-D view TORQUE ON A MAGNETIC DIPOLE m B Side view We will consider a dipole in a uniform magnetic field. We can use any shape we want for the dipole. Here we will select a square loop of wire. I out of page a I into page a I Wire loop Top view TORQUE ON A MAGNETIC DIPOLE a a 2 a a I Wire loop Top view TORQUE ON A MAGNETIC DIPOLE m F Side view Top view a B F a Pivot point I Pivot line Torque attempts to align dipole Wire loop moment m with B. TORQUE ON A MAGNETIC DIPOLE m F B a 2 Side view F r F Total torque 2F Pivot point a sin 2 F => Magnetic force on wire of length a Torque attempts to align dipole moment m with B. TORQUE ON A MAGNETIC DIPOLE m F a 2 Side view 2F F a sin 2 Pivot point F => Magnetic force on wire of length a F IBa Then Through postulate 1 for magnetic fields a 2 IB sin B TORQUE ON A MAGNETIC DIPOLE m F B a 2 Side view a 2 IB sin m B sin m B F Pivot point a a m a2 I I Wire loop ELEC 3105 BASIC EM AND POWER ENGINEERING Boundary conditions Inductance Magnetic energy Principle of virtual work 21 NORMAL COMPONENT OF B FIELD AT BOUNDARY B dA 0 S Gaussian surface B 2 Area A B normal 2 Interface Very thin T 0 2 1 B normal 1 B Area A 1 Net flux through a closed surface is zero. B2 normal B1 (normal ) The normal components of B are continuous across the interface 22 TANGENTIAL COMPONENT OF H FIELD AT BOUNDARY H H d I 0 B H 2 H tangential S 2 Square closed path Length L Interface Very thin 2 1 Length L T 0 H H tangential 1 1 Integral of H around closed path is equal to the current enclosed (I = 0) H 2 tangential H1 ( tangential) The tangential components of H are continuous across the interface THIS BOUNDARY CONDITION ASSUMES NO SURFACE CURRENT AT THE INTERFACE. 23 TANGENTIAL COMPONENT OF H FIELD AT BOUNDARY H d I H 2 Square closed path Length L X X Very thin X X X X H1 tangential HK2 (tangential) Surface Current X X X X X X X X X X 2 1 Length L T 0 H 2 H tangential S Interface B H H tangential 1 1 H 2 tangential H1 ( tangential) K The tangential components of H are discontinuous across the interface THIS BOUNDARY CONDITION ASSUMES A SURFACE CURRENT AT THE INTERFACE. 24 SUMMARY OF BOUNDARY CONDITIONS (GENERAL) 25 SUMMARY OF BOUNDARY CONDITIONS (CONDUCTORS) 26 SUMMARY OF BOUNDARY CONDITIONS (CONDUCTORS) 27 SELF INDUCTANCE Introduction A transformer is a device in which the current in one circuit induces an EMF in a second circuit through the changing magnetic field. B, H, and M relationship 17 To understand how current in one circuit induced EMF in another, we will first examine how a current in a circuit can induce an EMF in the same circuit. 28 SELF INDUCTANCE INDUCTANCE SELF Consider a single wire loop Enclosed surface S Current in loop produces a magnetic field B , giving a flux through the loop. B Bi Biot-Savard Law This expression is The Biot-Savard Law Consider a small segment of wire of overall length d Thus: i i The Biot-Savard law applied to the small segment gives an element of magnetic field dB at the point P. I d rˆ21 dBr1 o 4 r21 2 i dB P Magnetostatics Same result as postulate 2 for the magnetic field POSTULATE 2 FOR THE MAGNETIC FIELD A current element I d produces a magnetic field B which at a distance R is given by: I Rˆ dB o d 4 R 2 dB r 21 v I Units of {T,G,Wb/m2} 26 Lecture 15 slid 26 d 13 From Biot-Savard Law WRITE: Li 29 SELF INDUCTANCE Consider a single wire loop Enclosed surface S Current in loop produces a magnetic field B , giving a flux through the loop. B Li i i L is the self inductance of the loop d di v L dt dt v emf t 30 SELF INDUCTANCE Consider a single wire loop Enclosed surface S Current in loop produces a magnetic field B , giving a flux through the loop. B Li i It is difficult to compute L for a simple wire loop since the magnetic field produced by the loop is not constant across the surface of the loop. A possible solution is to find B at center of loop and then approximate: B S center i v v d di L dt dt 31 SELF INDUCTANCE A simple example for the calculation of a self inductance is the long solenoid. Magnetic field of a long solenoid Current out of page Axis of solenoid Magnetic field of a long solenoid Magnetic field of a long solenoid In the vicinity of the point P Bb 0 Current out of page N : number of turns enclosed by length L Current out of page P 1 3 2 4 11 Current into page 3 5 Axis of solenoid 41 B 0 P B 2 P Infinite coil of wire carrying a current I Evaluate B field here 5 4 resultant Current into page Expect B to lie along axis of the solenoid Implies that B field has no radial component. I.e. no component pointing towards42 or away from the solenoid axis. Magnetic field of a FINITE solenoid Magnetic field of a FINITE solenoid Current out of page NIr dI d L sin Axis of solenoid a sin r B dB P Current into page z dB L z a NIr 2 2r 3 finite solenoid start Cross-section cut through solenoid axis 2 La NI o 2 L 50 Magnetic field of a FINITE solenoid d 2 1 1 2 3 4 5 2 z r 3 d a NI o 2r L r d d L dB We can now sum (integrate) the expression for over the angular extent of the coil. I.e. sum over all the rings of the finite length solenoid. NI cos cos B NI 2 L B sin d 2 L NI cos cos zˆ B 2 L z a o z 2 z 28 2 •B is independent of distance from the axis of the long solenoid as we are inside the solenoid! •B is uniform inside the long solenoid. d dB Evaluate B field here o o finite coil of wire carrying a current I Radius of solenoid is a. dI a L infinite solenoid (36) sub in a o NI 1 2 o sin d z 1 dB o 1 z 34 2 35 32 SELF INDUCTANCE Current out of page Long solenoid of length B N turns of wire carrying current I AREA A NI B o B is uniform over the cross-section of the solenoid 33 SELF INDUCTANCE Long solenoid of length NI B o B AREA A Flux through one loop of area A 1 NIA o 34 SELF INDUCTANCE Long solenoid of length B NI B AREA A Flux through all N loops of solenoid o From LI N N Then L 1 N IA 2 o N A 2 o 35 SELF INDUCTANCE Long solenoid of length NI B o LI L N A 2 o AREA A Self inductance of a long solenoid of N turns with a current I in the windings. The solenoid has cross-sectional area A. 36 EXAMPLE: SELF INDUCTANCE 𝐼 Calculate the “self inductance” per unit length for a segment of a coax cable. Inner radius (a), outer radius (b). Example completed in class 37 Energy in Magnetic Field Consider a long solenoid in order to develop a general expression for the energy stored in a magnetic field. Magnetic field of a long solenoid Current out of page Axis of solenoid Magnetic field of a long solenoid Magnetic field of a long solenoid In the vicinity of the point P Bb 0 Current out of page N : number of turns enclosed by length L Current out of page P 1 3 2 4 11 Current into page 3 5 Axis of solenoid 41 B 0 P B 2 P Infinite coil of wire carrying a current I Evaluate B field here 5 4 resultant Current into page Expect B to lie along axis of the solenoid Implies that B field has no radial component. I.e. no component pointing towards42 or away from the solenoid axis. Magnetic field of a FINITE solenoid Magnetic field of a FINITE solenoid Current out of page NIr dI d L sin Axis of solenoid a sin r B dB P Current into page z dB L z a NIr 2 2r 3 finite solenoid start Cross-section cut through solenoid axis 2 La NI o 2 L 50 Magnetic field of a FINITE solenoid d 2 1 1 2 3 4 5 2 z r 3 d a NI o 2r L r d d L dB We can now sum (integrate) the expression for over the angular extent of the coil. I.e. sum over all the rings of the finite length solenoid. NI cos cos B NI 2 L B sin d 2 L NI cos cos zˆ B 2 L z a o z 2 z 28 2 •B is independent of distance from the axis of the long solenoid as we are inside the solenoid! •B is uniform inside the long solenoid. d dB Evaluate B field here o o finite coil of wire carrying a current I Radius of solenoid is a. dI a L infinite solenoid (36) sub in a o NI 1 2 o sin d z 1 dB o 1 z 34 2 35 38 Energy in Magnetic Field Current out of page May have core with constant permeability Long solenoid of length NI B AREA A N turns of wire carrying current I Find work done by current source in building up magnetic field: Power V I V d dI L dt dt 39 ENERGY IN MAGNETIC FIELD Energy in Magnetic d dI V L dt dt dW Power V I dt THEN I W LI dI Field d dW I dt dt THEN 0 THEN LI W 2 2 d dW I dt dt Energy stored 40 ENERGY IN MAGNETIC FIELD Energy in Magnetic LI W 2 W 2 Field L N A 2 Energy stored N AI 2 B 2 2 NI For core solenoid 1 N I W A 2 2 2 2 2 enclosed volume 1 W B A 2 2 41 ENERGY IN MAGNETIC FIELD Energy in Magnetic Field Total magnetic energy stored in solenoid W 1 B 2 A 2 W Energy density W VOLUME 2 W B VOLUME 2 1 B dv 2 2 vol EXPRESSION VALID FOR ALL 42 ENERGY IN MAGNETIC FIELD Energy in Magnetic Energy in Magnetic Field Energy in Magnetic Field Current out of page May have core with constant permeability Long solenoid of length NI B dW Power V I dt AREA A THEN dW L I W LI dI N turns of wire carrying current I d dI L dt dt d I dt dt THEN dW L THEN Find work done by current source in building up magnetic field: d dI V L Power V I dt dt Energy in Magnetic Field V 0 W LI 2 2 Field Total magnetic energy stored in solenoid Energy density 1 B A 2 2 W VOLUME 2 d I dt dt W W B VOLUME 2 Expression valid for all Energy stored Energy in Electric Field 43 Energy in Electric Field For electric fields, we argued that the energy was really stored in the potential energy of the charged particle’s positions, since it would require that much energy to take separate charges and form that distribution from a universe with equally distributed charges. Energy in Magnetic Field This is harder to do for magnetic fields since there are no magnetic charges. But one possible approach is to take current loops enclosing zero area, and consider the forces on the wires as we expand the loops so as to form the current distributions which generate the magnetic field. 44 PRINCIPLE OF VIRTUAL WORK (MAGNETIC) We can use the principle of virtual work to determine forces as we did for electric forces. Gives correct magnitude F mag U s mag Energy stored in magnetic field Position variable Forces in Electrostatics Conductor caries a surface charge of density Find force on plates of a parallel plate capacitor. Plate area A x L oE 2 U d 2 y F F E d s xLS yd U o E 2 xD y 2 L Force pulling metal insert into capacitor Be very careful using the principle of virtual work F o E 2 xD 2 45 PRINCIPLE OF VIRTUAL WORK (MAGNETIC) Magnetic Relays 46 PRINCIPLE OF VIRTUAL WORK (MAGNETIC) Use principle of virtual work to obtain expression for the magnetic force on the movable contact. I Magnetic Relays GAP Movable contact V Metal spring provides restoring force when current is zero Example completed in class 47 MUTUAL INDUCTANCE Enclosed surface S1 B Enclosed surface S2 2 1 i i 2 1 i v v Loop 1 Loop 2 1 2 We shall consider two current loops close together. 48 MUTUAL INDUCTANCE B Suppose current i1 flows in loop 1, creating a flux in the loop and a flux in loop 2. We will set the source current i2 zero for now. 1 12 1 S S 1 2 2 B da i 1 12 1 2 S2 v 1 Loop 1 Loop 2 Integral over loop 2 surface Magnetic field of loop 1 in the region of loop 2 Flux of loop 2 produced by current in loop 1 49 Now some math!!!! MUTUAL INDUCTANCE B da Using magnetic vector potential A da Using Stoke’s theorem A d Using definition of magnetic vector potential i d d 4 r d d Rearrange terms i 4 r 12 1 2 S2 1 12 2 S2 12 2 1 2 o 1 12 2 1 1 2 21 o 12 1 1 2 1 21 50 2 MUTUAL INDUCTANCE d d i 4 r o 12 1 1 2 1 i M 12 1 2 21 12 Constant that depends on loop geometry FLUX IN LOOP 2 DUE TO CURRENT IN LOOP 1 51 MUTUAL INDUCTANCE B Suppose current i2 flows in loop 2, creating a flux in the loop and a flux in loop 1. We will set the source current i1 zero for now. 2 21 1 S S 1 2 2 i 2 B da 21 2 1 S1 v 1 Loop 1 Loop 2 Integral over loop 1 surface Magnetic field of loop 2 in the region of loop 1 Flux of loop 1 produced by current in loop 2 Now some math!!!! MUTUAL INDUCTANCE B da Using magnetic vector potential A da Using Stoke’s theorem A d Using definition of magnetic vector potential i d d 4 r d d i Rearrange terms 4 r 21 2 1 S1 2 21 1 S1 21 1 2 1 o 2 21 1 2 2 1 12 o 21 2 2 1 2 1 12 53 MUTUAL INDUCTANCE d d i 4 r o 21 2 2 1 2 i M 21 2 1 12 21 Constant that depends on loop geometry FLUX IN LOOP 1 DUE TO CURRENT IN LOOP 2 54 MUTUAL INDUCTANCE d d i 4 r o 12 1 1 2 1 i M 12 1 d d i 4 r o 2 21 21 12 2 Conclusion M’s are geometrical factors 2 1 1 2 12 i M 21 2 21 M M M 12 21 MUTUAL INDUCTANCE BETWEEN LOOPS 55 MUTUAL INDUCTANCE General result Mutual Inductance Enclosed surface S1 B Enclosed surface S2 v 1 d d di di L M dt dt dt dt 1 21 1 2 1 2 1 i i v 2 1 2 i v v 1 d d di di M L dt dt dt dt 2 12 1 2 2 Sign convention 2 Loop 2 Loop 1 We shall consider two current loops close together. Indicates v2 positive when v1 is positive v 1 i i 1 2 primary v 2 56 ELEC 3105 BASIC EM AND POWER ENGINEERING START Extra 57 EXTRA ON INDUCTANCE 58 EXTRA ON INDUCTANCE 59 EXTRA ON INDUCTANCE 60 EXTRA ON INDUCTANCE 61 EXTRA ON INDUCTANCE 62 EXTRA ON INDUCTANCE 63 EXTRA ON INDUCTANCE 64 ELEC 3105 BASIC EM AND POWER ENGINEERING Particle accelerators 65 VAN DER GRAAFF GENERATOR This Van de Graaff generator of the first Hungarian linear particle accelerator achieved 700 kV in 1951 and 1000 kV in 1952. (Constructor: Simonyi Károly; Sopron, 1951) A Van de Graaff generator is an electrostatic generator which uses a moving belt to accumulate very high amounts of electrical potential on a hollow metal globe on the top of the stand. It was invented by American physicist Robert J. Van de Graaff in 1929. The potential difference achieved in modern Van de Graaff generators can reach 5 megavolts. A tabletop version can produce on the order of 100,000 volts and can store enough energy to produce a visible spark. A Van de Graaff generator operates by transferring electric charge from a moving belt to a terminal. The high voltages generated by the Van de Graaff generator can be used for accelerating subatomic particles to high speeds, making the generator a useful tool for PARTICLE ACCELERATOR LINIAC A linear particle accelerator (often shortened to linac) is a type of particle accelerator that greatly increases the velocity of charged subatomic particles or ions by subjecting the charged particles to a series of oscillating electric potentials along a linear beamline; this method of particle acceleration was invented by Leó Szilárd. It was patented in 1928 by Rolf Widerøe,[1] who also built the first operational device and was influenced by a publication of Gustav Ising.[2] Linacs have many applications: they generate X-rays and high energy electrons for medicinal purposes in radiation therapy, serve as particle injectors for higher-energy accelerators, and are used directly to achieve the highest kinetic energy for light particles (electrons and positrons) for particle physics. The design of a linac depends on the type of particle that is being accelerated: electrons, protons or ions. Linac range in size from a cathode ray tube (which is a type of linac) to the 3.2-kilometre-long (2.0 mi) linac at the SLAC National Accelerator Laboratory in Menlo PARTICLE ACCELERATOR CYCLOTRON The American physicist Ernest O. Lawrence won the 1939 Nobel Prize in physics for a breakthrough in accelerator design in the early 1930s. He developed the cyclotron, the first circular accelerator. A cyclotron is somewhat like a linac wrapped into a tight spiral. Instead of many tubes, the machine has only two hollow vacuum chambers, called dees, that are shaped like capital letter Ds back to back .A magnetic field, produced by a powerful electromagnet, keeps the particles moving in a circle. Each time the charged particles pass through the gap between the dees, they are accelerated. As the particles gain energy, they spiral out toward the edge of the accelerator until they gain enough energy to exit the SYNCHROCYCLOTRON AND ISOCHRONOUS CYCLOTRON A classic cyclotron can be modified to increase its energy limit. The historically first approach was the synchrocyclotron, which accelerates the particles in bunches. It uses a constant magnetic field B, but reduces the accelerating field's frequency so as to keep the particles in step as they spiral outward, matching their massdependent cyclotron resonance frequency. This approach suffers from low average beam intensity due to the bunching, and again from the need for a huge magnet of large radius and constant field over the larger orbit demanded by high energy. The second approach to the problem of accelerating relativistic particles is the isochronous cyclotron. In such a structure, the accelerating field's frequency (and the cyclotron resonance frequency) is kept constant for all energies by shaping the magnet poles so to increase magnetic field with radius. Thus, all particles get accelerated in isochronous time intervals. Higher energy particles travel a shorter distance in each orbit than they would in a classical cyclotron, thus remaining in phase with the accelerating field. The advantage of the isochronous cyclotron is that it can deliver continuous beams of higher average intensity, which is useful for some applications. The main disadvantages are the size and cost of the large magnet needed, and the difficulty in achieving the high magnetic field values required at the outer edge of the structure. Synchrocyclotrons have not been built since the isochronous cyclotron was developed. PARTICLE ACCELERATOR BETATRON A betatron is a cyclic particle accelerator developed by Donald Kerst at the University of Illinois in 1940 to accelerate electrons,[1][2][3] but the concepts ultimately originate from Rolf Widerøe,[4][5] whose development of an induction accelerator failed due to the lack of transverse focusing.[6] Previous development in Germany also occurred through Max Steenbeck in the 40s.[7] The betatron is essentially a transformer with a torus-shaped vacuum tube as its secondary coil. An alternating current in the primary coils accelerates electrons in the vacuum around a circular path. The betatron was the first important machine for producing high energy electrons. PARTICLE ACCELERATOR TRIUMPH TRIUMF is Canada's national laboratory for particle and nuclear physics. Its headquarters are located on the south campus of the University of British Columbia in Vancouver, British Columbia. TRIUMF houses the world's largest cyclotron,[1] a source of 500 MeV protons, which was named an IEEE Milestone in 2010.[2] TRIUMF's activities involve particle physics, nuclear physics, nuclear medicine, and materials science. There are over 450 scientists, engineers, and staff on the TRIUMF site, as well as 150 students and postdoctoral fellows. The lab attracts over 1000 national and international researchers every year TRIUMF has generated over $1B in economic impact activity over the last decade. TRIUMF scientists and university-based physicists develop and implement Natural Sciences and Engineering Research Council’s (NSERC) longrange plan for subatomic physics. TRIUMF uses these plans to develop its own priorities. TRIUMF supports only those projects that have been independently peer reviewed and endorsed by the international scientific community. TRIUMF has over 50 international agreements for collaborative scientific research. Asteroid 14959 TRIUMF is named in honour of the laboratory PARTICLE ACCELERATOR FUSOR A fusor is a device that uses an electric field to heat ions to conditions suitable for nuclear fusion. The machine has a voltage between two metal cages inside a vacuum. Positive ions fall down this voltage drop, building up speed. If they collide in the center, they can fuse. This is a type of Inertial electrostatic confinement device. A Farnsworth–Hirsch fusor is the most common type of fusor.[2] This design came from work by Philo T. Farnsworth in (1964) and Robert L. Hirsch in 1967.[3][4] A variant of fusor had been proposed previously by: William Elmore, James L. Tuck, and Ken Watson at the Los Alamos National Laboratory[5] though they never built the machine. Fusors have been built by various institutions. These include academic institutions such as the University of Wisconsin–Madison,[6] the Massachusetts Institute of Technology[7] and government entities, such as the Atomic Energy Organization of Iran and the Turkish Atomic Energy Authority.[8][9] Fusors have also been developed commercially, as sources for neutrons by DaimlerChrysler Aerospace[10] and as a method for generating medical isotopes.[11][12][13] Fusors have also become very popular for hobbyists and amateurs. A growing number of amateurs have performed nuclear fusion using simple fusor machines.