Physics Electricity and Magnetism Lecture 09 - Charges & Currents in Magnetic Fields Y&F Chapter 27, Sec. 1-8

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1 Phsics Electricit and Magnetism Lecture 09 - Charges & Currents in Magnetic Fields Y&F Chapter 27, Sec. 1-8 What Produces Magnetic Field? Properties of Magnetic ersus Electric Fields Force on a Charge Moing through Magnetic Field Magnetic Field Lines A Charged Particle Circulating in a Magnetic Field Cclotron Frequenc The Cclotron, the Mass Spectrometer, the Earth s Field Crossed Electric and Magnetic Fields The e/m Ratio for Electrons Magnetic Force on a Current-Carring Wire Torque on a Current Loop: the Motor Effect The Magnetic Dipole Moment Summar

Magnetic Field E and g Electrostatic & graitational forces act through ector fields Now: Magnetic force (quite different): magnetic field Force law first: How are charges & currents affected b a gien

Then: How to create field (net lecture) Currents in loops of wire Intrinsic spins of e -,p + currents magnetic dipole moment Spins can align permanentl to form natural magnets Unification: Mawell s

2 Magnetic Field E and g Electrostatic & graitational forces act through ector fields Now: Magnetic force (quite different): magnetic field Force law first: How are charges & currents affected b a gien field? Then: How to create field (net lecture) Currents in loops of wire Intrinsic spins of e -,p + currents magnetic dipole moment Spins can align permanentl to form natural magnets Unification: Mawell s equations, electromagnetic waes Permanent Magnets : North- and South- seeking poles Natural magnets known since antiquit Earth s magnetic field, compass Ferromagnetic materials make a magnet when cooled in field (Fe, Ni, Co) - - Spins align into domains (magnets) Other materials (plastic, copper, wood, ) slightl or not affected (para- and dia- magnetism) REPELS ATTRACTS UNIFORM N S DIPOLES ALIGN IN FIELD

Electric field lines isualizing the direction and magnitude of E. Magnetic force acts at a distance through magnetic field.

3 Electric Field ersus Magnetic Field Electric force acts at a distance through electric field. Vector field, E. Source: electric charge. Positie charge (+) and negatie charge (-). Opposite charges attract Like charges repel. Electric field lines isualizing the direction and magnitude of E. Magnetic force acts at a distance through magnetic field. Vector field, Source: moing electric charge (current, een in substances such as permanent magnets). North pole (N) and south pole (S) Opposite poles attract Like poles repel. Magnetic field lines isualizing the direction and magnitude of.

4 Differences between magnetic & electrostatic field Test charge and electric field E F q Single electric poles eist Cut up a bar magnet E Test monopole and magnetic field? F p Magnetic poles are alwas found in pairs. A single magnetic pole has neer been found. small, complete magnets There is no magnetic monopole......dipoles are the basic units Electrostatic q Gauss Law E da enc S 0 Magnetic Gauss Law da 0 S Magnetic flu through each and eer Gaussian surface = 0 Magnetic field eerts force on moing charges (current) onl

5 Magnetic Force on a Charged Particle Define b the magnetic force F it eerts on a charged particle moing with a elocit F qsin( ) q F LORENTZ FORCE Units: 1 Tesla Newtons Coulomb m/ s 1 GAUSS = 10-4 Tesla See Lecture 01 for cross product definitions and eamples Tpical Magnitudes: Earth s field: 10-4 T. ar Magnet: 10-2 T. Electromagnet: 10-1 T. F is proportional to speed as well as charge q and field. F is geometricall comple depends on cross product o F = 0 if is parallel to. o F is otherwise normal to plane of both and. o F reerses sign for opposite sign of charge o Source is also q [current length]. Electric force can do work on a charged particle UT magnetic force cannot do work on moing particles since F. = 0.

Magnetic force geometr eamples: A simple geometr charge +q along z direction along + direction CONVENTION Head Tail z F m 0kˆ 0î î kˆ ĵ Fm q q 0 0 ĵ ĵ î kˆ More simple eamples 0 F charge

6 Magnetic force geometr eamples: A simple geometr charge +q along z direction along + direction CONVENTION Head Tail z F m 0kˆ 0î î kˆ ĵ Fm q q 0 0 ĵ ĵ î kˆ More simple eamples 0 F charge +q F m m is down F m = q 0 0 F m 0 charge -q into page F m is still down F m = q 0 0 F 0 charge +q F m is out of the page F m = q 0 0 charge +q or -q parallel to 0 F m = q 0 0 sin(0) F m 0

7 A numerical eample Electron beam moing in plane of sketch = 10 7 m/s along + = 10-3 T. out of page along + m F e force for +e a) Find the force on the electron : o Fm - e sin( 90 ) kˆ 15 F m kˆ Negatie sign means force is opposite to result of using the RH rule b) Acceleration of electron : 15 Fm N a kˆ m Kg e 10 Direction is the same as that of the force 15 kˆ 2 [m/s ]

8 When is magnetic force = zero? 9-1: A particle in a magnetic field is found to hae zero magnetic force on it. Which of the situations below is NOT possible? A. The particle is neutral.. The particle is stationar. C. The motion of the particle is parallel to the magnetic field. D. The motion of the particle is opposite to the magnetic field. E. All of them are possible. F q sin( )

9 Direction of Magnetic Force 9-2: The figures shows fie situations in which a positiel charged particle with elocit traels through a uniform magnetic field. For which situation is the direction of the magnetic force along the + ais? Hint: Use Right Hand Rule. A C z z z D E z z

Magnetic Units and Field Line Eamples SI unit of magnetic field:

lines interpret similarl to E The tangent to a magnetic field line

lines represents the magnitude of the magnetic field is stronger

At surface of neutron star Near big electromagnet Inside sunspot

10 Magnetic Units and Field Line Eamples SI unit of magnetic field: tesla (T) 1T = 1 N/[Cm/s] = 1 N/[Am] = 10 4 gauss Magnetic field lines interpret similarl to E The tangent to a magnetic field line at an point gies the direction of at that point; The spacing of the lines represents the magnitude of the magnetic field is stronger where the lines are closer together, and conersel. At surface of neutron star Near big electromagnet Inside sunspot Near small bar magnet At Earth s surface In interstellar space 10 8 T 1.5 T 10-1 T 10-2 T 10-4 T T

11 Charged Particles Circle at Constant Speed in a Uniform Magnetic Field: Cclotron Frequenc Uniform in z direction, in - plane tangent to path F is normal to both &... so Power = F. = 0. Magnetic force can not change particle s speed or KE F is a centripetal force, motion is UCM A charged particle moing in a plane perpendicular to a field circles in the plane with constant speed Set Radius of the path Period Cclotron angular frequenc m q r m r q F 2 c c 2r 2 c 2m q q m and ω do not depend on elocit. If not normal to particle spirals around causes circling is cons tant Fast particles moe in large circles and slow ones in small circles All particles with the same charge-to-mass ratio hae the same period. The rotation direction for a positie and negatie particles is opposite. para z F m F m F m CW rotation

Cclotron particle accelerator Earl nuclear phsics research, iomedical applications magnetic field gap Inject charged particles in center Charged Dees reerse polarit as particles cross gap to

12 Cclotron particle accelerator Earl nuclear phsics research, iomedical applications magnetic field gap Inject charged particles in center Charged Dees reerse polarit as particles cross gap to accelerate them. Particles spiral out in magnetic field as the gain KE and are detected c r m q Frequenc of polarit reersal needed does not depend on speed or radius of path - it can be constant! 2 c q m

Earth s field shields us from the Solar Wind and produces

wind particles (cclotron effect) protecting the Earth and

Some solar wind particles spiral around the Earth s

13 Earth s field shields us from the Solar Wind and produces the Aurora Earth s magnetic field deflects charged solar wind particles (cclotron effect) protecting the Earth and making life possible (magnetosphere). Some solar wind particles spiral around the Earth s magnetic field lines producing the Aurora at high latitudes. The can also be trapped.

or molecules are separated, forming a spectrum r 1 2mV q q m 2V 2 r 2 Mass spectrogram for the

14 Another cclotron effect deice: Mass spectrometer Separates particles with different charge/mass ratios r 2qV m m q ionized atoms or molecules accelerated through potential V ionized isotopes or molecules are separated, forming a spectrum r 1 2mV q q m 2V 2 r 2 Mass spectrogram for the surface of Titan. Relatie abundances are plotted as functions of the ratio of mass m to charge z.

15 Circulating Charged Particle 9-3: The figures show circular paths of two particles haing the same speed in a uniform magnetic field, which is directed into the page. One particle is a proton; the other is an electron (much less massie). Which figure is phsicall reasonable? A C r m q D E

16 Charged particle in both E and fields Add the electrostatic and magnetic forces F tot qe q ma also F tot Eample: Crossed E and fields E Equilibrium when... F tot Does F = ma change if ou obsere this while moing at constant elocit? 0 qe out of paper E up and normal to + charge normal to both E & q F e F m qe q (up) (dow n) E / CONVENTION OUT Independent of charge q Charges moe through fields un-deflected Use to select particles with a particular elocit IN F e F m FD of q E OPPOSED FORCES

17 Measuring e/m ratio for the electron (J. J. Thompson, 1897) electron gun E e L CRT screen First:. Appl E field onl in direction. = deflection of beam from center of screen (position for E = 0) F qe ma q - e measure notethat Net:. Find a b measuring and flight time t = L / (constant) t 1 2 at L / 2 e m a E Use crossed and E fields to measure Let point into the page, perpendicular to both E and V F M points along, opposite to FE (negatie charge) F M e F E ee at equilibrium E / Adjust until beam deflection = 0 (F E cancels F M ) to find and time t t L E a 2 2 t 2 E L e m E 2 2 L C/kg

18 Force due to crossed E and fields 9-4: The figure shows four directions for the elocit ector of a positiel charged particle moing through a uniform electric field E (into the page) and a uniform magnetic field (pointing to the right). The speed is E/. Which direction of elocit produces the greatest magnitude of the net force? A E D C

Force on a straight wire carring current in a field Free electrons (negatie): Drift elocit d is opposite to current (along wire) Lorentz force on an electron = - e d (normal to wire) Motor effect:

19 Force on a straight wire carring current in a field Free electrons (negatie): Drift elocit d is opposite to current (along wire) Lorentz force on an electron = - e d (normal to wire) Motor effect: wire is pushed or pulled b the charges d F m d df m nˆ nˆ d unit ector alongcurrent forcedueto charges in length d dq d CONVENTION OUT dq is the charge moing in wire whose length d = - d dt Recall: dq = - i dt, Note: d is in direction of current (+ charges) df m i ( d dt) i d Integrate along the whole length L of the wire (assume is constant ) L lengthof F m wire, parallelto current i L IN il uniform F m

20 Motor effect on a wire: which direction is the pull? F m i L i L replaces q F F F is in A current-carring loop eperiences A torque (but zero net force) F = 0 F = 0 i F is out

21 Eample: A 3.0 A. current flows along + in a wire 1.0 m long aligned parallel to the -ais. Find the magnitude and direction of the magnetic force on the wire, assuming that the magnetic field is uniform and gien b: 3.0î 5.0ĵ il î Fm i L î ( 3. 0î 5. 0ĵ) F î î m 0 [Tesla] [m] kˆ î ĵ kˆ [N] Onl the component of perpendicular to the current-length contributes to the force z F i L Eample: Same as aboe, but now the field has a z-component 3.0î 5.0ĵ 4.0kˆ il 3.0 F i L î ( 3. 0î 5. 0ĵ î [Tesla] [m] m 0 î î 0 î ĵ kˆ î kˆ F m kˆ 12. 0ĵ [N] ĵ kˆ) Whateer the direction of the field, there can neer be a force component along the current-length direction

22 Torque on a current-carring wire loop in a magnetic field z Upper sketch: -ais toward iewer Loop can rotate about -ais field is along z ais i L Appl F m to each side of the loop F 1 = F 3 = ia: Forces cancel again but net torque is not zero! Moment arms for F 1 & F 3 equal b.sin()/2 Force F 1 produces CW torque equal to 1 = i.a..b.sin()/2 Same for F 3 Torque ector is down into paper along rotation ais Net torque iabsin( ) z into paper Lower sketch: F 2 = F 4 = i.b.cos(): Forces cancel, same line of action zero torque

23 Torque on a current-carring loop: Motor Effect z () z torque iabsin( ) i A nˆ A area of loop ab Maimum torque: Sinusoidal ariation: Stable when nˆ ma ia nˆ ( ) ma sin( ) parallels Restoring torque oscillations Field PRODUCED b a current loop is a magnetic dipole field

24 ELECTRIC MOTOR: Reerse current I when torque changes sign ( = 0 ) Use mechanical commutator for AC or DC MOVING COIL GALVANOMETER asis of most 19 th & 20 th centur analog instruments: oltmeter, ohmmeter, ammeter, speedometer, gas gauge,. Coil spring calibrated to balance torque at proper mark on scale Multiple turns of wire increase torque N turns assumed in flat, planar coil Real motors use multiple coils (smooth torque) NiA nˆ or A NiA sin( ) area of loop ab

25 Current loops are basic magnetic dipoles Represent loop as a ector Magnetic Dimensions dipole moment 2 ampere - m N numberof turnsin the loop Magnetic dipole moment measures strength of response to eternal field strength of loop as source of a dipole field A sin( ) area of Newton-m Tesla loop Ni ab A nˆ Joule Tesla U M d sin( )d ELECTRIC DIPOLE MAGNETIC DIPOLE MOMENT p qd NiA SAMPLE VALUES [J / T] Small bar magnet ~ 5 J/T TORQUE POTENTIAL ENERGY e U e pe p E m U m Earth ~ J/T Proton (intrinsic) ~ J/T Electron (intrinsic) ~ J/T

26 Torque and potential energ of a dipole 9-5: In which configuration is the torque on the dipole at it s maimum alue? m 9-6: In which configuration is the potential energ of the dipole at it s smallest alue? U m A C D E

27 Summar: Lecture 9 Chapter 28 Magnetic Fields

Chapter 27: Magnetic Field and Magnetic Forces

Chapter 27: Magnetic Field and Magnetic Forces Iron ore found near Magnesia Compass needles align N-S: magnetic Poles North (South) Poles attracted to geographic North (South) Like Poles repel, Opposites