DEPARTMENT OF PHYSICS AND ASTRONOMY PA113/Unit 3 Electricity and Magnetism Course PA113 – Unit 3

PA113/Unit 3 UNIT 3 – Introductory Lecture  The Magnetic Field –Chapter 28  Sources of the Magnetic Field –Chapter 29

PA113/Unit 3 Importance of Magnetic Fields  Practical Uses –Electric motors, Loud speakers, Navigation (Earth’s magnetic field)  In Experimental Physics –Mass spectrometers, Particle accelerators, Plasma confinement  In the Universe –Stars (e.g. the Sun), Interstellar space, Intergalactic structure, Jets

PA113/Unit 3 Importance of Magnetic Fields

PA113/Unit 3 Importance of Magnetic Fields  Units – SI Tesla (T) = (N C -1 )/(m s -1 ) or N A -1 m -1 – 1 Gauss (G) = T  Examples –Terrestrial B field ~ 4x10 -5 T –Solenoid ~ T –Permanent magnet ~ T –Atomic interactions ~ 10 T –Superconducting magnet ~ 10 2 T –White dwarfs ~ T –Neutron stars < 10 8 T

PA113/Unit 3 Ch28 – The Magnetic Field  28-1 Force exerted by a Magnetic Field  28-2 Motion of a point charge in a Magnetic Field  28-3 Torques on current loops and magnets  28-4 The Hall Effect

PA113/Unit 3 Vector Notation  The DOT product  The CROSS product

PA113/Unit The Force Exerted by a Magnetic Field  Key Concept – Magnetic fields apply a force to moving charges Current element

PA113/Unit The Force Exerted by a Magnetic Field

PA113/Unit 3 Representation of Magnetic Field  Like electric field, can be represented by field lines –Field direction indicated by direction of lines –Field strength indicated by density of lines  But, unlike electric field –Magnetic field lines perpendicular to force –No isolated magnetic poles, so no points in space where field lines begin or end

PA113/Unit Motion of a Point Charge in a Magnetic Field  Key Concept – Force is perpendicular to field direction and velocity  Therefore, magnetic fields do no work on particles  There is no change in magnitude of velocity, just direction

PA113/Unit 3 Motion of a Point Charge in a Magnetic Field

PA113/Unit Motion of a Point Charge in a Magnetic Field  Radius of circular orbit  Cyclotron period  Cyclotron frequency

PA113/Unit Torques on Current Loops and Magnets  Key concept – a current loop experiences no net force in a uniform B field but does experience a torque

PA113/Unit Torques on Current Loops and Magnets Magnetic dipole moment

PA113/Unit 3 Potential Energy of a Magnetic Dipole in a Magnetic Field  Potential energy  Work done….. Integrate Zero at θ = 90 o

PA113/Unit The Hall Effect V h = v d Bw

PA113/Unit 3 Ch29 – Sources of the Magnetic Field  29-1 The Magnetic Field of moving point charges  29-2 The Magnetic Field of Currents –Biot-Savart Law  29-3 Gauss’ Law for Magnetism  29-4 Ampère’s Law  29-5 Magnetism in matter

PA113/Unit The Magnetic Field of Moving Point Charges  Point charge q moving with velocity v produces a field B at point P μ o = permeability of free space μ o = 4  x T·m·A -1

PA113/Unit The Magnetic Field of Currents: The Biot-Savart Law  Key concept – current as a series of moving charges – replace qv by Idl Add each element to get total B field

PA113/Unit 3  Key concept – The net flux of magnetic field lines through a closed surface is zero (i.e. no magnetic monopoles) Magnetic flux 29-3 Gauss’ Law for Magnetism

PA113/Unit Gauss’ Law for Magnetism Electric dipoleMagnetic dipole (or current loop)

PA113/Unit Ampère’s Law  Key concept – like Gauss’ law for electric field, uses symmetry to calculate B field around a closed curve C N.B. This version assumes the currents are steady

PA113/Unit Magnetism in Matter  Magnetization, M =  m B app /  0   m is the magnetic susceptibility  Paramagnetic –M in same direction as B, dipoles weakly add to B field (small +ve  m )  Diamagnetic –M in opposite direction to B, dipoles weakly oppose B field (small -ve  m )  Ferromagnetic –Large +ve  m, dipoles strongly add to B-field. Can result in permanent magnetic field in material.

PA113/Unit 3 End of lecture 1 End of lecture 1

DEPARTMENT OF PHYSICS AND ASTRONOMY PA113/Unit 3 Electricity and Magnetism Course 113 – Unit 3

PA113/Unit 3 UNIT 3 – Problem solving Lecture  The Magnetic Field –Chapter 28  Sources of the Magnetic Field –Chapter 29

PA113/Unit 3 Problem Solving  Read the book!!!!!  Look at some examples  Try out some questions  Draw a diagram – include vector nature of the field (r and v or dl )

PA113/Unit 3 You must know how to…  Calculate force on a moving charge –Or current element  Understand the properties of a dipole –Torque and magnetic moment  Calculate the B field using 1.The Biot-Savart law 2.Ampère’s Law  Understand Gauss’ Law for Magnetism

PA113/Unit Example – the Biot-Savart Law applied to a current loop

PA113/Unit 3 Field due to a current loop

PA113/Unit 3 Field due to a current loop

PA113/Unit 3 Field due to a current loop 2πR

PA113/Unit 3 Magnetic field lines of 2 loops

PA113/Unit 3 Many loops – a solenoid

PA113/Unit 3 The B field in a very long solenoid Can use the Biot-Savart Law or Ampère’s Law Length L N turns n = N/L Radius R Current I di=nIdx Field in a very long solenoid: B =  0 nI

PA113/Unit 3 Field around and inside a wire Classic example of the use of Ampère’s Law

PA113/Unit 3 Direction of field around a wire

PA113/Unit 3 End of lecture 2

DEPARTMENT OF PHYSICS AND ASTRONOMY PA113/Unit 3 Electricity and Magnetism Course 113 – Unit 3

PA113/Unit 3 UNIT 3 – Follow-up Lecture  The Magnetic Field –Chapter 28  Sources of the Magnetic Field –Chapter 29

PA113/Unit 3 Ch28 – The Magnetic Field  28-1 Force exerted by a Magnetic Field  28-2 Motion of a point charge in a Magnetic Field  28-3 Torques on current loops and magnets  28-4 The Hall Effect

PA113/Unit The Force Exerted by a Magnetic Field  Key Concept – Magnetic fields apply a force to moving charges Current element

PA113/Unit Motion of a Point Charge in a Magnetic Field  Radius of circular orbit  Cyclotron period  Cyclotron frequency

PA113/Unit Torques on Current Loops and Magnets Magnetic dipole moment

PA113/Unit 3 Ch29 – Sources of the Magnetic Field  29-1 The Magnetic Field of moving point charges  29-2 The Magnetic Field of Currents –Biot-Savart Law  29-3 Gauss’ Law for Magnetism  29-4 Ampère’s Law  29-5 Magnetism in matter

PA113/Unit The Magnetic Field of Currents: The Biot-Savart Law  Key concept – current as a series of moving charges – replace qv by Idl Add each element to get total B field

PA113/Unit 3  Key concept – The net flux of magnetic field lines through a closed surface is zero (i.e. no magnetic monopoles) Magnetic flux 29-3 Gauss’ Law for Magnetism

PA113/Unit Ampère’s Law  Key concept – like Gauss’ law for electric field, uses symmetry to calculate B field around a closed curve C N.B. This version assumes the currents are steady

PA113/Unit 3 Example Example

PA113/Unit 3 Field of a tightly wound toroid If b-a < r then B varies little – principle of fusion reactors

PA113/Unit 3 Why use fusion? Chemical reaction C+0 2  CO 2 (e.g. Coal) goes at ~700 K and gives ~10 7 J kg -1 Fission, such as U n  Ba Kr n goes at ~10 3 K and gives ~10 12 J kg -1 Fusion, such as in the Sun, H 2 + H 3  He 4 + n goes at ~10 8 K and gives ~10 14 J kg -1

PA113/Unit 3 Conditions required

PA113/Unit 3 Typical Fusion Reaction Chains The SunThe laboratory

PA113/Unit 3 Tokamak Fusion Test Reactor Operated from 1982 – 1997 Max Temp = 510 million K; Max power = 10.7 MW

PA113/Unit 3 Reactor Results

PA113/Unit 3 End of lecture 3 End of lecture 3

PA113/Unit 3 Definition of the Ampère  Force between 2 straight parallel conducting wires