Aim of the lecture Introduction to Magnets Magnetic field, B

1. Lecture 7 • Aim of the lecture • Introduction to • Magnets • Magnetic field, B • Interaction between current and B field • Force • Right Hand Rule •   Main learning outcomes • familiarity with • Magnets • Magnetic Field • Lorentz Force

2. Magnets: • Everyone (should be) familiar with magnets • Every permanent magnet has • Two ‘poles’ • Called North and South poles • Two North poles or two South poles repel each other • Opposite poles attract each other

3. These filed lines can be visualised using ‘iron filings’, as in the picture This origin of this force can be represented with field lines, Just as for electric charges. The field is a magnetic field (as opposed to an electric field)

4. Magnets like this are called • Permanent Magnets • Come in many shape and sizes

5. The earth is a big dipolemagnet

6. And the sun

7. In fact without the earth’s magnetic field • Life on earth would not exist • The field protects us from most • cosmic rays (high energy charged particles from space) • why – see later

8. Some animals, such as this loggerhead turtle, caneven directly detect the earth’s field and use it to navigate

9. All these magnets are characterised by being • Dipoles • Always have a North and a South • Very similar to an electric dipole:

10. Magnetic Dipole Electric Dipole • Far away ( >3 times pole separation) from dipole • Electric and Magnetic Field lines have same pattern • Close to dipole • Field lines differ because • Magnetic lines are in a magnetic medium • If the electric dipole charges had a dielectric between them • The field lines could look identical for both

11. Important Points • There are no observed magnetic monopoles • Only magnetic dipoles • Magnetic dipoles always have a magnetic medium between them • Magnetic fields are analogous to Electric Fields • For a dipole they differ in shape close to magnet only because of the magnetic medium that makes up the magnet Interesting fact: There is no theoretical reason why monopoles should notexist. Particle Physicists look for them. It is thought they may just be very(very!) heavy and therefore difficult to make.

12. Classical atomic hydrogen model • Electron in orbit around proton • Electon is charged • Electron ‘moving’ • Therefore a current loop • Therefore a magnetic dipole moment If all the ‘small’ dipoles in an object were to point the same way It would make a macroscopic permanent magnet (a dipole) This is how we have permanent dipole magnets But no magnetic monopoles All the magnetic fields we observe are cause by moving charges

13. Note: this is NOT a good model of the reality of the origin of atomic magnetic dipole moments Quantum mechanics needs to be used It is the spin angular momentum which matters most Spin is an entirely quantum mechanical effect It is ‘sort of’ the ‘internal’ spin of the electron More is beyond the scope of this course James Clerk Maxwell If we had magnetic monopoles, then a monopole moving in a loop would produce a dipole ELECTRIC field. Electricity and Magnetism are both aspects of the same force, Electromagnetism – and it is described by Maxwells Equations

14. Permanent Magnets can be made only from a few materials • Iron (Fe) • Nickel (Ni) • Cobalt (Co) • gadolinium and dysprosium (at low temperature) Today’s permanent magnets are made of alloys. Alloy materials include • Aluminium-Nickel-Cobalt (Alnico) • Neodymium-Iron-Boron (Neodymium magnets or "super magnets", a member of the rare earth category)• Samarium-Cobalt (a member of the rare earth category)• Strontium-Iron (Ferrite or Ceramic)

15. The reason that these materials exhibit this property is • They have unpaired electron spins in the atom • Which spontaneously align in the bulk material • The spin is associated with a dipole magnetic moment (see later) • So all the ‘small’ dipoles align and add up to a macroscopic one This is real data! It shows measurementsof the spin of individualatoms in a material Atomic separation is just a few angstoms (10-10m) When spins align the‘mini’ dipoles add up to give a permanent magnet

16. Magnetic Field Magnetic Field is given the symbol B The unit of magnetic field strength is the Tesla, T In fact there is a complication. There are two types of magnetic field, The B-field and the H-field. We will not consider this in the course, the H-field is different only inside magnetic media In free space the B-field and the H-field only differ by a constant. This is also usually true inside a magnetic medium as well. Note that B is a vector quantity B

17. The Tesla is a big unit An MMR machine for medical use has a field ~2T The earth’s field is ~ 30mT – 60mT at the surface

18. The worlds strongest continuouslyoperable magnet is 45T Stronger is possible, but themagnets explode when operated, so the strong field is only available for a short time. Picture shows a 1000T magnet, Which provides 1ms of operation.

19. Some other examples for reference

20. More on units Because the SI unit, the Tesla, is so large, it is common for magnetic fields to be quoted in a different (non-SI) unit, the Gauss, G Nicola Tesla 104 G = 1 T Carl Feidrich Gauss

21. Magnetic Field and Current The origin of permanent magnetic field is The dipole moments of atoms These are related to ‘moving’ electric charges in the atom A current (moving charge) has a magnetic field associated with it B Electron flow I Conventional current flow

22. Fingers point in the direction ofthe B field. The Right-hand Thumb Rule applies to Conventional current (ie flow of positive charge)

23. Remember this is conventional current, not electron flow.

24. B = m0I 2pr m0 is called the permeability of free space it is a constant of nature • Imagine sitting on one of the moving charges. • Then you would see no magnetic field as from your frame of reference the charge would be stationary. • You would see an electric field. • So electric and magnetic fields are • VERY closely related. A magnetic field is actually just a relativistic transformation of an electric field - this is well beyond the scope of this course – but interesting!

25. Solenoids By coiling a wire into a solenoid A uniform B field can be created inside This is an example of how particular magnetic field geometriescan be created. A uniform field in this case.

26. n – number of turns of wire I – current in wire m0 permeability of free space k is the relative permeability, it is a property ofthe material, and is usually given the symbol mr

27. The ATLAS detectorat the LHC has a largesolenoid to enable itto measure chargedparticles

28. The ATLAS solenoidcreates a 1.5T uniform B field in a very large volume. The energy stored is huge. Magnetic fields store energy, just like electric fields – see later B

29. B These are coils of a ‘torroidal’ magnet in ATLAS - a different field configurationused to measure the muon particles.

30. Forces on charged particles in a magnetic field A charged particle moving in a B field experiences a force: The direction of the force is perpendicular to the motion The direction of the force is perpendicular to the B field

31. The right hand rule can be used to relate the directions

32. Electric and Magnetic forcescombined This force is called The Lorentz Force The force is given by: Prof. Lorentz B-field charge velocity The vector equation above uses a cross product, (NOT a dot product) to combine the two vectors.

33. Because the force is always perpendicular to the velocity • speed never changes • Particle will circle in uniform B A x (blue here) indicates thatthe B field is pointing INTO page A . would mean it was pointingout of the page This is the basis for keepingcharged particles circling in anaccelerator like the LHC.

34. This is the basis of an electric motor Force on a Current • A current is simply a ‘line’ of moving charge, so • A wire will have a force on it General case is: = ILB for simple case • Where • I is current, • L length of wire in the field • B is the (uniform) B-field

35. why Force on a charge = qvxB A current = quantity of charge crossing a point per unit time = Qv where Q is the linear charge density = qv / s where q is total charge, s is linear length so Is = qv Where we have defined I to be a vector current having adirection as well as a magnitude. Hence we can write F = qvxB as IsxB The force on a short length, dL can then be written as F = I(L)dLxB where I(L) is the magnitude of the current and dL points in the direction of the current flow

36. Arbitrary shaped Wires • For a current loop in a uniform B-field • The net force will be zero • Independent of its shape • Because the integral round a closed loop must be 0 • But the torque is not zero (otherwise motors would not work!)

37. Actual motor has • several coils • Several magnets • ‘Capacitors’ to stop interference Basic DC motor Slip rings and ‘brushes’organised to ensure thatforce is always in same(rotational) direction on coil

38. Maxwell's Equations We will explain theseby the end of the course

39. The ‘easiest’ of Maxwell’s equations is divB= 0 also written as . B= 0 If F = Ui + Vj + Wk where I,j,k are orthogonal unit vectors, then • What this says is that there is no source of magnetic field in the theory, • it is the same as saying that there are no magnetic monopoles • At any point in space the number of B-field lines ‘in’ = number ‘out’ Like conservation of charge, quantity of current into node = quantity out

40. If there is no electric charge present, then can also say . E= 0 but this is not the general case. Maxwell’s equations in free space (no electric charges present and in vacuum)

41. What this equation is saying is that the source of the electric field is the charge. • It relates the charge density r to E • The divergence of the electric field is equal to the local charge density • Integrating the equation shows that the charge contained inside closed surface is proportional to the total electric field penetrating the surface.

42. To finish thesecond half! We will returnto the coffeecup later.