1. 1 29-30 Magnetism - content Magnetic Force – Parallel conductors Magnetic Field Current elements and the general magnetic force and field law Lorentz Force Origin of magnetic force Application of magnetic field formula ”Amperes” circuital law Application of the circuital law Magnetic dipoles Magnetic Loadspeaker Recording and playback unit

2. 2 Ampere 1820-1825 measured interactions between currents in closed conductors 29-30 Magnetism is the interactions between charge in motion, i.e. currents.

3. 3 L  I2I2 I1I1 Starting point is the parallel currents Sign of current according to direction => anti-parallel currents repell parallel currents attract

4. 4 Magnetic field L in current direction

5. 5 dadadldl dd dI = J. da Magnified current element J Current element A current element is a vector defined as

6. 6 Magnetic field from a current element   We will show that contribution from a current element is Total field in point B is then For

7. 7 General magnetic force law Since q1q1 q2q2 r v2v2 v1v1 Law of Biot-Savart 1820

8. 8 Field Theory

9. 9 The Hall effect A current carrying conductor in a magnetic field  V = V 2 -V 1 = E H L = v d BL. L A Hall probe can be used to ”measure” the magnetic field.

10. 10 (Interaction between moving free charges) v v’ fmfm f m R fefe fefe e-e- e-e- Consider two electron beams: f m V V’ fefe fefe e-e- e-e- From this we conclude: R Use

11. 11 V V Observer at rest Observer in motion V Relative rest (Relative motion) Electromagnetism Electric Force Magnetic Force R

12. 12 (Origin of magnetic effect – interactions take time) v R=ct 0 R vt v R*=ct Assume Interaction speed c Invariance of interaction speed In motion, interaction occurs over a larger distance, R*, and the strength decreases. Coulombs law changes to which is electric plus magnetic force

13. 13 1. Field on axis from a circular current loop Calculations of the magnetic field

14. 14 2. Field from an ”infinite” current plane x   r y y Q K is current line density (A/m) Consider plane to consist of parallel threads of infinitesimal thickness From one thread

15. 15 3. The solenoid field A solenoid is an infinitely long coil. It is built up by parallel loops: On the axis Sum all contributions from the loops ( see example 30.4 in Benson) to get where N is number of turns and L is length of solenoid equivalent to two parallel planes

16. 16 ”Ampere’s” circuital law for the magnetic field dlr I I C If C is a circle with radius r I I C dl r For an arbitrary integration curve Current enclosed by curve C

17. 17 Verification of Amperes circuital law 1. Current carrying plate I = KL L B Integration path C 2. Solenoid since solenoid approximation means neglecting all field outside coil L

18. 18 Application of Circuit Law Coaxial cable with homogenous current over cross sectional area: a. Identify symmetry: cylindrical, i.e. circles around axis. 1. Current density b. Choose integration path as circles around axis I I Integration path where S is the surface bounded by C

19. 19 I I Integration path 2. Coaxial cable with homogenous current over cross sectional area: r

20. 20 I I Integration path 3. Current density Coaxial cable with homogenous current over cross sectional area: r

21. 21 I I Integration path 4. Coaxial cable with homogenous current over cross sectional area:

22. 22 Magnetic dipoles Compare a solenoid with a permanent bar magnet A current loop is the infinitesimal magnetic dipole. What is its dipole moment?

23. 23 Torque and energy for interacting magnetic dipole Torque is Magnetic dipole moment is defined so that and vectorially Energy Work to rotate from aligned to anti-aligned is So that magnetic energy is Equivialent with electric dipole formulas. (Minus sign is conventional, but not correct)

24. 24 Earth Magnetism

25. 25 Magnetism in Biology Magnetite found in animals Bacteria Pigeon bird Solomon fish