29-30 Magnetism - content

1. Magnetic Loadspeaker Recording and playback unit 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

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

3. I1 I2 L r Starting point is the parallel currents • Sign of current according to direction => • anti-parallel currents repell • parallel currents attract

4. Magnetic field L in current direction

5. Magnified current element dt da dl J dI = J. da Current element A current element is a vector defined as

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

7. General magnetic force law Law of Biot-Savart 1820 Since v2 v1 q2 r q1

8. Field Theory

9. The Hall effect A current carrying conductor in a magnetic field DV = V2-V1 = EHL = vdBL. A Hall probe can be used to ”measure” the magnetic field. L

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

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

12. (Origin of magnetic effect – interactions take time) • Assume • Interaction speed c • Invariance of interaction speed R*=ct vt v v R=ct0 R In motion, interaction occurs over a larger distance, R*, and the strength decreases. Coulombs law changes to which is electric plus magnetic force

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

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

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

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

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

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

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

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

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. 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 Equivialent with electric dipole formulas. (Minus sign is conventional, but not correct) So that magnetic energy is

24. Earth Magnetism

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