Magnetism

1. Magnetism It is a Strangely Attractive Topic

2. History • Term comes from the ancient Greek city of Magnesia, at which many natural magnets were found. We now refer to these natural magnets as lodestones (also spelled loadstone; lode means to lead or to attract) which contain magnetite, a natural magnetic material Fe3O4. • Chinese as early as 121 AD knew that an iron rod which had been brought near one of these natural magnets would acquire and retain the magnetic property…and that such a rod when suspended from a string would align itself in a north-south direction.

3. Finally, the Science • Not until 1819 was a connection between electrical and magnetic phenomena shown. Danish scientist Hans Christian Oersted observed that a compass needle in the vicinity of a electrical current carrying wire was deflected! • In 1831, Michael Faraday discovered that a momentary current existed in a circuit when the current in a nearby circuit was started or stopped • Shortly thereafter, he discovered that motion of a magnet toward or away from a circuit or vice versa could produce the same effect.

4. The Connection is Made SUMMARY: Oersted showed that magnetic effects could be produced by moving electrical charges; Faraday and Henry showed that electric currents could be produced by moving magnets So.... All magnetic phenomena result from forces between electric charges in motion.

5. For Every North, There is a South Every magnet has at least one north pole and one south pole.  By convention, we say that the magnetic field lines leave the North end of a magnet and enter the South end of a magnet.  If you take a bar magnet and break it into two pieces, each piece will again have a North pole and a South pole.  If you take one of those pieces and break it into two, each of the smaller pieces will have a North pole and a South pole.  No matter how small the pieces of the magnet become, each piece will have a North pole and a South pole.  S N S N S N

6. No Monopoles Allowed It has not been shown to be possible to end up with a single North pole or a single South pole, which is a monopole ("mono" means one or single, thus one pole).  S N

7. How we can recognize poles in a magnet? The ends of a magnet are where the magnetic effect is the strongest. These are called “poles.” Each magnet has 2 poles – 1 north, 1 south.

8. The Concept of “Fields” Michael Faraday realized that ... A magnet has a magnetic Field distributed through- -out the surrounding space

9. Magnetic Field Lines Magnetic field lines describe the structure of magnetic fields in three dimensions. They can be draw as: If at any point in the region near to magnet we place an ideal compass needle, then the needle will always point along the field line. Field lines closer where the magnetic force is strong, and far to each other where it is weak. For instance, in a compact bar magnet or "dipole," field lines spread out from one pole and converge towards the other, and of course, the magnetic force is strongest near the poles where they come together.

10. Properties: 1. Like poles repels 2. Opposite poles attract

11. 3. Magnets have magnetic fields 4. We will say that a moving charge sets up in the space around it a magnetic field 5. It is the magnetic field which exerts a force on any other charge moving through it 6. Magnetic fields are vector quantities….that is, they have a magnitude and a direction

12. Defining Magnetic Field magnitude and Direction Magnetic Field vectors written as B Magnitude of the B-vector is proportional to the force acting on the moving charge, magnitude of the moving charge, the magnitude of its velocity, and the angle between v and the B-field. Unit is the Tesla or the Gauss (1 T = 10,000 G).

13. Force on a moving Charge in a magnetic field Given by Right Hand Rule Put your fingers in the direction of motion of the charge, curl them in the direction of the magnetic field. Your thumb now points in the direction of the magnetic force acting on the charge (positive). if the charge is negative, the force direction is opposite that of your thumb. This force will bend the path of the moving charge appropriately.

14. The Lorentz Force in magnetic field

15. Work done in moving a test charge in a static magnetic field • For an element of the path dl, the work done: • F.dl • is  to the direction of motion (v). • so work done =0 • thus, a static magnetic field can not change the kinetic energy of a moving charge. • Magnetic forces may alter the direction in which a particle moves, but they cannot speed it up or slow it down.

16. Lorentz Force in electric and magnetic field • If a point charge is moving in a region where both electric and magnetic fields exist, then it experiences a total force given by • The Lorentz force equation is useful for determining the equation of motion for electrons in electromagnetic deflection systems such as CRTs.

17. Magnetic Flux • If magnetic lines of flux are parallel to given surface whose area is A,  will be zero. If magnetic lines of flux are perpendicular to given surface whose area is A,  will be maximum.

18. Gauss’ law in magnetostatics Compare to Gauss’ law for electric field No magnetic ‘charge’, so right-hand side = 0 for magnetic field. Basic magnetic element is the dipole Net magnetic flux through any closed surface is always zero:

19. Force on a current carrying wire The current in a wire is the charge per unit time passing a given point. negative charges move to the left count the same as positive ones to the right. This gives the clue that phenomena involving moving charges depend on the product of charge and velocity: if you change the sign of q and v, you get the same answer, so it doesn't really matter which you have.

20. Let a line charge  travelling down a wire at speed v constitutes a current choose a segment of length vt, So charge vt, passes point P in a time interval t So here we should consider current as a vector (specially for surface and volume currents):

21. The magnetic force on a segment of current-carrying wire is evidently as I and dl both point in the same direction, Magnitude of I is constant along a wire

22. The force is perpendicular to B and idl. The direction of B is determined by right-hand rule: The direction of B is determined by right-hand rule for linear currents: If the wire is grasped in the palm of the right hand such that thumb pointing in the direction of the current then the fingers encircling the wire point in the direction of the magnetic field.

23. Force on a current carrying wire.

24. Stationary charges produce: electric fields: electrostatics Steady current produce: magnetic fields:magnetostatics Steady currents flows in a wire it’s magnitude must be the same all along the line (otherwise charge would be piling up somewhere, then it would not be a steady current). In magnetostatics

25. The Magnetic Field of a Steady Current The magnetic field of a steady line current is given by the Biot-Savart law: permeability of free space Here the integration is along the current path, in the direction of the flow; dl' is an element of length along the wire, is the vector from the source to the point r So, the magnetic field “circulates” around the wire

26. Magnetic field inside a current carrying loop.

27. For surface currents the Biot-Savart law becomes For volume currents the Biot-Savart law becomes For a point charge can we define Biot-Savart law? No Why? A point charge does not constitute a steady Current.

28. P r R q x I dx Þ Þ Þ \ Magnetic Field of  straight current carrying wire • Calculate field at point P using Biot-Savart Law: Which way is B?

29. P r R q x I dx Þ \

30. I (b) B = (m0I)/(2R) (c) B = (m0I)/(2pR) (a) B = 0 r Idx R To calculate the magnetic field at the center, we must use the Biot-Savart Law: • Two nice things makes our calculation very easy: • Idx is always perpendicular to r • r is a constant (=R) What is the magnitude of the magnetic field at the center of a loop of radius R, carrying current I?

31. The Curl of B Let’s assume an infinite straight wire through which current is coming out of the page. The integral of B around a circular path of radius s, centered at the wire, is

32. Now suppose we have a bundle of straight wires. Each wire that passes through our loop contributes 0I, and those outside contribute nothing. The line integral will then be where Ienc stands for the total current enclosed by the integration path. If the flow of charge is represented by a volume current density J, the enclosed current is the integral taken over the surface bounded by the loop.

33. So, we get: Applying Stokes' theorem (line integral to surface integral): The equation for the curl of B is called Ampere's law (in differential form). This is the integral version of Ampere's law For currents with appropriate symmetry, Ampere's law in integral form can be used easily for calculating the magnetic field.

34. The Divergence of B The Biot-Savart law for a volume current This formula gives the magnetic field at a point r = (x, y, z) in terms of an integral over the current distribution J(x', y', z'). B is a function of (x,y,z), J is a function of (x', y', z'),

35. Here the integration is over the primed coordinates. The curl are to be taken with respect to the unprimed coordinates because B is a function of (x, y, z). Applying the divergence to above equation, , because J doesn't depend on the unprimed variables (x, y, z) and  Monopole can not exist. So