26. Magnetism: Force & Field

2 Topics The Magnetic Field and Force The Hall Effect Motion of Charged Particles Origin of the Magnetic Field Laws for Magnetism Magnetic Dipoles Magnetism

3 Introduction An electric field is a disturbance in space caused by electric charge. A magnetic field is a disturbance in space caused by moving electric charge. An electric field creates a force on electric charges. A magnetic field creates a force on moving electric charges.

4 Magnetic Field and Force It has been found that the magnetic force depends on the angle between the velocity of the electric charge and the magnetic field

5 The force on a moving charge can be written as where B represents the magnetic field Magnetic Field and Force

6 The SI unit of magnetic field is the tesla (T) = 1 N /(A.m). But often we use a smaller unit: the gauss (G)1 G = T Magnetic Field and Force

The Hall Effect

8 h Consider a magnetic field into the page and a current flowing from left to right. Free positive charges will be deflected upwards and free negative charges downwards.

9 The Hall Effect Hall Voltage Eventually, the induced electric force balances the magnetic force: h Hall coefficient t is the thickness

Motion of Charged Particles in a Magnetic Field

11 Motion of Charged Particles in a Magnetic Field The magnetic force on a point charge does no work. Why? The force merely changes the direction of motion of the point charge.

12 Newton’s 2 nd Law So radius of circle is Motion of Charged Particles in a Magnetic Field

13 Since, the cyclotron period is Its inverse is the cyclotron frequency Motion of Charged Particles in a Magnetic Field

14 The Van Allen Belts

15 Wikimedia Commons

Origin of the Magnetic Field

17 The Biot-Savart Law A point charge produces an electric field. When the charge moves it produces a magnetic field, B:  0 is the magnetic constant: As drawn, the field is into the page

18 The Biot-Savart Law When the expression for B is extended to a current element, IdL, we get the Biot-Savart law: The total field is found by integration:

19 Biot-Savart Law: Example P The magnetic field due to an infinitely long current can be computed from the Biot-Savart law: x

20 Biot-Savart Law: Example Note: if your right-hand thumb points in the direction of the current, your fingers will curl in the direction of the resulting magnetic field I

Laws of Magnetism

22 Magnetic Flux Just as we did for electric fields, we can define a flux for a magnetic field: But there is a profound difference between the two kinds of flux…

23 Gauss’s Law for Magnetism Isolated positive and negative electric charges exist. However, no one has ever found an isolated magnetic north or south pole, that is, no one has ever found a magnetic monopole Consequently, for any closed surface the magnetic flux into the surface is exactly equal to the flux out of the closed surface

24 Gauss’s Law for Magnetism This yields Gauss’s law for magnetism Unfortunately, however, because this law does not relate the magnetic field to its source it is not useful for computing magnetic fields. But there is a law that is…

25 Ampere’s Law I If one sums the dot product around a closed loop that encircles a steady current I then Ampere’s law holds: That law can be used to compute magnetic fields, given a problem of sufficient symmetry

26 Ampere’s Law: Example What’s the magnetic field a distance z above an infinite current sheet of current density  per unit length in the y direction? From symmetry, the magnetic field must point in the positive y direction above the sheet and in the negative y direction below the sheet. x y z

27 Ampere’s Law: Example Ampere’s law states that the line integral of the magnetic field along any closed loop is equal to  0 times the current it encircles: x y z Draw a rectangular loop of height 2a in z and length b in y, symmetrically placed about the current sheet.

28 Ampere’s Law: Example The only contribution to the integral is from the upper and lower segments of the loop. From symmetry the magnitude of the magnetic field is constant and the same on both segments. Therefore, the integral is just 2Bb. The encircled current is I = b. So, Ampere’s law gives 2Bb =  0  b and therefore B =  0  / 2 x y z

Magnetic Force on a Current

30 Force on each charge: Force on wire segment: Magnetic Force on a Current n = number of charges per unit volume

31 Magnetic Force on a Current Note the direction of the force on the wire For a current element IdL the force is

32 Magnetic Force Between Conductors Since the force on a current-carrying wire in a magnetic field is two parallel wires, with currents I 1 and I 2 exert a magnetic force on each other. The force on wire 2 is: d

Magnetic Dipoles

34 Magnetic Moment A current loop experiences no net force in a uniform magnetic field. But it does experience a Ftorque F B The force is F = IaB

35 Magnetic Moment Magnitude of torque where A = ab For a loop with N turns, the torque is

36 Magnetic Moment It is useful to define a new vector quantity called the magnetic dipole moment then we can write the torque as

37 Example: Tilting a Loop

38 Example: Tilting a Loop

39 Magnetic Moment The magnetic torque that causes the dipole to rotate does work and tends to decrease the potential energy of the magnetic dipole If we agree to set the potential energy to zero at 90 o then the potential energy is given by

Magnetization

41 Magnetization Atoms have magnetic dipole moments due to orbital motion of the electrons magnetic moment of the electron When the magnetic moments align we say that the material is magnetized.

42 Types of Materials Materials exhibit three types of magnetism: paramagnetic diamagnetic ferromagnetic

43 Paramagnetism Paramagnetic materials have permanent magnetic moments moments randomly oriented at normal temperatures adds a small additional field to applied magnetic field

44 Paramagnetism Small effect (changes B by only 0.01%) Example materials Oxygen, aluminum, tungsten, platinum

45 Diamagnetism Diamagnetic materials no permanent magnetic moments magnetic moments induced by applied magnetic field B applied field creates magnetic moments opposed to the field

46 Diamagnetism Common to all materials. Applied B field induces a magnetic field opposite the applied field, thereby weakening the overall magnetic field But the effect is very small: B m ≈ B app

47 Diamagnetism Example materials high temperature superconductors copper silver

48 Ferromagnetism Ferromagnetic materials have permanent magnetic moments align at normal temperatures when an external field is applied and strongly enhances applied magnetic field

49 Ferromagnetism Ferromagnetic materials (e.g. Fe, Ni, Co, alloys) have domains of randomly aligned magnetization (due to strong interaction of magnetic moments of neighboring atoms)

50 Ferromagnetism Applying a magnetic field causes domains aligned with the applied field to grow at the expense of others that shrink Saturation magnetization is reached when the aligned domains have replaced all others

51 Ferromagnetism In ferromagnets, some magnetization will remain after the applied field is reduced to zero, yielding permanent magnets Such materials exhibit hysteresis

52 Summary Magnetic Force Perpendicular to velocity and field Does no work Changes direction of motion of charged particle Motion of Point Charge Helical path about field

53 Summary Magnetic Dipole Moment A current loop experiences no net magnetic force in a uniform field But it does experience a torque

54 Summary The magnetism of materials is due to the magnetic dipole moments of atoms, which arise from: the orbital motion of electrons and the intrinsic magnetic moment of each electron

55 Summary Three classes of materials DiamagneticM = –const B ext, small effect (10 -4 ) Paramagnetic M = +const B ext small effect (10 -2 ) FerromagneticM ≠ const B ext large effect (1000)