Magnetic force and magnetic fields

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1 magnetar Magnetic force and magnetic fields Feb 28, 2012

2 Magnetic field Iron filings may be used to show the pattern of the magnetic field lines. A compass can be used to trace the field lines. The lines outside the magnet point from the North pole to the South pole.

Source of magnetic field Magnetic fields are

magnetic field is found to circulate around the

3 Source of magnetic field Magnetic fields are associated with charge in motion and with elementary charged particle spin. If charge moves in a conductor, for example, a magnetic field is found to circulate around the direction of the moving charge.details in Chapter 30 Magnetic fields superpose like electric fields.

The north-seeking pole of a compass points to the north geographic pole. This would correspond to the Earth s south magnetic pole.

4 The Earth s magnetic field The source of the Earth s magnetic field is likely convection induced electrical currents in the Earth s core. The direction of the Earth s magnetic field wanders and reverses. The north-seeking pole of a compass points to the north geographic pole. This would correspond to the Earth s south magnetic pole. The south-seeking pole points to the south geographic pole. This would correspond to the Earth s north magnetic pole. The field shields the Earth from charged particles emitted by the Sun.

5 Pole shifts File:Earth_Magnetic_Field_Declinat ion_from_1590_to_1990.gif During a pole reversal, the field is reduced by a factor of ten. The shielding of the Earth from solar radiation is reduced.

6 Ferromagnetism The magnetic field of a permanent magnet is associated with alignment of the spins of charged electrons in atoms. The electrons behave like a collection of tiny permanent magnets. When these tiny magnets are aligned, not randomly oriented, a net macroscopic magnetization and magnetic field is observed.

The force on a charge q with velocity v in a magnetic field B is given by the cross

7 Magnetic force The magnetic field B can be defined by the magnetic force on charged particles. The force on a charge q with velocity v in a magnetic field B is given by the cross product of velocity and field FB = q v x B The magnitude of the magnetic force on a charged particle is F B = q v B sin!. Unit T = Wb m -2 = N C -1 m -1 s.

8 Strength of magnetic fields 1 T = 10,000 gauss ~duncan/magnetar.html

9 Motion in a uniform magnetic field Consider a particle moving in an external magnetic field with its velocity v perpendicular to the field B. The force is always directed toward the center of the circular path and does no work so the KE and speed are constant. The magnetic force causes a centripetal acceleration, changing the direction of the velocity of the particle. Equate the magnetic and centripetal forces: F B = qvb=ma = mv 2 /r Solve for r=mv/(qb) = p/(qb)

10 Example Electrons are observed bending clockwise in a magnetic field perpendicular to the page with radius of curvature r= p/eb. From the radius and field, the momentum p may be determined. Given the momentum per unit charge created by a known electric field, the quantity q/m may be determined. Does B point into or out of the page?

11 Example An electron moves in a circle of radius r=1 mm in a magnetic field B= 1 T. What is its speed? Assume m =9e-31 kg and q=-e = -1.6e-19 C.

12 Example An electron moves in a circle of radius r=1 mm in a magnetic field B= 1 T. r What is its speed? Assume m =9e-31 kg and q=-e = -1.6e-19 C. (For v~c, there are relativistic corrections to our formulae.)

13 Example (cont) What constant electric field strength E would be required to accelerate an electron from rest over a distance d=1 m to a speed c/2 where c is light speed? (Again, for v~c, there are corrections to this result.)

-V Volts Example (cont) 0 Volts d E v For a constant electric field strength E required to accelerate an electron from rest over a distance d=1 m to a speed c/2 where c is light speed If the

14 -V Volts Example (cont) 0 Volts d E v For a constant electric field strength E required to accelerate an electron from rest over a distance d=1 m to a speed c/2 where c is light speed If the accelerating field is achieved with equal and opposite charges on two parallel plates separated by d=1 m, a) what is the voltage between the plates and b) what is the surface charge density on the plates?

Charge and mass of the electron Thomson s e/m experiment Electrons are accelerated from the cathode. They are deflected by electric and magnetic fields.

15 Charge and mass of the electron Thomson s e/m experiment Electrons are accelerated from the cathode. They are deflected by electric and magnetic fields.the beam of electrons strikes a fluorescent screen. e/m was measured. The electric field required to levitate singly charged oil drops was used by Millikan to determine e. Together, m is determined.

16 General motion in a uniform magnetic field The component of velocity along the field is constant so the general motion is a helix. The angular frequency qb/m is called the cyclotron frequency.

17 Combined E and B fields Motion in combined electric and magnetic fields is governed the total force F = qe +qvxb If E and B are uniform and at right angles as shown, the total force vanishes for v =E/B. Only particles with this speed pass straight through such a velocity selector.

Mass spectrometer A mass spectrometer separates ions according to their mass-to-charge ratio. In one design, a beam of ions passes through a velocity selector and enters a second magnetic field.

18 Mass spectrometer A mass spectrometer separates ions according to their mass-to-charge ratio. In one design, a beam of ions passes through a velocity selector and enters a second magnetic field. After entering the second magnetic field, the ions move in a semicircle of radius r before striking a detector at P. If the ions are positively charged, they deflect to the left. If the ions are negatively charged, they deflect to the right.

19 Sector mass spectrometer In a simple spectrometer, ions are created in a gas plasma and accelerated. The deflection by the magnetic field for fixed energy depends on q/m. Different isotopes (nuclear masses) are separated. More sophisticated high resolution spectrometers use high speed beams.

Cyclotron In a cyclotron, an alternating voltage

20 Cyclotron In a cyclotron, an alternating voltage is applied between two cup-shaped dees accelerating ions in the gap. The ions then travel in a circular orbit in a uniform constant magnetic field. The voltage alternates at the cyclotron frequency so that with each gap crossing, the energy is increased.

to achieve improved beam stability and higher

21 Sector focused cyclotron Advanced cyclotrons use multiple magnets and multiple acceleration gaps to achieve improved beam stability and higher energy (520 MeV at TRIUMPH). 520,000,000 Volts equivalent

22 Large hadron collider Two proton beams counter circulate and collide. During acceleration, protons are given repeated kicks at a fixed location while B-field increases to kept beams in the tunnel. r= 4.3 km, B(max)- 8 T, E = 7 TeV = 7,000,000,000,000 Volts equivalent

23 Magnetic force on a wire When an electric current flows in a conducting wire in a magnetic field, the magnetic force acts of the electrons. The electrons are bound electrically to the conductor so the force is transferred to the wire.

24 Force on a wire The total force is the product of the force on one charge and the number of charges. For electron drift speed vd = v d and density n, the total force on a length L of wire of cross sectional area A is F=nLA(-e)v d xb=ilnxb

25 Generalization For a vector element of length ds, df= I dsxb. The total force on the current carrying wire is obtained by integrating over elements of the wire.

26 Torque on a current loop There is a force on sides 2 & 4 since they are perpendicular to the field: F 2 = F 4 = I a B The direction of F2 is out of the page. The direction of F 4 is into the page. The forces are equal and in opposite directions, but not along the same line of action. The forces produce a torque around point O.

The net torque about point O will be # = IAB sin!

27 Torque on a current loop Assume the magnetic field makes an angle of "! < 90 o with a line perpendicular to the plane of the loop. The net torque about point O will be # = IAB sin!. The vector torque may be written in terms of a vector A perpendicular to the loop of magnitude equal to the area and direction given by the right hand rule.

28 Magnetic dipole moment The product m= IA is defined as the magnetic dipole moment of the loop, often called the simply the magnetic moment and denoted by Greek mu.. SI units: A m2 Torque N (or Greek tau ) in terms of magnetic moment: N = mxb Valid for a loop of any shape

29 Magnetic energy The magnetic moment of this loop is up. The torque is trying to align the moment with the field. The magnetic potential energy is U = -m. B

Hall effect When a current carrying conductor is placed in a magnetic field, a potential difference is generated in a direction perpendicular to both the current and the magnetic field.

30 Hall effect When a current carrying conductor is placed in a magnetic field, a potential difference is generated in a direction perpendicular to both the current and the magnetic field. This phenomena is known as the Hall effect. It arises from the deflection of charge carriers to one side of the conductor as a result of the magnetic forces they experience. The Hall effect gives information regarding the sign of the charge carriers and their density. It can also be used to measure magnetic fields.

31 Hall voltage $V H = E H d = v d B d d is the width of the conductor v d is the drift velocity If B and d are known, v d can be found. R H = 1 / nq is called the Hall coefficient. A properly calibrated conductor can be used to measure the magnitude of an unknown magnetic field.

32 The magnetic field around a current carrying wire Pick form of current Current comes out of the page

33 The magnetic field around a current carrying wire loop Current circulates in the center.

Chapter 29. Magnetic Fields

Chapter 29. Magnetic Fields Chapter 29 Magnetic Fields A Brief History of Magnetism 13 th century BC Chinese used a compass Uses a magnetic needle Probably an invention of Arabic or Indian origin 800 BC Greeks Discovered magnetite