Key Contents. Magnetic fields and the Lorentz force. Magnetic force on current. Ampere s law. The Hall effect

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1 Magnetic Fields

2 Key Contents Magnetic fields and the Lorentz force The Hall effect Magnetic force on current The magnetic dipole moment Biot-Savart law Ampere s law The magnetic dipole field

3 What is a Magnetic Field? As we have discussed, one major goal of physics is the study of how an electric field can produce an electric force on a charged object; A closely related goal is the study of how a megnetic field can produce a magnetic force on a moving charged particle or on a magnetic object such as a magnet; A magnetic field is the magnetic effect of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude (or strength); as such it is a vector field.

4 What Produces a Magnetic Field Because an electric field E is produced by an electric charge, we might reasonably expect that a a magnetic field B is produced by a magnetic charge; Although individual magnetic charges (called magnetic monopoles) are predicted by certain theories, their existence has not been confirmed. How then are magnetic fields produced? There are two ways

5 What Produces a Magnetic Field? A magnetic field is generated when electric charge carriers such as electrons move through space or within an electrical conductor. The geometric shapes of the magnetic flux lines produced by moving charge carriers (electric current) are similar to the shapes of the flux lines in an electrostatic field. But there are differences in the ways electrostatic and magnetic fields interact with the environment. In everyday life, magnetic fields are most often encountered as a force created by permanent magnets, which pull on ferromagnetic materials such as iron, cobalt, or nickel, and attract or repel other magnets. Magnetic fields are widely used throughout modern technology, particularly in electrical engineering and electromechanics. The Earth produces its own magnetic field, which is important in navigation, and it shields the Earth's atmosphere from solar wind. Rotating magnetic fields are used in both electric motors and generators. Magnetic forces give information about the charge carriers in a material through the Hall effect. The interaction of magnetic fields in electric devices such as transformers is studied in the discipline of magnetic circuits.

6 The definition of B How can we define a magnetic field if there is no such thing as a magnetic charge? We must define B in another way in terms of the magnetic force F B exerted on a moving electrically charged test particle. The magnetic force on a charged particle, F B, is found to be: Here q is the charge of the particle, v is its velocity, and B the magnetic field in the region. The magnitude of this force is then: Here f is the angle between vectors v and B. # The Lorentz force: F = q(e+ v B)

7 The definition of B The SI unit for B is newton per coulomb-meter per second. For convenience, this is called the tesla (T): An earlier (non-si) unit for B is the gauss (G), and

8 The Lorentz Force F q v B

9 Magnetic Field Lines

10 Example, Magnetic Force on a Moving Charged Particle

11 Crossed Fields, Discovery of an Electron When the two fields in Fig are adjusted so that the two deflecting forces acting on the charged particle cancel, we have Thus, the crossed fields allow us to measure the speed of the charged particles passing through them. The deflection of a charged particle, moving through an electric field E without B, between two plates, at the far end of the plates is (s= v 0 t at 2 ) Here, v is the particle s speed, m its mass, q its charge, and L is the length of the plates.

12 Crossed Fields, The Hall Effect Fig A strip of copper carrying a current i is immersed in a magnetic field. (a)the situation immediately after the magnetic field is turned on. The curved path that will then be taken by an electron is shown. (b) The situation at equilibrium, which quickly follows. Note that negative charges pile up on the right side of the strip, leaving uncompensated positive charges on the left. Thus, the left side is at a higher potential than the right side. (c) For the same current direction, if the charge carriers were positively charged, they would pile up on the right side, and the right side would be at the higher potential.

13 Crossed Fields, The Hall Effect A Hall potential difference V is associated with the electric field across strip width d, and the magnitude of that potential difference is V =Ed. When the electric and magnetic forces are in balance (Fig. 28-8b), where v d is the drift speed. But, Where J is the current density, A the cross-sectional area, e the electronic charge, and n the number of charges per unit volume. Therefore, n d V i A B e Here, l=( A/d), the thickness of the strip. One may measure the number density of the charge carriers and also determine the sign (polarity) of the charge!

14 A Circulating Charged Particle Consider a particle of charge magnitude q and mass m moving perpendicular to a uniform magnetic field B, at speed v. The magnetic force continuously deflects the particle, and since B and v are always perpendicular to each other, this deflection causes the particle to follow a circular path. The magnetic force acting on the particle has a magnitude of q vb. Fig Electrons circulating in a chamber containing gas at low pressure (their path is the glowing circle). A uniform magnetic field, B, pointing directly out of the plane of the page, fills the chamber. Note the radially directed magnetic force F B ; for circular motion to occur, F B must point toward the center of the circle, (Courtesy John Le P.Webb, Sussex University, England)

15 A Circulating Charged Particle For uniform circular motion Fig Electrons circulating in a chamber containing gas at low pressure (their path is the glowing circle). A uniform magnetic field, B, pointing directly out of the plane of the page, fills the chamber. Note the radially directed magnetic force F B ; for circular motion to occur, F B must point toward the center of the circle, (Courtesy John Le P.Webb, Sussex University, England)

16 Helical Paths The velocity vector, v, of such a particle resolved into two components, one parallel to and one perpendicular to it: The parallel component determines the pitch p of the helix (the distance between adjacent turns (Fig b)). The perpendicular component determines the radius of the helix. The more closely spaced field lines at the left and right sides indicate that the magnetic field is stronger there. When the field at an end is strong enough, the particle reflects from that end. If the particle reflects from both ends, it is said to be trapped in a magnetic bottle or magnetic mirror.

17 Example Helical Motion of a Charged Particle in a Magnetic Field

18 Example, Uniform Circular Motion of a Charged Particle in a Magnetic Field

19 Magnetic Force on a Current-Carrying Wire

20 Magnetic Force on a Current-Carrying Wire

21 Example, Magnetic Force on a Wire Carrying Current

22 Torque on a Current Loop The figure shows a simple motor consisting of a single curent-carrying loop immersed in a magnetic field. The two magnetic forces produce a torque on the loop tending to rotate it about its central axis.

23 Torque on a Current Loop For N loops, when A=ab, the area of the loop, the total torque is:

24 The Magnetic Dipole Moment, m Definition: Here, N is the number of turns in the coil, i is the current through the coil, and A is the area enclosed by each turn of the coil. Direction: The direction of m is that of the normal vector to the plane of the coil.

25 The Magnetic Dipole Moment, m The definition of torque can be rewritten as: Just as in the electric case, the magnetic dipole in an external magnetic field has an energy that depends on the dipole s orientation in the field: A magnetic dipole has its lowest energy (-mb cos 0=mB) when its dipole moment m is lined up with the magnetic field. It has its highest energy (-mb cos 180 =+mb) when m is directed opposite the field.

26 Calculating the Magnetic Field due to a Current Symbol m 0 is a constant, called the permeability constant, whose value is In vector form

27 Magnetic Field due to a Long Straight Wire The magnitude of the magnetic field at a perpendicular distance R from a long (infinite) straight wire carrying a current i is given by Fig Iron filings that have been sprinkled onto cardboard collect in concentric circles when current is sent through the central wire. The alignment, which is along magnetic field lines, is caused by the magnetic field produced by the current. (Courtesy Education Development Center)

28 Magnetic Field due to a Long Straight Wire

29 Magnetic Field due to a Current in a Circular Arc of Wire

30 Example, Magnetic field at the center of a circular arc of a circle

31 Example, Magnetic field off to the side of two long straight currents

32 Force Between Two Parallel Wires

33 Ampere s Law Curl your right hand around the Amperian loop, with the fingers pointing in the direction of integration. A current through the loop in the general direction of your outstretched thumb is assigned a plus sign, and a current generally in the opposite direction is assigned a minus sign.

34 Ampere s Law Magnetic Field Outside a Long Straight Wire Carrying Current

35 Ampere s Law Magnetic Field Inside a Long Straight Wire Carrying Current

36 Example Ampere s Law to find the magnetic field inside a long cylinder of current.

37 Solenoids and Toroids Fig A vertical cross section through the central axis of a stretched-out solenoid. The back portions of five turns are shown, as are the magnetic field lines due to a current through the solenoid. Each turn produces circular magnetic field lines near itself. Near the solenoid s axis, the field lines combine into a net magnetic field that is directed along the axis. The closely spaced field lines there indicate a strong magnetic field. Outside the solenoid the field lines are widely spaced; the field there is very weak.

38 Solenoids Fig Application of Ampere s law to a section of a long ideal solenoid carrying a current i. The Amperian loop is the rectangle abcda. Here n be the number of turns per unit length of the solenoid

39 Magnetic Field of a Toroid where i is the current in the toroid windings (and is positive for those windings enclosed by the Amperian loop) and N is the total number of turns. This gives

40 Example, The field inside a solenoid

41 A Current Carrying Coil as a Magnetic Dipole

42 A Current Carrying Coil as a Magnetic Dipole

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