2 Magnetic Poles Every magnet, regardless of its shape, has two poles Called north and south polesPoles exert forces on one anotherSimilar to the way electric charges exert forces on each otherLike poles repel each otherN-N or S-SUnlike poles attract each otherN-SThe force between two poles varies as the inverse square of the distance between themA single magnetic pole has never been isolated
3 Magnetic Fields Magnetic Field is Created by the Magnets A vector quantity, Symbolized by BDirection is given by the direction a north pole of a compass needle points in that locationMagnetic field lines can be used to show how the field lines, as traced out by a compass, would look
4 Sources of the Magnetic Field Chapter 32Sources of the Magnetic Field
5 Sources of Magnetic Field Real source of Magnetic Field –moving electric charges orelectric currentInside every magnet – electric currents
6 Sources of Magnetic Field Inside every magnet – electric currentsNSno magnetic field
7 Biot-Savart Law The magnetic field is dB at some point P The length element is dsThe wire is carrying a steady current of IThe magnitude of dB is proportional to the current and to the magnitude ds of the length element dsThe magnitude of dB is proportional to sin q, where q is the angle between the vectors ds andThe vector dB is perpendicular to both ds and to the unit vector directed from ds toward PThe magnitude of dB is inversely proportional to r2, where r is the distance from ds to P
8 Biot-Savart LawThe constant mo is called the permeability of free spacemo = 4p x 10-7 T. m / A
9 Biot-Savart Law: Total Magnetic Field dB is the field created by the current in the length segment dsTo find the total field, sum up the contributions from all the current elements IdsThe integral is over the entire current distribution
10 Magnetic Field compared to Electric Field DistanceThe magnitude of the magnetic field varies as the inverse square of the distance from the sourceThe electric field due to a point charge also varies as the inverse square of the distance from the chargeDirectionThe electric field created by a point charge is radial in directionThe magnetic field created by a current element is perpendicular to both the length element ds and the unit vectorSourceAn electric field is established by an isolated electric chargeThe current element that produces a magnetic field must be part of an extended current distribution
11 Magnetic Field of a Long Straight Conductor The thin, straight wire is carrying a constant currentIntegrating over all the current elements gives
12 Magnetic Field of a Long Straight Conductor If the conductor is an infinitely long, straight wire, q1 = 0 and q2 = pThe field becomes
13 Magnetic Field of a Long Straight Conductor The magnetic field lines are circles concentric with the wireThe field lines lie in planes perpendicular to to wireThe magnitude of B is constant on any circle of radius a
14 Magnetic Field for a Curved Wire Segment Find the field at point O due to the wire segmentI and R are constantsq will be in radians
15 Magnetic Field for a Curved Wire Segment Consider the previous result, with q = 2pThis is the field at the center of the loopROI
16 Example 1Determine the magnetic field at point A.A
17 Example 2Determine the magnetic field at point A.A
18 Example 3Two parallel conductors carry current in opposite directions. One conductor carries a current of 10.0 A. Point A is at the midpoint between the wires, and point C is a distance d/2 to the right of the 10.0-A current. If d = 18.0 cm and I is adjusted so that the magnetic field at C is zero, find (a) the value of the current I and (b) the value of the magnetic field at A.d/2
19 Example 3Two parallel conductors carry current in opposite directions. One conductor carries a current of 10.0 A. Point A is at the midpoint between the wires, and point C is a distance d/2 to the right of the 10.0-A current. If d = 18.0 cm and I is adjusted so that the magnetic field at C is zero, find (a) the value of the current I and (b) the value of the magnetic field at A.d/2
20 Example 4The loop carries a current I. Determine the magnetic field at point A in terms of I, R, and L.
21 Example 4The loop carries a current I. Determine the magnetic field at point A in terms of I, R, and L.
22 Example 4The loop carries a current I. Determine the magnetic field at point A in terms of I, R, and L.
23 Example 5A wire is bent into the shape shown in Fig.(a), and the magnetic field is measured at P1 when the current in the wire is I. The same wire is then formed into the shape shown in Fig. (b), and the magnetic field is measured at point P2 when the current is again I. If the total length of wire is the same in each case, what is the ratio of B1/B2?
24 Example 5A wire is bent into the shape shown in Fig.(a), and the magnetic field is measured at P1 when the current in the wire is I. The same wire is then formed into the shape shown in Fig. (b), and the magnetic field is measured at point P2 when the current is again I. If the total length of wire is the same in each case, what is the ratio of B1/B2?
26 Ampere’s lawAmpere’s law states that the line integral of Bds around any closed path equals where I is the total steady current passing through any surface bounded by the closed path.
27 Field due to a long Straight Wire Need to calculate the magnetic field at a distance r from the center of a wire carrying a steady current IThe current is uniformly distributed through the cross section of the wire
28 Field due to a long Straight Wire The magnitude of magnetic field depends only on distance r from the center of a wire.Outside of the wire, r > R
29 Field due to a long Straight Wire The magnitude of magnetic field depends only on distance r from the center of a wire.Inside the wire, we need I’, the current inside the amperian circle
30 Field due to a long Straight Wire The field is proportional to r inside the wireThe field varies as 1/r outside the wireBoth equations are equal at r = R
31 Magnetic Field of a Toroid Find the field at a point at distance r from the center of the toroidThe toroid has N turns of wire
32 Magnetic Field of a Solenoid A solenoid is a long wire wound in the form of a helixA reasonably uniform magnetic field can be produced in the space surrounded by the turns of the wire
33 The field lines in the interior are Magnetic Field of a SolenoidThe field lines in the interior areapproximately parallel to each otheruniformly distributedclose togetherThis indicates the field is strong and almost uniform
34 Magnetic Field of a Solenoid The field distribution is similar to that of a bar magnetAs the length of the solenoid increasesthe interior field becomes more uniformthe exterior field becomes weaker
35 Magnetic Field of a Solenoid An ideal solenoid is approached when:the turns are closely spacedthe length is much greater than the radius of the turnsConsider a rectangle with side ℓ parallel to the interior field and side w perpendicular to the fieldThe side of length ℓ inside the solenoid contributes to the fieldThis is path 1 in the diagram
36 Magnetic Field of a Solenoid Applying Ampere’s Law givesThe total current through the rectangular path equals the current through each turn multiplied by the number of turnsSolving Ampere’s law for the magnetic field isn = N / ℓ is the number of turns per unit lengthThis is valid only at points near the center of a very long solenoid
37 Interaction of Charged Particle with Magnetic Field Chapter 32Interaction of Charged Particle with Magnetic Field
38 Interaction of Charged Particle with Magnetic Field The properties can be summarized in a vector equation:FB = q v x BFB is the magnetic forceq is the chargev is the velocity of the moving chargeB is the magnetic field
39 Interaction of Charged Particle with Magnetic Field The magnitude of the magnetic force on a charged particle is FB = |q| vB sin qq is the smallest angle between v and BFB is zero when v and B are parallel or antiparallelq = 0 or 180oFB is a maximum when v and B are perpendicularq = 90o
40 Direction of Magnetic Force The fingers point in the direction of vB comes out of your palmCurl your fingers in the direction of BThe thumb points in the direction of v x B which is the direction of FB
41 Direction of Magnetic Force Thumb is in the direction of vFingers are in the direction of BPalm is in the direction of FBOn a positive particleYou can think of this as your hand pushing the particle
42 Direction of force Motion Differences Between Electric and Magnetic FieldsDirection of forceThe electric force acts along the direction of the electric fieldThe magnetic force acts perpendicular to the magnetic fieldMotionThe electric force acts on a charged particle regardless of whether the particle is movingThe magnetic force acts on a charged particle only when the particle is in motion
43 Work The electric force does work in displacing a charged particle Differences Between Electric and Magnetic FieldsWorkThe electric force does work in displacing a charged particleThe magnetic force associated with a steady magnetic field does no work when a particle is displacedThis is because the force is perpendicular to the displacement
44 Work in Magnetic FieldThe kinetic energy of a charged particle moving through a magnetic field cannot be altered by the magnetic field aloneWhen a charged particle moves with a velocity v through a magnetic field, the field can alter the direction of the velocity, but not the speed or the kinetic energy
45 Magnetic FieldThe SI unit of magnetic field is tesla (T)
46 Force on a WireThe magnetic force is exerted on each moving charge in the wireF = q vd x BThe total force is the product of the force on one charge and the number of chargesF = (q vd x B)nALIn terms of current:F = I L x BL is a vector that points in the direction of the currentIts magnitude is the length L of the segment
47 Force on a Wire Consider a small segment of the wire, ds The force exerted on this segment is F = I ds x BThe total force is
48 Force on a Wire: Uniform Magnetic Field B is a constantThen the total force isFor closed loop:The net magnetic force acting on any closed current loop in a uniform magnetic field is zero
49 Magnetic Force between two parallel conductors Two parallel wires each carry a steady currentThe field B2 due to the current in wire 2 exerts a force on wire 1 of F1 = I1ℓ B2Substituting the equation for B2 gives
50 Magnetic Force between two parallel conductors Parallel conductors carrying currents in the same direction attract each otherParallel conductors carrying current in opposite directions repel each otherThe result is often expressed as the magnetic force between the two wires, FBThis can also be given as the force per unit length: