Chapter 22, Magnetism Magnets Poles of a magnet (north and south ) are the ends where objects are most strongly attracted. Like poles repel each other and unlike poles attract each other Magnetic poles cannot be isolated An unmagnetized piece of iron can be magnetized by stroking it with a magnet Magnetism can be induced Soft and Hard magnetic materials. If a magnet is broken in half, each half has two poles: Magnetic Fields A vector quantity, symbolized by B Direction is given by the direction a north pole of a compass needle points in that location A compass can be used to show the direction of the magnetic field lines Magnetic field lines cannot cross 1
Magnetic Field Lines Iron filings are used to show the pattern of the magnetic field lines Earth s Magnetic Field The Earth s geographic north pole corresponds to a magnetic south pole and the geographic south pole corresponds to a magnetic north pole The direction of the Earth s magnetic field reverses every few million years The Earth s magnetic field resembles that achieved by burying a huge bar magnet deep in the Earth s interior The most likely source of the Earth s magnetic field is believed to be electric currents in the liquid part of the core 2
Magnetism Applications Abound Moving Charge In Magnetic Field An electric field surrounds any stationary electric charge A magnetic field surrounds any magnetic material A magnetic field surrounds any moving electric charge When moving through a magnetic field, a charged particle experiences a magnetic force F This magnetic force can be used to define the magnitude of the magnetic field B qvb sin F qv sin 3
Units of Magnetic Field The SI unit of magnetic field is the Tesla (T) Wb m T 2 N C (m / s) Wb is a Weber The cgs unit is a Gauss (G) 1 T = 10 4 G Conventional laboratory magnets 25000 G or 2.5 T Superconducting magnets 300000 G or 30 T Earth s magnetic field 0.5 G or 5 x 10-5 T N A m Direction of Magnetic Force (Right Hand Rule) Experiments show that the direction of the magnetic force is always perpendicular to both v and B F max occurs when v is perpendicular to B F = 0 when v is parallel to B F qvb sin right-hand rule Lorentz Force F qe qv B 4
Motion of Charged Particles in a Constant B Field Because the magnetic force is always perpendicular to the direction of motion, the path of a particle is circular. Also, while an electric field can do work on a particle, a magnetic field cannot the particle s speed remains constant. For a particle of mass m and charge q, moving at a speed v in a magnetic field B, the radius of the circle it travels can be solved by equating the centripetal force with the magnetic force. One gets (v is the perpendicular velocity) Example 29. An electron and a proton move in circular orbits in a plane perpendicular to a uniform magnetic field B. Find the ratio of the radii of their circular orbits when the electron and the proton have (a) the same momentum and (b) the same kinetic energy. 5
Magnetic Force on a Current Carrying Conductor A force is exerted on a currentcarrying wire placed in a magnetic field The direction of the force is given by right hand rule #1 The total force is the sum of all the magnetic forces on all the individual charges producing the current F = B I l sin θ θ is the angle between B and I If the magnetic field is confined to a small region, what is l in the above examples? Loops of Current and Magnetic Torque In the current loop shown, the vertical sides experience forces that are equal in magnitude and opposite in direction. They do not operate at the same point, so they create a torque around the vertical axis of the loop. The total torque is the sum of the torques from each force: Or, since A = hw, 6
Torque on a Multi-turn Current Loops = N B I A sin Applies to any shape loop N is the number of turns in the coil A is the area of the loop A = l a Calculate the torque on a single loop, using lower wire as the origin: F 2 has no contribution (lever arm zero) Two side wires have no contribution as they cancel each other. F 1 = l I B, with a lever arm of a, contributes a torque of = l I B a sin= A I B sin, which is the total torque. Maximum torque is experienced when the surface of the loop is parallel to the magnetic field, i.e. when the magnetic flux is at a minimum! Electric Motor An electric motor consists of a rigid current-carrying loop that rotates when placed in a magnetic field The torque acting on the loop will tend to rotate the loop to smaller values of θ until the torque becomes 0 at θ = 0 If the loop turns past this point and the current remains in the same direction, the torque reverses and turns the loop in the opposite direction To provide continuous rotation in one direction, the current in the loop must periodically reverse In ac motors, this reversal naturally occurs In dc motors, a split-ring commutator and brushes are used Actual motors would contain many current loops and commutators 7
Magnetic Fields: Long Straight Wire A current-carrying wire produces a magnetic field Direction of the magnetic field is given by the Right Hand Rule The magnitude of the field at a distance r from a wire carrying a current of I is B I o 2r µ o = 4 x 10-7 T m / A µ o is called the permeability of free space (Supplementary) Origin of magnetic field: generation by moving charges o q B o o E v 2 4 r rˆ v Ampère s Law Ampère s Circuital Law B Δl = µ o I Sum over the closed path Sum all the products of B Δl around an arbitrary closed path Note: this path need not be contained in a plane. The current is the total current flowing (charge per unit time) across the enclosed surface, which need not be flat. Ampère s Law can be used easily to find B for a long straight wire B I o 2r 8
Magnetic Force Between Two Parallel Conductors I1 oi 2 F1 I1B2 2 d The force on wire 1 is due to the current in wire 1 and the magnetic field produced by wire 2 The force per unit length is: o I1 I 2 d F 2 Parallel conductors carrying currents in the same direction attract each other Parallel conductors carrying currents in the opposite directions repel each other Isn t there an electric force between the the two wires because of the moving charges? No! Defining Ampere and Coulomb The force between parallel conductors can be used to define the Ampere (A) If two long, parallel wires 1 m apart carry the same current, and the magnitude of the magnetic force per unit length is 2 x 10-7 N/m, then the current is defined to be 1 A The SI unit of charge, the Coulomb (C), can be defined in terms of the Ampere If a conductor carries a steady current of 1 A, then the quantity of charge that flows through any cross section in 1 second is 1 C 9
Examples 56. Find the direction and magnitude of the force that each wire experiences in the figure, using vector addition. Magnetic Field of a Current Loop The strength of a magnetic field produced by a wire can be enhanced by forming the wire into a loop All the segments, Δx, contribute to the field, increasing its strength B center o q Qv o oi rˆ v 2 4 r 2r 2 r 2r 10
Magnetic Field of a Current Loop solenoid Magnetic Field in a Solenoid The magnitude of the field inside a solenoid is constant at all points far from its ends B = µ o n I n is the number of turns per unit length n = N / l This result can be obtained by applying Ampère s Law to the solenoid LB LB out N N I B o on I L o I 11
Magnetic Effects of Electrons Orbits & Spin An individual atom should act like a magnet because of the motion of the electrons about the nucleus Each electron circles the atom once in about every 10-16 seconds This would produce a current of 1.6 ma and a magnetic field of about 20 T at the center of the circular path However, the magnetic field produced by one electron in an atom is often canceled by an oppositely revolving electron in the same atom Electrons also have spin (like top) The field due to the spinning is generally stronger than the field due to the orbital motion Electrons usually pair up with their spins opposite each other, so their fields cancel each other Magnetic Effects of Electrons -- Domains In some materials, the spins do not naturally cancel Such materials are called ferromagnetic Large groups of atoms in which the spins are aligned are called domains When an external field is applied, the domains that are aligned with the field tend to grow at the expense of the others This causes the material to become magnetized In hard magnetic materials, the domains remain aligned after the external field is removed With a core in a loop, the magnetic field is enhanced since the domains in the core material align, increasing the magnetic field 12
Magnetic Field Near Gap Of Capacitor? Ampere s Law: B Δl = µ o I +µ o I d displacement current Hall Effect 13
Example 40. The force on the rectangular loop of wire in the magnetic field in the figure can be used to measure field strength. The field is uniform, and the plane of the loop is perpendicular to the field. (A) What is the direction of the magnetic force on the loop? (b) If a current of 5.00 A is used, what is the force per testla on the 20.0-cm-wide loop? 84. One long straight wire is to be held directly above another by repulsion between their currents. The lower wire carries 100 A and the wire 7.50 cm above it is 10-gauge (2.588 mm diameter) copper wire. What current must flow in the upper wire, neglecting the Earth s field? (b) What is the smallest current if the Earth s 3.00x10-5 T field is parallel to the ground and is not neglected? (c) Is the supported wire in a stable or unstable equilibrium if displaced vertically? If displaced horizontally? Chapter 22 Summary All magnets have two poles, north and south. Magnetic fields can be visualized using magnetic field lines. These lines point away from north poles and toward south poles. The Earth produces its own magnetic field. A magnetic field exerts a force on an electric charge only if it is moving: A right-hand rule gives the direction of the magnetic force on a positive charge. If a charged particle is moving at an angle to a magnetic field, it moves in a helix. Electric currents create magnetic fields; the direction can be determined using a right-hand rule. Ampère s law: Magnetic field of a long, straight wire: Force between current-carrying wires: Magnetic field of a solenoid: 14