Chapter 19 Magnetism

Magnets Poles of a magnet are the ends where objects are most strongly attracted Poles of a magnet are the ends where objects are most strongly attracted Two poles, called north and south Two poles, called north and south Like poles repel each other and unlike poles attract each other Like poles repel each other and unlike poles attract each other Similar to electric charges Similar to electric charges Magnetic poles cannot be isolated Magnetic poles cannot be isolated If a permanent magnetic is cut in half repeatedly, you will still have a north and a south pole If a permanent magnetic is cut in half repeatedly, you will still have a north and a south pole This differs from electric charges This differs from electric charges There is some theoretical basis for monopoles, but none have been detected There is some theoretical basis for monopoles, but none have been detected

More About Magnetism An unmagnetized piece of iron can be magnetized by stroking it with a magnet An unmagnetized piece of iron can be magnetized by stroking it with a magnet Somewhat like stroking an object to charge an object Somewhat like stroking an object to charge an object Magnetism can be induced Magnetism can be induced If a piece of iron, for example, is placed near a strong permanent magnet, it will become magnetized If a piece of iron, for example, is placed near a strong permanent magnet, it will become magnetized

Types of Magnetic Materials Soft magnetic materials, such as iron, are easily magnetized Soft magnetic materials, such as iron, are easily magnetized They also tend to lose their magnetism easily They also tend to lose their magnetism easily Hard magnetic materials, such as cobalt and nickel, are difficult to magnetize Hard magnetic materials, such as cobalt and nickel, are difficult to magnetize They tend to retain their magnetism They tend to retain their magnetism

Magnetic Fields A vector quantity A vector quantity Symbolized by B Symbolized by B Direction is given by the direction a north pole of a compass needle points in that location Direction is given by the direction a north pole of a compass needle points in that location Magnetic field lines can be used to show how the field lines, as traced out by a compass, would look Magnetic field lines can be used to show how the field lines, as traced out by a compass, would look

Magnetic Field Lines, sketch A compass can be used to show the direction of the magnetic field lines (a) A sketch of the magnetic field lines (b)

Magnetic Field Lines, Bar Magnet Iron filings are used to show the pattern of the electric field lines Iron filings are used to show the pattern of the electric field lines The direction of the field is the direction a north pole would point The direction of the field is the direction a north pole would point

Magnetic Field Lines, Unlike Poles Iron filings are used to show the pattern of the electric field lines Iron filings are used to show the pattern of the electric field lines The direction of the field is the direction a north pole would point The direction of the field is the direction a north pole would point Compare to the electric field produced by an electric dipole Compare to the electric field produced by an electric dipole

Magnetic Field Lines, Like Poles Iron filings are used to show the pattern of the electric field lines Iron filings are used to show the pattern of the electric field lines The direction of the field is the direction a north pole would point The direction of the field is the direction a north pole would point Compare to the electric field produced by like charges Compare to the electric field produced by like charges

Magnetic and Electric Fields An electric field surrounds any stationary electric charge An electric field surrounds any stationary electric charge A magnetic field surrounds any moving electric charge A magnetic field surrounds any moving electric charge A magnetic field surrounds any magnetic material A magnetic field surrounds any magnetic material

Earth’s Magnetic Field The Earth’s geographic north pole corresponds to a magnetic south pole The Earth’s geographic north pole corresponds to a magnetic south pole The Earth’s geographic south pole corresponds to a magnetic north pole The Earth’s geographic south pole corresponds to a magnetic north pole Strictly speaking, a north pole should be a “north-seeking” pole and a south pole a “south-seeking” pole Strictly speaking, a north pole should be a “north-seeking” pole and a south pole a “south-seeking” pole

Earth’s Magnetic Field The Earth’s magnetic field resembles that achieved by burying a huge bar magnet deep in the Earth’s interior The Earth’s magnetic field resembles that achieved by burying a huge bar magnet deep in the Earth’s interior

Dip Angle of Earth’s Magnetic Field If a compass is free to rotate vertically as well as horizontally, it points to the earth’s surface If a compass is free to rotate vertically as well as horizontally, it points to the earth’s surface The angle between the horizontal and the direction of the magnetic field is called the dip angle The angle between the horizontal and the direction of the magnetic field is called the dip angle The farther north the device is moved, the farther from horizontal the compass needle would be The farther north the device is moved, the farther from horizontal the compass needle would be The compass needle would be horizontal at the equator and the dip angle would be 0° The compass needle would be horizontal at the equator and the dip angle would be 0° The compass needle would point straight down at the south magnetic pole and the dip angle would be 90° The compass needle would point straight down at the south magnetic pole and the dip angle would be 90°

More About the Earth’s Magnetic Poles The dip angle of 90° is found at a point just north of Hudson Bay in Canada The dip angle of 90° is found at a point just north of Hudson Bay in Canada This is considered to be the location of the south magnetic pole This is considered to be the location of the south magnetic pole The magnetic and geographic poles are not in the same exact location The magnetic and geographic poles are not in the same exact location The difference between true north, at the geographic north pole, and magnetic north is called the magnetic declination The difference between true north, at the geographic north pole, and magnetic north is called the magnetic declination The amount of declination varies by location on the earth’s surface The amount of declination varies by location on the earth’s surface

Earth’s Magnetic Declination

Source of the Earth’s Magnetic Field There cannot be large masses of permanently magnetized materials since the high temperatures of the core prevent materials from retaining permanent magnetization There cannot be large masses of permanently magnetized materials since the high temperatures of the core prevent materials from retaining permanent magnetization The most likely source of the Earth’s magnetic field is believed to be electric currents in the liquid part of the core The most likely source of the Earth’s magnetic field is believed to be electric currents in the liquid part of the core

Reversals of the Earth’s Magnetic Field The direction of the Earth’s magnetic field reverses every few million years The direction of the Earth’s magnetic field reverses every few million years Evidence of these reversals are found in basalts resulting from volcanic activity Evidence of these reversals are found in basalts resulting from volcanic activity The origin of the reversals is not understood The origin of the reversals is not understood

Magnetic Fields When moving through a magnetic field, a charged particle experiences a magnetic force When moving through a magnetic field, a charged particle experiences a magnetic force This force has a maximum value when the charge moves perpendicularly to the magnetic field lines This force has a maximum value when the charge moves perpendicularly to the magnetic field lines This force is zero when the charge moves along the field lines This force is zero when the charge moves along the field lines

Magnetic Fields, cont One can define a magnetic field in terms of the magnetic force exerted on a test charge One can define a magnetic field in terms of the magnetic force exerted on a test charge Similar to the way electric fields are defined Similar to the way electric fields are defined

Units of Magnetic Field The SI unit of magnetic field is the Tesla (T) The SI unit of magnetic field is the Tesla (T) Wb is a Weber Wb is a Weber The cgs unit is a Gauss (G) The cgs unit is a Gauss (G) 1 T = 10 4 G 1 T = 10 4 G

A Few Typical B Values Conventional laboratory magnets Conventional laboratory magnets G or 2.5 T G or 2.5 T Superconducting magnets Superconducting magnets G or 30 T G or 30 T Earth’s magnetic field Earth’s magnetic field 0.5 G or 5 x T 0.5 G or 5 x T

Finding the Direction of Magnetic Force Experiments show that the direction of the magnetic force is always perpendicular to both v and B 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 max occurs when v is perpendicular to B F = 0 when v is parallel to B F = 0 when v is parallel to B

Right Hand Rule #1 Hold your right hand open Hold your right hand open Place your fingers in the direction of B Place your fingers in the direction of B Place your thumb in the direction of v Place your thumb in the direction of v The direction of the force on a positive charge is directed out of your palm The direction of the force on a positive charge is directed out of your palm If the charge is negative, the force is opposite that determined by the right hand rule If the charge is negative, the force is opposite that determined by the right hand rule

QUICK QUIZ 19.1 A charged particle moves in a straight line through a certain region of space. The magnetic field in that region (a) has a magnitude of zero, (b) has a zero component perpendicular to the particle's velocity, or (c) has a zero component parallel to the particle's velocity.

QUICK QUIZ 19.1 ANSWER (b). The force that a magnetic field exerts on a charged particle moving through it is given by F = qvB sin θ = qvB, where B is the component of the field perpendicular to the particle’s velocity. Since the particle moves in a straight line, the magnetic force (and hence B, since qv ≠ 0) must be zero.

QUICK QUIZ 19.2 The north-pole end of a bar magnet is held near a stationary positively charged piece of plastic. Is the plastic (a) attracted, (b) repelled, or (c) unaffected by the magnet?

QUICK QUIZ 19.2 ANSWER (c). The magnetic force exerted by a magnetic field on a charge is proportional to the charge’s velocity relative to the field. If the charge is stationary, as in this situation, there is no magnetic force.

Magnetic Force on a Current Carrying Conductor A force is exerted on a current-carrying wire placed in a magnetic field A force is exerted on a current-carrying wire placed in a magnetic field The current is a collection of many charged particles in motion The current is a collection of many charged particles in motion The direction of the force is given by right hand rule #1 The direction of the force is given by right hand rule #1

Force on a Wire The blue x’s indicate the magnetic field is directed into the page The blue x’s indicate the magnetic field is directed into the page The x represents the tail of the arrow The x represents the tail of the arrow Blue dots would be used to represent the field directed out of the page Blue dots would be used to represent the field directed out of the page The represents the head of the arrow The represents the head of the arrow In this case, there is no current, so there is no force In this case, there is no current, so there is no force

Force on a Wire, cont B is into the page B is into the page Point your fingers into the page Point your fingers into the page The current is up the page The current is up the page Point your thumb up the page Point your thumb up the page The force is to the left The force is to the left Your palm should be pointing to the left Your palm should be pointing to the left

Force on a Wire, final B is into the page B is into the page Point your fingers into the page Point your fingers into the page The current is down the page The current is down the page Point your thumb down the page Point your thumb down the page The force is to the right The force is to the right Your palm should be pointing to the right Your palm should be pointing to the right

Force on a Wire, equation The magnetic force is exerted on each moving charge in the wire The magnetic force is exerted on each moving charge in the wire The total force is the sum of all the magnetic forces on all the individual charges producing the current 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 F = B I ℓ sin θ θ is the angle between B and I θ is the angle between B and I The direction is found by the right hand rule, pointing your thumb in the direction of I instead of v The direction is found by the right hand rule, pointing your thumb in the direction of I instead of v

Torque on a Current Loop Applies to any shape loop Applies to any shape loop N is the number of turns in the coil N is the number of turns in the coil

Electric Motor An electric motor converts electrical energy to mechanical energy An electric motor converts electrical energy to mechanical energy The mechanical energy is in the form of rotational kinetic energy The mechanical energy is in the form of rotational kinetic energy An electric motor consists of a rigid current-carrying loop that rotates when placed in a magnetic field An electric motor consists of a rigid current-carrying loop that rotates when placed in a magnetic field

Electric Motor, 2 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° 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 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

Electric Motor, 3 To provide continuous rotation in one direction, the current in the loop must periodically reverse To provide continuous rotation in one direction, the current in the loop must periodically reverse In ac motors, this reversal naturally occurs In ac motors, this reversal naturally occurs In dc motors, a split-ring commutator and brushes are used In dc motors, a split-ring commutator and brushes are used Actual motors would contain many current loops and commutators Actual motors would contain many current loops and commutators

Electric Motor, final Just as the loop becomes perpendicular to the magnetic field and the torque becomes 0, inertia carries the loop forward and the brushes cross the gaps in the ring, causing the current loop to reverse its direction Just as the loop becomes perpendicular to the magnetic field and the torque becomes 0, inertia carries the loop forward and the brushes cross the gaps in the ring, causing the current loop to reverse its direction This provides more torque to continue the rotation This provides more torque to continue the rotation The process repeats itself The process repeats itself

Force on a Charged Particle in a Magnetic Field Consider a particle moving in an external magnetic field so that its velocity is perpendicular to the field Consider a particle moving in an external magnetic field so that its velocity is perpendicular to the field The force is always directed toward the center of the circular path The force is always directed toward the center of the circular path The magnetic force causes a centripetal acceleration, changing the direction of the velocity of the particle The magnetic force causes a centripetal acceleration, changing the direction of the velocity of the particle

Force on a Charged Particle Equating the magnetic and centripetal forces: Equating the magnetic and centripetal forces: Solving for r: Solving for r: r is proportional to the momentum of the particle and inversely proportional to the magnetic field r is proportional to the momentum of the particle and inversely proportional to the magnetic field

QUICK QUIZ 19.3 As a charged particle moves freely in a circular path in the presence of a constant magnetic field applied perpendicular to the particle's velocity, its kinetic energy (a) remains constant, (b) increases, or (c) decreases.

QUICK QUIZ 19.3 ANSWER (a). The magnetic force acting on the particle is always perpendicular to the velocity of the particle, and hence to the displacement the particle is undergoing. Under these conditions, the force does no work on the particle and the particle’s kinetic energy remains constant.

QUICK QUIZ 19.4 Two charged particles are projected into a region in which a magnetic field is perpendicular to their velocities. After they enter the magnetic field, you can conclude that (a) the charges are deflected in opposite directions, (b) the charges continue to move in a straight line, (c) the charges move in circular paths, or (d) the charges move in circular paths but in opposite directions.

QUICK QUIZ 19.4 ANSWER (c). Anytime the velocity of a charged particle is perpendicular to the magnetic field, it will follow a circular path. The two particles will move in opposite directions around their circular paths if their charges have opposite signs, but their charges are unknown so (d) is not an acceptable answer.

Magnetic Fields – Long Straight Wire A current-carrying wire produces a magnetic field A current-carrying wire produces a magnetic field The compass needle deflects in directions tangent to the circle The compass needle deflects in directions tangent to the circle The compass needle points in the direction of the magnetic field produced by the current The compass needle points in the direction of the magnetic field produced by the current

Direction of the Field of a Long Straight Wire Right Hand Rule #2 Right Hand Rule #2 Grasp the wire in your right hand Grasp the wire in your right hand Point your thumb in the direction of the current Point your thumb in the direction of the current Your fingers will curl in the direction of the field Your fingers will curl in the direction of the field

Magnitude of the Field of a Long Straight Wire The magnitude of the field at a distance r from a wire carrying a current of I is The magnitude of the field at a distance r from a wire carrying a current of I is µ o = 4  x T m / A µ o = 4  x T m / A µ o is called the permeability of free space µ o is called the permeability of free space

Ampère’s Law André-Marie Ampère found a procedure for deriving the relationship between the current in a arbitrarily shaped wire and the magnetic field produced by the wire André-Marie Ampère found a procedure for deriving the relationship between the current in a arbitrarily shaped wire and the magnetic field produced by the wire Ampère’s Circuital Law Ampère’s Circuital Law  B || Δℓ = µ o I  B || Δℓ = µ o I Sum over the closed path Sum over the closed path

Ampère’s Law, cont Choose an arbitrary closed path around the current Choose an arbitrary closed path around the current Sum all the products of B || Δℓ around the closed path Sum all the products of B || Δℓ around the closed path

Ampère’s Law to Find B for a Long Straight Wire Use a closed circular path Use a closed circular path The circumference of the circle is 2  r The circumference of the circle is 2  r This is identical to the result previously obtained This is identical to the result previously obtained

Magnetic Force Between Two Parallel Conductors The force on wire 1 is due to the current in wire 1 and the magnetic field produced by wire 2 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: The force per unit length is:

Force Between Two Conductors, cont Parallel conductors carrying currents in the same direction attract each other Parallel conductors carrying currents in the same direction attract each other Parallel conductors carrying currents in the opposite directions repel each other Parallel conductors carrying currents in the opposite directions repel each other

Defining Ampere and Coulomb The force between parallel conductors can be used to define the Ampere (A) 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 N/m, then the current is defined to be 1 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 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 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 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

QUICK QUIZ 19.5 If I 1 = 2 A and I 2 = 6 A in the figure below, which of the following is true: (a) F 1 = 3F 2, (b) F 1 = F 2, or (c) F 1 = F 2 /3?

QUICK QUIZ 19.5 ANSWER (b). The two forces are an action- reaction pair. They act on different wires, and have equal magnitudes but opposite directions.

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 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 All the segments, Δx, contribute to the field, increasing its strength

Magnetic Field of a Current Loop – Total Field

Magnetic Field of a Solenoid If a long straight wire is bent into a coil of several closely spaced loops, the resulting device is called a solenoid If a long straight wire is bent into a coil of several closely spaced loops, the resulting device is called a solenoid It is also known as an electromagnet since it acts like a magnet only when it carries a current It is also known as an electromagnet since it acts like a magnet only when it carries a current

Magnetic Field of a Solenoid, 2 The field lines inside the solenoid are nearly parallel, uniformly spaced, and close together The field lines inside the solenoid are nearly parallel, uniformly spaced, and close together This indicates that the field inside the solenoid is nearly uniform and strong This indicates that the field inside the solenoid is nearly uniform and strong The exterior field is nonuniform, much weaker, and in the opposite direction to the field inside the solenoid The exterior field is nonuniform, much weaker, and in the opposite direction to the field inside the solenoid

Magnetic Field in a Solenoid, 3 The field lines of the solenoid resemble those of a bar magnet The field lines of the solenoid resemble those of a bar magnet

Magnetic Field in a Solenoid, Magnitude The magnitude of the field inside a solenoid is constant at all points far from its ends The magnitude of the field inside a solenoid is constant at all points far from its ends B = µ o n I B = µ o n I n is the number of turns per unit length n is the number of turns per unit length n = N / ℓ n = N / ℓ The same result can be obtained by applying Ampère’s Law to the solenoid The same result can be obtained by applying Ampère’s Law to the solenoid

Magnetic Field in a Solenoid from Ampère’s Law A cross-sectional view of a tightly wound solenoid A cross-sectional view of a tightly wound solenoid If the solenoid is long compared to its radius, we assume the field inside is uniform and outside is zero If the solenoid is long compared to its radius, we assume the field inside is uniform and outside is zero Apply Ampère’s Law to the red dashed rectangle Apply Ampère’s Law to the red dashed rectangle

Magnetic Effects of Electrons - - Orbits An individual atom should act like a magnet because of the motion of the electrons about the nucleus 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 seconds Each electron circles the atom once in about every 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 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 However, the magnetic field produced by one electron in an atom is often canceled by an oppositely revolving electron in the same atom

Magnetic Effects of Electrons – Orbits, cont The net result is that the magnetic effect produced by electrons orbiting the nucleus is either zero or very small for most materials The net result is that the magnetic effect produced by electrons orbiting the nucleus is either zero or very small for most materials

Magnetic Effects of Electrons - - Spins Electrons also have spin Electrons also have spin The classical model is to consider the electrons to spin like tops The classical model is to consider the electrons to spin like tops It is actually a relativistic and quantum effect It is actually a relativistic and quantum effect

Magnetic Effects of Electrons – Spins, cont The field due to the spinning is generally stronger than the field due to the orbital motion 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 Electrons usually pair up with their spins opposite each other, so their fields cancel each other That is why most materials are not naturally magnetic That is why most materials are not naturally magnetic

Magnetic Effects of Electrons - - Domains In some materials, the spins do not naturally cancel In some materials, the spins do not naturally cancel Such materials are called ferromagnetic Such materials are called ferromagnetic Large groups of atoms in which the spins are aligned are called domains 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 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 This causes the material to become magnetized

Domains, cont Random alignment, a, shows an unmagnetized material Random alignment, a, shows an unmagnetized material When an external field is applied, the domains aligned with B grow, b When an external field is applied, the domains aligned with B grow, b

Domains and Permanent Magnets In hard magnetic materials, the domains remain aligned after the external field is removed In hard magnetic materials, the domains remain aligned after the external field is removed The result is a permanent magnet The result is a permanent magnet In soft magnetic materials, once the external field is removed, thermal agitation cause the materials to quickly return to an unmagnetized state In soft magnetic materials, once the external field is removed, thermal agitation cause the materials to quickly return to an unmagnetized state With a core in a loop, the magnetic field is enhanced since the domains in the core material align, increasing the magnetic field With a core in a loop, the magnetic field is enhanced since the domains in the core material align, increasing the magnetic field

Chapter 20 Induced Voltages and Inductance

Induced emf A current can be produced by a changing magnetic field A current can be produced by a changing magnetic field First shown in an experiment by Michael Faraday First shown in an experiment by Michael Faraday A primary coil is connected to a battery A primary coil is connected to a battery A secondary coil is connected to an ammeter A secondary coil is connected to an ammeter

Faraday’s Experiment The purpose of the secondary circuit is to detect current that might be produced by the magnetic field The purpose of the secondary circuit is to detect current that might be produced by the magnetic field When the switch is closed, the ammeter deflects in one direction and then returns to zero When the switch is closed, the ammeter deflects in one direction and then returns to zero When the switch is opened, the ammeter deflects in the opposite direction and then returns to zero When the switch is opened, the ammeter deflects in the opposite direction and then returns to zero When there is a steady current in the primary circuit, the ammeter reads zero When there is a steady current in the primary circuit, the ammeter reads zero

Faraday’s Conclusions An electrical current is produced by a changing magnetic field An electrical current is produced by a changing magnetic field The secondary circuit acts as if a source of emf were connected to it for a short time The secondary circuit acts as if a source of emf were connected to it for a short time It is customary to say that an induced emf is produced in the secondary circuit by the changing magnetic field It is customary to say that an induced emf is produced in the secondary circuit by the changing magnetic field

Magnetic Flux The emf is actually induced by a change in the quantity called the magnetic flux rather than simply by a change in the magnetic field The emf is actually induced by a change in the quantity called the magnetic flux rather than simply by a change in the magnetic field Magnetic flux is defined in a manner similar to that of electrical flux Magnetic flux is defined in a manner similar to that of electrical flux Magnetic flux is proportional to both the strength of the magnetic field passing through the plane of a loop of wire and the area of the loop Magnetic flux is proportional to both the strength of the magnetic field passing through the plane of a loop of wire and the area of the loop

Magnetic Flux, 2 You are given a loop of wire You are given a loop of wire The wire is in a uniform magnetic field B The wire is in a uniform magnetic field B The loop has an area A The loop has an area A The flux is defined as The flux is defined as Φ B = B  A = B A cos θ Φ B = B  A = B A cos θ θ is the angle between B and the normal to the plane θ is the angle between B and the normal to the plane

Magnetic Flux, 3 When the field is perpendicular to the plane of the loop, as in a, θ = 0 and Φ B = Φ B, max = BA When the field is parallel to the plane of the loop, as in b, θ = 90° and Φ B = 0 The flux can be negative, for example if θ = 180° SI units of flux are T m² = Wb (Weber)

Magnetic Flux, final The flux can be visualized with respect to magnetic field lines The flux can be visualized with respect to magnetic field lines The value of the magnetic flux is proportional to the total number of lines passing through the loop The value of the magnetic flux is proportional to the total number of lines passing through the loop When the area is perpendicular to the lines, the maximum number of lines pass through the area and the flux is a maximum When the area is perpendicular to the lines, the maximum number of lines pass through the area and the flux is a maximum When the area is parallel to the lines, no lines pass through the area and the flux is 0 When the area is parallel to the lines, no lines pass through the area and the flux is 0

Electromagnetic Induction – An Experiment When a magnet moves toward a loop of wire, the ammeter shows the presence of a current (a) When a magnet moves toward a loop of wire, the ammeter shows the presence of a current (a) When the magnet is held stationary, there is no current (b) When the magnet is held stationary, there is no current (b) When the magnet moves away from the loop, the ammeter shows a current in the opposite direction (c) When the magnet moves away from the loop, the ammeter shows a current in the opposite direction (c) If the loop is moved instead of the magnet, a current is also detected If the loop is moved instead of the magnet, a current is also detected

Electromagnetic Induction – Results of the Experiment A current is set up in the circuit as long as there is relative motion between the magnet and the loop A current is set up in the circuit as long as there is relative motion between the magnet and the loop The same experimental results are found whether the loop moves or the magnet moves The same experimental results are found whether the loop moves or the magnet moves The current is called an induced current because is it produced by an induced emf The current is called an induced current because is it produced by an induced emf

Faraday’s Law and Electromagnetic Induction The instantaneous emf induced in a circuit equals the time rate of change of magnetic flux through the circuit The instantaneous emf induced in a circuit equals the time rate of change of magnetic flux through the circuit If a circuit contains N tightly wound loops and the flux changes by ΔΦ during a time interval Δt, the average emf induced is given by Faraday’s Law: If a circuit contains N tightly wound loops and the flux changes by ΔΦ during a time interval Δt, the average emf induced is given by Faraday’s Law:

Faraday’s Law and Lenz’ Law The change in the flux, ΔΦ, can be produced by a change in B, A or θ The change in the flux, ΔΦ, can be produced by a change in B, A or θ Since Φ B = B A cos θ Since Φ B = B A cos θ The negative sign in Faraday’s Law is included to indicate the polarity of the induced emf, which is found by Lenz’ Law The negative sign in Faraday’s Law is included to indicate the polarity of the induced emf, which is found by Lenz’ Law The polarity of the induced emf is such that it produces a current whose magnetic field opposes the change in magnetic flux through the loop The polarity of the induced emf is such that it produces a current whose magnetic field opposes the change in magnetic flux through the loop That is, the induced current tends to maintain the original flux through the circuit That is, the induced current tends to maintain the original flux through the circuit

Applications of Faraday’s Law – Ground Fault Interrupters The ground fault interrupter (GFI) is a safety device that protects against electrical shock The ground fault interrupter (GFI) is a safety device that protects against electrical shock Wire 1 leads from the wall outlet to the appliance Wire 1 leads from the wall outlet to the appliance Wire 2 leads from the appliance back to the wall outlet Wire 2 leads from the appliance back to the wall outlet The iron ring confines the magnetic field, which is generally 0 The iron ring confines the magnetic field, which is generally 0 If a leakage occurs, the field is no longer 0 and the induced voltage triggers a circuit breaker shutting off the current If a leakage occurs, the field is no longer 0 and the induced voltage triggers a circuit breaker shutting off the current

Applications of Faraday’s Law – Electric Guitar A vibrating string induces an emf in a coil A vibrating string induces an emf in a coil A permanent magnet inside the coil magnetizes a portion of the string nearest the coil A permanent magnet inside the coil magnetizes a portion of the string nearest the coil As the string vibrates at some frequency, its magnetized segment produces a changing flux through the pickup coil As the string vibrates at some frequency, its magnetized segment produces a changing flux through the pickup coil The changing flux produces an induced emf that is fed to an amplifier The changing flux produces an induced emf that is fed to an amplifier

Applications of Faraday’s Law – Apnea Monitor The coil of wire attached to the chest carries an alternating current The coil of wire attached to the chest carries an alternating current An induced emf produced by the varying field passes through a pick up coil An induced emf produced by the varying field passes through a pick up coil When breathing stops, the pattern of induced voltages stabilizes and external monitors sound an alert When breathing stops, the pattern of induced voltages stabilizes and external monitors sound an alert

Application of Faraday’s Law – Motional emf A straight conductor of length ℓ moves perpendicularly with constant velocity through a uniform field A straight conductor of length ℓ moves perpendicularly with constant velocity through a uniform field The electrons in the conductor experience a magnetic force The electrons in the conductor experience a magnetic force F = q v B F = q v B The electrons tend to move to the lower end of the conductor The electrons tend to move to the lower end of the conductor

QUICK QUIZ 20.1 The figure below is a graph of magnitude B versus time t for a magnetic field that passes through a fixed loop and is oriented perpendicular to the plane of the loop. Rank the magnitudes of the emf generated in the loop at the three instants indicated (a, b, c), from largest to smallest.

QUICK QUIZ 20.1 ANSWER (b), (c), (a). At each instant, the magnitude of the induced emf is proportional to the rate of change of the magnetic field (hence, proportional to the slope of the curve shown on the graph).

Lenz’ Law Revisited – Moving Bar Example As the bar moves to the right, the magnetic flux through the circuit increases with time because the area of the loop increases As the bar moves to the right, the magnetic flux through the circuit increases with time because the area of the loop increases The induced current must in a direction such that it opposes the change in the external magnetic flux The induced current must in a direction such that it opposes the change in the external magnetic flux

Lenz’ Law, Bar Example, cont The flux due to the external field in increasing into the page The flux due to the external field in increasing into the page The flux due to the induced current must be out of the page The flux due to the induced current must be out of the page Therefore the current must be counterclockwise when the bar moves to the right Therefore the current must be counterclockwise when the bar moves to the right

Lenz’ Law, Bar Example, final The bar is moving toward the left The bar is moving toward the left The magnetic flux through the loop is decreasing with time The magnetic flux through the loop is decreasing with time The induced current must be clockwise to to produce its own flux into the page The induced current must be clockwise to to produce its own flux into the page

Lenz’ Law Revisited, Conservation of Energy Assume the bar is moving to the right Assume the bar is moving to the right Assume the induced current is clockwise Assume the induced current is clockwise The magnetic force on the bar would be to the right The magnetic force on the bar would be to the right The force would cause an acceleration and the velocity would increase The force would cause an acceleration and the velocity would increase This would cause the flux to increase and the current to increase and the velocity to increase… This would cause the flux to increase and the current to increase and the velocity to increase… This would violate Conservation of Energy and so therefore, the current must be counterclockwise This would violate Conservation of Energy and so therefore, the current must be counterclockwise

Lenz’ Law, Moving Magnet Example A bar magnet is moved to the right toward a stationary loop of wire (a) As the magnet moves, the magnetic flux increases with time The induced current produces a flux to the left, so the current is in the direction shown (b)

Lenz’ Law, Final Note When applying Lenz’ Law, there are two magnetic fields to consider When applying Lenz’ Law, there are two magnetic fields to consider The external changing magnetic field that induces the current in the loop The external changing magnetic field that induces the current in the loop The magnetic field produced by the current in the loop The magnetic field produced by the current in the loop

Generators Alternating Current (AC) generator Alternating Current (AC) generator Converts mechanical energy to electrical energy Converts mechanical energy to electrical energy Consists of a wire loop rotated by some external means Consists of a wire loop rotated by some external means There are a variety of sources that can supply the energy to rotate the loop There are a variety of sources that can supply the energy to rotate the loop These may include falling water, heat by burning coal to produce steam These may include falling water, heat by burning coal to produce steam

AC Generators, cont Basic operation of the generator Basic operation of the generator As the loop rotates, the magnetic flux through it changes with time As the loop rotates, the magnetic flux through it changes with time This induces an emf and a current in the external circuit This induces an emf and a current in the external circuit The ends of the loop are connected to slip rings that rotate with the loop The ends of the loop are connected to slip rings that rotate with the loop Connections to the external circuit are made by stationary brushed in contact with the slip rings Connections to the external circuit are made by stationary brushed in contact with the slip rings

AC Generators, final The emf generated by the rotating loop can be found by The emf generated by the rotating loop can be found by ε =2 B ℓ v  =2 B ℓ sin θ If the loop rotates with a constant angular speed, ω, and N turns If the loop rotates with a constant angular speed, ω, and N turns ε = N B A ω sin ω t ε = ε max when loop is parallel to the field ε = ε max when loop is parallel to the field ε = 0 when when the loop is perpendicular to the field ε = 0 when when the loop is perpendicular to the field

DC Generators Components are essentially the same as that of an ac generator Components are essentially the same as that of an ac generator The major difference is the contacts to the rotating loop are made by a split ring, or commutator The major difference is the contacts to the rotating loop are made by a split ring, or commutator

DC Generators, cont The output voltage always has the same polarity The output voltage always has the same polarity The current is a pulsing current The current is a pulsing current To produce a steady current, many loops and commutators around the axis of rotation are used To produce a steady current, many loops and commutators around the axis of rotation are used The multiple outputs are superimposed and the output is almost free of fluctuations The multiple outputs are superimposed and the output is almost free of fluctuations

Motors Motors are devices that convert electrical energy into mechanical energy Motors are devices that convert electrical energy into mechanical energy A motor is a generator run in reverse A motor is a generator run in reverse A motor can perform useful mechanical work when a shaft connected to its rotating coil is attached to some external device A motor can perform useful mechanical work when a shaft connected to its rotating coil is attached to some external device

Motors and Back emf The phrase back emf is used for an emf that tends to reduce the applied current The phrase back emf is used for an emf that tends to reduce the applied current When a motor is turned on, there is no back emf initially When a motor is turned on, there is no back emf initially The current is very large because it is limited only by the resistance of the coil The current is very large because it is limited only by the resistance of the coil

Motors and Back emf, cont As the coil begins to rotate, the induced back emf opposes the applied voltage As the coil begins to rotate, the induced back emf opposes the applied voltage The current in the coil is reduced The current in the coil is reduced The power requirements for starting a motor and for running it under heavy loads are greater than those for running the motor under average loads The power requirements for starting a motor and for running it under heavy loads are greater than those for running the motor under average loads