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PHY204H, Study Guide for Exam 3

by: Kelly Domogala

PHY204H, Study Guide for Exam 3 PHY 204H

Marketplace > University of Rhode Island > Physics 2 > PHY 204H > PHY204H Study Guide for Exam 3
Kelly Domogala
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Covers the material on our third exam: Chapter 26- The Magnetic Field, Chapter 27- Sources of the Magnetic Field and Chapter 28- Magnetic Induction.
Elementary Physics II
Leonard Kahn
Study Guide
PHY204, PHY204H, phy, Physics, physics 2, Magnetism, Study Guide
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This 16 page Study Guide was uploaded by Kelly Domogala on Sunday April 3, 2016. The Study Guide belongs to PHY 204H at University of Rhode Island taught by Leonard Kahn in Spring 2016. Since its upload, it has received 34 views. For similar materials see Elementary Physics II in Physics 2 at University of Rhode Island.


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Date Created: 04/03/16
Physics Study Guide Exam 3: Chapters 26, 27 and 28 Chapter 26: The Magnetic Field • Every magnet of any shape has two poles, called the north and the south pole, where the force exerted by the magnet is strongest. • The like poles of two magnets repel each other and the unlike poles of two magnets attract each other.▯ • The north pole of a compass needle points toward the south pole of a given magnet o What we call the north pole of Earth is actually a south magnetic pole • Magnetic poles always occur in pairs The Force Exerted by a Magnetic Field • When a particle that has charge q and velocity vis in a region with a magnetic field B, a force acts on the particle that is proportional to q, to v, to B, and to the sine of the angle between the directions of vand B. o The force is perpendicular to both the velocity and the magnetic field. • Magnetic Force on a Moving Charged Particle: ????   =  ????????  ????  ???? • Because F is perpendicular to both v and B, F is perpendicular to the plane defined by these two vectors. o The direction of v X Bis given by the right-hand rule as vis rotated into B • The SI unit of magnetic field is the tesla (T) o Like the farad, the tesla is a large unit. • A commonly used unit, de- rived from the cgs system, is the gauss (G), which is related to the tesla as follows: o 1 G = 10^-4 T • When a current-carrying wire is in a region that has a magnetic field, there is a force on the wire that is equal to the sum of the magnetic forces on the individual charge carriers in the wire. • Magnetic Force on a Straight Segment of Current-Carrying Wire: ????   =  ????????  ????  ????   o Lis a vector whose magnitude is the length of the segment and whose direction is the same as that of the current. o Assumed that the wire segment is straight and that the magnetic field does not vary over its length • Magnetic Force on a Current Element: ????????   =  ????????????  ????  ???? o Bis the magnetic field vector at the location of the segment. o The quantity I dl︎is called a current element. o We find the total magnetic force on a current- carrying wire by summing (integrating) the magnetic forces due to all the current elements in the wire. • The magnetic field can be represented by magnetic field lines. o The direction of the field is indicated by the direction of the field lines and the magnitude of the field is indicated by the density (number per unit area) of the lines on surface perpendicular to the lines. o Electric field lines are in the direction of the electric force on a positive charge, but the magnetic field lines are perpendicular to the magnetic force on a moving charge. ▯ o Electric field lines begin on positive charges and end on negative charges; magnetic field lines neither begin nor end. ▯ o Do not think the field lines for the magnetic field of a magnet begin on magnetic south poles and end on magnetic north poles. In reality, they neither begin nor end. Instead they enter the magnet at one end and exit the magnet at the other end. Motion of a Point Charge in a Magnetic Field • The magnetic force on a charged particle moving through a region with a magnetic field is always perpendicular to the velocity of the particle. • Magnetic forces do no work on particles and do not change their kinetic energy. • In the special case where the velocity of a charged particle is perpendicular to a uniform magnetic field, the particle moves in a circular orbit. The magnetic force provides the force in the centripetal direction that is necessary for circular motion. • The period of the particle’s circular orbit is called the cyclotron period ????????(????????) • Cyclotron Period: ???? = ???????? = ???????????? ???? ???????? ???? ???????? ???? • Cyclotron Frequency: ???? = ???? = ????????????      ???? = ???????????? ???????? • Suppose that a charged particle is in a region that has a uniform magnetic field and is moving with a velocity that is not perpendicular to B. There is no magnetic force component, and thus no acceleration component, parallel to B, so the component of the velocity that is parallel to Bremains constant. The magnetic force on the particle is perpendicular to B, so the change in motion of the particle due to this force is the same as that just discussed. The path of the particle is thus a helix. The Velocity Selector • The magnetic force on a charged particle moving in a uniform magnetic field can be balanced by an electric force if the magnitudes and directions of the magnetic field and the electric field are properly chosen. • Because the electric force is in the direction of the electric field (for particles with positive charge) and the magnetic force is perpendicular to the magnetic field, the electric and magnetic fields in the region through which the particle is moving must be perpendicular to each other if the forces are to balance. o Such a region is said to have crossed fields. ▯ • The two forces balance is ???????? = ????????????, if ????▯=   • For given magnitudes of the electric and magnetic fields, the forces balance only for particles that have the exact speed given by v=E/B. Any particle that has this speed, regardless of its mass or charge, will traverse the space undeflected. o A particle that has a greater speed will be deflected toward the direction of the magnetic force, and a particle that has a lesser speed will be deflected in the direction of the electric force. o This arrangement of fields is often used as a velocity selector, which is a device that allows only particles with the speed v=E/B to pass. The Mass Spectrometer • The mass spectrometer was developed as a means of measuring the masses of isotopes. • The mass-to-charge ratio of an ion of known speed can be determined by measuring the radius of the circular path taken by the ion in a known magnetic field. • The ions are accelerated by an electric field and enter a uniform magnetic field. ▯ ▯ ▯▯ • = ▯ ▯ ∆▯ The Cyclotron • The cyclotron was invented by E. O. Lawrence and M. S. Livingston in 1934 to accelerate particles, such as protons or deuterons, to large kinetic energies. • The particles move in two semicircular metal containers called dees ???? • The kinetic energy of a particle leaving a cyclotron: ???? = ???????? = ???? ???? ???? ???? ????(???? ???? )???????? ???? ???? Torques on Current Loops and Magnets • A current-carrying loop experiences no net force in a uniform magnetic field, but it does experience a net torque. • The orientation of the loop can be described conveniently by a unit vector n that is normal to the plane of the loop o If the fingers of the right hand curl around the loop in the direction of the current, the thumb points in the direction of n. • Torque for a loop: ???? = ???????????????????????????????? o A is the area of the loop o N is the number of turns of the loop o The torque tends to twist the loops so that n is in the same direction as B • The torque can be written conveniently in terms of the magnetic dipole moment ????  (also referred to simply as the magnetic moment) of the current loop • Magnetic Dipole Moment of a Current Loop: ???? = ???????????????? • Torque on a Current Loop: ???? = ????  ????  ???? Potential Energy of a Magnetic Dipole in a Magnetic Field • When a torque is exerted on a rotating object, work is done. • Potential Energy of a Magnetic Dipole: ???? = −???????????????????????? = −???? ∙ ???? o At an angle ???? to the direction of a magnetic field • When a permanent magnet, such as a compass needle or a bar magnet, is placed in a region where there is a magnetic field B, the field exerts a torque on the magnet that tends to rotate the magnet so that it lines up with the field. • The bar magnet is characterized by a magnetic moment ????, a vector that points in the same direction as an arrow drawn from the south pole of the magnet to the north pole of the magnet. A short bar magnet thus behaves like a current loop. The Hall Effect • When these charges are traveling in a conducting wire, they will be pushed to one side of the wire. This results in a separation of charge in the wire—a phenomenon called the Hall effect. • This phenomenon allows us to determine the sign of the charge on the charge carriers and the number of charge carriers per unit volume n in a conductor. • The Hall effect also provides a convenient method for measuring magnetic fields. • The potential difference between the top of the strip and the bottom of the strip is called the Hall voltage. • If the width of the strip is w, the potential difference is H w. ???? ???? o The Hall voltage is therefore: ???? = ???????????? = ???????????????? = ???? ???????????? • The number of charge carriers per unit volume in the strip is ???? ???? ???? = ???????????????? Chapter 27: Sources of the Magnetic Field The Magnetic Field of Moving Point Charges ???????????????????????? • Magnetic Field of a Moving Point Charge: ???? = ???? ???????????? o ????  is a unit vector that points to the field point P from the charge q moving with velocity v, and ???? is a constant of ▯ proportionality called the magnetic constant (permeability of free space) ▯???? o ???? =????????????????????????  ???? ∙ ????/???? The Magnetic Field of Currents: The Biot-Savart Law • Biot-Savart Law: ???????? = ???????????????????????????? ???????????????? • The magnetic field is perpendicular to both ???? and v, in the case of a point charge, or  ???? and dl in the case of a current element. • The magnetic field due to the total current in a circuit can be calculated by using the Biot–Savart law to find the field due to each current element, and then summing (integrating) over all the current elements in the circuit. ????????????(????????????) ???????????? • B at the Center of a Current Loop: ???? = ???????????????? = ???????? ???? ???????????????????? • B on the Axis of a Current Loop: ???? = ???? ???? ???????? ???? ▯???????? ???? • Magnetic-Dipole Field on the Axis of the Dipole: ???? = ???????????????? ???????? ???????? • A current loop behaves as a magnetic dipole because it experiences a torque ????  ????  ???? when placed in an external magnetic field and it also produces a magnetic dipole field at field points far from the current loop. B Due to a Current in a Solenoid • A solenoid is a conducting wire wound into a helix of closely spaced turns • The magnetic field of a solenoid is essentially that of a set of N identical current loops placed side by side • n = N/L is the number of turns per unit length ???? ????▯???????? ????▯???????? • B on the Axis of a Solenoid: ???? = ???? ????????(???? ???? ???? − ???? ????) ????▯???????? ▯???? ????▯???????? ▯???? • B Inside a Long Solenoid: ???? ???????????? o B at either end of a long solenoid is half the value of B at points deep inside the solenoid B Due to a Current in a Straight Wire ???????????? • B Due to a Straight-Wire Segment: ???? = (???????????????? ???? ???????????????? ) ???? ???????????? ???????????????? • B Due to an Infinitely Long, Straight Wire: ???? = ???????????? Magnetic Force Between Parallel Wires • Two parallel currents attract each other. • Two antiparallel currents will repel each other • The ampere is that constant current which, if maintained in two straight, parallel conductors of infinite length and of negligible circular cross sections placed one meter apart in a vacuum, would ▯-7 produce a force between the conductors equal to 2 X 10 newton per meter of length. • A current balance is a device that can be used to calibrate an ammeter from the definition of the ampere. Gauss’s Law for Magnetism • Magnetic field lines appear to diverge from the north-pole end of a bar magnet and appear to converge on the south-pole end. Inside the magnet, however, the magnetic field lines neither diverge from a point near the north-pole end, nor do they converge on a point near the south-pole end. Instead, the magnetic field lines pass through the bar magnet from the south-pole end to the north-pole end. • If a Gaussian surface encloses one end of a bar magnet, the number of magnetic field lines that penetrate the surface from the inside is exactly equal to the number of magnetic field lines that penetrate the surface from the outside. That is, the net flux of the magnetic field B through any closed surface S is always zero. • Gauss’s Law for Magnetism: ???? = ???? ∙ ???????????? = ???????????????? = ???? o ???? is the component of B normal to surface S at area element ▯ dA • It is the mathematical statement that no point in space exists from which magnetic field lines diverge, or to which magnetic field lines converge. That is, isolated magnetic poles do not exist. • The fundamental unit of magnetism is the magnetic dipole. • The field lines of Bare continuous loops. Ampere’s Law • Ampère’s law relates the tangential component Bof the magnetic field summed (integrated) around a closed curve C to the current Ithat passes through any surface bounded by C.▯ • Ampere’s Law: ???????????? =???? ???????? o I is the net current that penetrates any surface S bounded by the curve C. o The positive tangential direction for the path integral along C is related to the choice for the positive direction for the current I through S by the right-hand rule • Ampère’s law holds as long as the currents are steady and continuous. o This means the current does not change in time and that charge is not accumulating anywhere. • B for a Long Straight Wire: ???? ???? ???? ???? ???? = ????  ????????????  ???? ≥ ????  ????????????  ???? = ???? ????????  ????????????  ???? ≤ ????   ???????????? ???????????? • A Toroid consists of loops of wire wound around a doughnut- shaped form • B Inside a Tightly Wound Toroid: ???? ???????? ???? = ????  ????????????  ???? < ???? < ????  ????????????  ????  ????????????  ???? < ????  ????????  ???? > ???? ???????????? Magnetism in Matter • Atoms have magnetic dipole moments due to the motion of their electrons and due to the intrinsic magnetic dipole moment associated with the spin of the electrons. • Unlike the situation with electric dipoles, the alignment of magnetic dipoles parallel to an external magnetic field tends to increase the field. • Inside the current loop, the magnetic field lines are parallel to the magnetic dipole moment. o Thus, inside a magnetically polarized material, the magnetic dipoles create a magnetic field that is parallel to the magnetic dipole moment vectors. • Materials fall into three categories—paramagnetic, ferromagnetic, and diamagnetic—according to the behavior of their magnetic moments in an external magnetic field. • Paramagnetism arises from the partial alignment of the electron spins (in metals) or from the atomic or molecular magnetic moments by an applied magnetic field in the direction of the field. In paramagnetic materials, the magnetic dipoles do not interact strongly with each other and are normally randomly oriented. In the presence of an applied magnetic field, the dipoles are partially aligned in the direction of the field, thereby increasing the field. The increase in the total magnetic field is very small. • Ferromagnetism is much more complicated. Because of a strong interaction between neighboring magnetic dipoles, a high degree of alignment occurs even in weak external magnetic fields, which causes a very large increase in the total field. Even when there is no external magnetic field, a ferromagnetic material may have its magnetic dipoles aligned, as in permanent magnets. • Diamagnetism arises from the orbital magnetic dipole moments induced by an applied magnetic field. These magnetic moments are opposite the direction of the applied magnetic field, thereby decreasing the field. This effect actually occurs in all materials; however, because the induced magnetic moments are very small compared to the permanent magnetic moments, diamagnetism is often masked by paramagnetic or ferromagnetic effects. Diamagnetism is thus observed only in materials whose atoms have no permanent magnetic moments. Magnetization and Magnetic Susceptibility • When some material is placed in a strong magnetic field, such as that of a solenoid, the magnetic field of the solenoid tends to align the magnetic dipole moments (either permanent or induced) inside the material and the material is said to be magnetized. • We describe a magnetized material by its magnetization, M , which is defined as the net magnetic dipole moment per unit volume of the material ▯▯ o ???? = ▯▯ • Surface currents are called amperian currents • The proportionality constant xm is a dimensionless number called the magnetic susceptibility • ???? = ???? ▯▯▯+ ???? ▯ = ???? ▯▯▯ 1 + ????▯ = ????▯???? ▯▯▯ o ???? = 1 + ???? is called the relative permeability of the material ▯ ▯ • For paramagnetic materials, xmis a small positive number that depends on temperature. For diamagnetic materials (other than superconductors), it is a small negative constant independent of temperature. Atomic Magnetic Moments • Classical Relation Between Magnetic Moment and Angular ???? Momentum: ???? = ???? ???????? • Magnetic Moment Due to the Orbital Motion of an Electron: ???????? ???? ???? ???? = − ???????? ????= −???? ???????? ▯???????? ???? ▯???????? • The Bohr Magneton: ???? = ???????????????????????????? ???? ∙ ???? = ????.????????????????????  ????  /???? = ????.???????????????????? ▯????  ????????  /  ???? o Is the quantum unit of magnetic moment ???????? ???? ???? • Magnetic Moment Due to Electron Spin: ???? = −???????? = −???????? ???? ???????? ???? ???? • If all the atoms in a sample of material have their magnetic moments aligned, the magnetic moment per unit volume of the sample is the product of the number of atoms per unit volume n and the magnetic moment m of each atom. For this extreme case, the saturation magnetization M iss o ???? = ???????? Paramagnetism • Paramagnetism occurs in materials whose atoms have permanent magnetic moments that interact with each other only very weakly, resulting in a very small, positive magnetic susceptibilitymx . When there is no external magnetic field, these magnetic moments are randomly oriented. In the presence of an external magnetic field, the magnetic moments tend to line up parallel to the field, but this is counteracted by the tendency for the magnetic moments to be randomly oriented due to thermal motion. The degree to which the moments line up with the field depends on the strength of the field and on the temperature. • Curie’s Law: ???? = ???????????????????????????? ???? ???????? ???? o kT = 2.59X10^-2 eV Ferromagnetism • Ferromagnetism arises from a strong interaction between the electrons in a partially full band in a metal or between the localized electrons that form magnetic moments on neighboring atoms. This interaction, called the exchange interaction, lowers the energy of a pair of electrons with parallel spins. • Ferromagnetic materials have very large positive values of magnetic susceptibility x (as measured under conditions described, which m follow). In samples of these substances, a small external magnetic field can produce a very large degree of alignment of the atomic magnetic dipole moments. • The region of space over which the magnetic dipole moments are aligned is called a magnetic domain • At temperatures above a critical temperature, called the Curie temperature, thermal agitation is great enough to break up this alignment and ferromagnetic materials become paramagnetic. • When an external magnetic field is applied, the boundaries of the domains may shift or the direction of alignment within a domain may change so that there is a net macroscopic magnetic moment in the direction of the applied field. Because the degree of alignment is large for even a small external field, the magnetic field produced in the material by the dipoles is often much greater than the external field. • Let us consider what happens when we magnetize a long iron rod by placing it inside a solenoid and gradually increase the current in the solenoid windings. o As the current is gradually increased from zero, B increases from zero along the part of the curve from the origin O to point P 1 The flattening of this curve near point P ind1cates that the magnetization M is approaching its saturation value M ,sat which all the atomic magnetic moments are aligned. Above saturation, B increases only because the magnetizing field B appuonI increases. When B appis gradually decreased from point P ,1there is not a corresponding decrease in the magnetization. The shift of the domains in a ferromagnetic material is not completely reversible, and some magnetization remains even when B appis reduced to zero, as indicated in the figure. This effect is called hysteresis, from the Greek word hysteros meaning later or behind, and the curve is called a hysteresis curve. The value of the magnetic field at point P 4 when B appis zero is called the remnant field B rem . At that point, the iron rod is a permanent magnet. If the current in the solenoid is now reversed so that B appis in the opposite direction, the magnetic field B is gradually brought to zero at point c. The remaining part of the hysteresis curve is obtained by further increasing the current in the opposite direction until point P is reached, which corresponds to saturation in 2 the opposite direction, and then decreasing the current to zero at point P and increasing it again in its original direction. 3 • Because the magnetization M depends on the previous history of the material, and because it can have a large value even when the applied field is zero, it is not simply related to the applied fieldapp • Permeability of a material: ???? = ???? + ???? ???? = ???? ???? ???? ???? ???? ???? • For paramagnetic and diamagnetic materials, x is much less than 1 so the permeability m and the magnetic constant (permeability of empty space) m ar0 very nearly equal. • Because B does not vary linearly with B , the relative permeability app is not constant. The maximum value of K occmrs at a magnetization that is considerably less than the saturation magnetization. • Note that the maximum values of K aremmuch greater than 1. • The area enclosed by the hysteresis curve is proportional to the energy dissipated as heat in the irreversible process of magnetizing and demagnetizing. If the hysteresis effect is small, so that the area inside the curve is small, indicating a small energy loss, the material is called magnetically soft. o The hysteresis curve for a magnetically soft material: the remnant field B is nearly zero, and the energy loss per cycle rem is small. Magnetically soft materials are used for transformer cores to allow the magnetic field B to change without incurring large energy losses as the field changes. • A large remnant field is desirable in a permanent magnet. o Magnetically hard materials are used for permanent magnets Diamagnetism • Diamagnetic materials are those materials that have very small negative values of magnetic susceptibility x m • The change in the magnetic moment of the charges is in the direction opposite that of the external applied field. Because the permanent magnetic moments of the two charges are equal and oppositely directed they add to zero, leaving only the induced magnetic moments, which are both in the direction opposite to the direction of the applied magnetic field. • A material is diamagnetic if its atoms have zero net angular momentum and therefore no permanent magnetic moment. • The induced magnetic moments that cause diamagnetism have magnitudes of the order of 10 Bohr magnetons. Because this is much smaller than the permanent magnetic moments of the atoms of paramagnetic or ferromagnetic materials, the diamagnetic effect in these atoms is masked by the alignment of their permanent magnetic moments. However, because this alignment decreases with temperature, all materials are theoretically diamagnetic at sufficiently high temperatures. • When a superconductor is placed in an external magnetic field, electric currents are induced on the superconductor’s surface so that the net magnetic field in the superconductor is zero. • A superconductor is thus a perfect diamagnet with a magnetic susceptibility of -1. Chapter 28: Magnetic Induction • A changing magnetic field a changing magnetic flux through a surface bounded by a closed stationary loop of wire induces a current in the wire. The emfs and currents caused by such changing magnetic fluxes are called induced emfs and induced currents. The process itself is referred to as induction. Faraday and Henry also discovered that in a static magnetic field a changing magnetic flux through a surface bounded by a moving loop of wire induces an emf in the wire. An emf caused by the motion of a conductor in a region with a magnetic field is called a motional emf. Magnetic Flux • Magnetic Flux: ???? = ???? ∙ ???????????? = ???? ???????? • The unit of magnetic flux is a weber (Wb) • Because B is proportional to the number of field lines per unit area, the magnetic flux is proportional to the number of field lines through an element of area. • If the surface is flat and has an area A, and if B is uniform (has the same magnitude and direction) everywhere on the surface, the magnetic flux through the surface is ???? = ???? ∙ ???????????? = ???????????????????????? = ???? ???? ???? o ???? is the angle between the direction of B and ???? • If the coil has N turns, the flux through the surface is N multiplied by the flux through each turn. That is, ???? = ???????????????????????????? o A is the area of the flat surface bounded by a single turn. Inducted EMF and Faraday’s Law • If the magnetic flux through a surface bounded by a wire (a conducting path) changes, an emf equal in magnitude to the rate of change of the flux is induced in the wire. • The flux can be changed by increasing or decreasing B, by increasing or decreasing A, or by changing the angle u. o If the magnetic field is due to a permanent magnet, the magnitude of the magnetic field can be increased or decreased by moving a permanent magnet toward or away from the surface. o If the magnetic field is due to a current in a circuit, the magnitude of the magnetic field can be increased or decreased by increasing or decreasing the current. o The flux through the surface can also be changed by varying the angle ????. To vary ????, we can change either the orientation of the surface or the direction of the magnetic field. In each case, if along the perimeter of the surface there is a conducting path, such as a metal wire, an emf E is induced along the path that is equal in magnitude to the rate of change of the magnetic flux through the surface. • Faraday’s Law: ???? = − ???????? ???????? o The minus sign in Faraday’s law has to do with the direction of the induced emf (clockwise or counterclockwise • It is electric forces associated with a nonconservative electric field E doing the work on the mobile charges. • The electric field associated with a changing magnetic field is nonconservative. • Induced EMF For a Stationary Circuit in a Changing Magnetic ???? ???????? Field: ???? = ???? ∙ ???????? = − ???????? ???? ∙ ???????????? = − ???????? Lenz’s Law • Lenz’s Law: The induced emf is in such a direction as to oppose, or tend to oppose, the change that produces it • Alternative Statement of Lenz’s Law: When a magnetic flux through a surface changes, the magnetic field due to any induced current produces a flux of its own—through the same surface and opposite in sign to the initial change in flux. • There is an induced emf only while the flux is changing. The emf does not depend on the magnitude of the flux itself, but only on its rate of change. If there is a large steady flux through a circuit, there is no induced emf. • A self-induced emf opposes the change in the current. It is therefore called a back emf. Motional EMF • Motional emf is any emf induced by the motion of a conductor in a region in which there exists a magnetic field • Magnitude of EMF For a Rod Moving Perpendicular to Both the Length of the Rod and B: ???? = ???????????? Generators and Motors • A simple generator of alternating current is a coil rotating in a uniform magnetic field. The ends of the coil are connected to rings called slip rings that rotate with the coil. Electrical contact is made with the coil by stationary graphite brushes in contact with the rings. • When the coil is mechanically rotated, the flux through it will change, and an emf will be induced in the coil according to Faraday’s law. If the initial angle between n and Bis zero, then the angle at some later time t is given by ???? = ???????? where w is the angular frequency of rotation. • The EMF in the coil will be: ???????? ???? ???? = − ???????? = −???????????? ???????????????????????????? = ???????????????????????????????????? • ???? = ???????????????? ???????????? • The same coil in a magnetic field that can be used to generate an alternating emf can also be used as an ac motor. Instead of mechanically▯rotating the coil to generate an emf, we apply an alternating current to▯the coil from another ac generator▯ • A current▯loop in a magnetic field experiences a torque that tends to rotate the▯ loop such that its magnetic moment Mpoints in the direction of B and ▯the plane of the loop is perpendicular to B.▯ Eddy Currents • A changing flux often induces circulating currents, which are called eddy currents, in a piece of bulk metal like the core of a transformer. The heat produced by such current constitutes a power loss in the transformer. • Eddy currents are frequently undesirable because power is lost due to Joule heating by the current, and this dissipated energy must be transferred to the environment. Inductance • Consider a coil carrying a current I. The current in the coil produces a magnetic field Bthat varies from point to point, but at each point in space the value of B is proportional to I. The magnetic flux of B through the coil is therefore also proportional to I. • Definition of Self-Inductance: ???? = ???????? o L, the proportionality constant, is called the self-inductance of the coil • The self-inductance depends on the geometric shape of the coil. • The SI unit of inductance is the henry (H) ???? ???? • Self-Inductance of a Long Solenoid: ???? = ???? = ???? ???? ???????? o The self-inductance of a long solenoid is proportional to the square of the number of turns per unit length n and to the volume Al ???????? ???????? • Self-Induced EMF: ???? = − = −???? ???????? ???????? • The self-induced emf is often called a back emf. • A coil or solenoid that has enough turns to have a significant self- inductance is called an inductor. • In circuits, an inductor is denoted by the symbol . ???????? • Potential Difference Across an Inductor: ???????? = ???? − ???????? = −???? − ???????? ???????? o r is the internal resistance of the inductor • For an ideal inductor, r=0 Mutual Inductance • When two or more circuits are close to each other, the magnetic flux through one circuit depends not only on the current in that circuit but also on the current in the nearby circuits • Definition-Mutual Inductance: ???? ???????? = ???????? ???? o M is called the mutual inductance of the two circuits • The mutual inductance depends on the geometrical arrangement of the two circuits. For instance, if the circuits are far apart, the flux of Bthrough circuit 2 will be small and the mutual inductance will be small. • Inductance for two tightly would coaxial solenoids: ???? ???? = ????????= ???? ???? ???????????????????? ???? ???????? o The area used to compute the flux through the outer solenoid is not the area of the surface bounded by a loop of that ▯ solenoid, ????????▯, but rather is the area of the surface bounded by a loop of the inner solenoid, ???????? . This is because the magnetic ▯ field due to the inner solenoid is zero outside the inner solenoid. Magnetic Energy • An inductor stores magnetic energy ???? • Energy Stored in an Inductor: ???? = ???????? ???? ???????? • Magnetic Energy Density: ???? = ???? ???? ???????????? o The energy per unit volume is the magnetic energy density ???? ???? Circuits • A circuit containing a resistor and an inductor is called an RL circuit. Because all circuits have resistance and self-inductance at room temperature, the analysis of an RL circuit can be applied to some extent to all circuits. ▯▯ ▯ ▯ ▯ ▯ ▯ • ???? = 1 − ???? ▯ = ???? ▯1 − ???? )▯ ▯ o ???? = ▯▯ is the current as ???? → ∞ ▯ ▯ ▯ o ???? =  ▯s the time constant of the circuit Works Cited Tipler, Paul Allen, and Gene Mosca. Physics for Scientists and Engineers . New York: W.H. Freeman, 2008. Print.


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