《物理双语教学课件》Chapter 21 Induction and Inductance, Maxwell1599s Equations 自感互感 麦克斯韦方程_课件适合双语教学

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Chapter 21 Induction and Inductance, Maxwell’s Equations

21.1 Faraday’s Law of Induction 1.Two experiments

(1)The first experiment is shown in figure.We come to the conclusion from the experiment that the current produced in the loop is called induced current, the work done per unit charge in producing that current is called an induced electromotive force(emf), and the proce of producing the current and emf is called induction.(2)The second experiment is shown in the right figure.2.Faraday’s law of induction: An emf is induced in the left-hand loop in above figures when the number of magnetic field lines that pa through the loop is changing.3.Magnetic flux:

(1)Suppose a loop enclosing an area A is placed in a magnetic field.Then the magnetic flux through the loop is BBdA,where

dA

is a vector of magnitude dA that is perpendicular to a differential area

dA.(2)The SI unit for magnetic flux is the tesla-square meter, which is called the weber(abbreviated Wb).4.We can state Faraday’s law in a more quantitative and useful way: the magnitude of the emf induced in a conducting loop is equal to the rate at which the magnetic flux through that loop changes with time.It means sign indicating that opposition.5.Soon after Faraday proposed his law of induction, Heinrich Friedrich Lenz devised a rule – now known as Lenz’s law – for determining the direction of an induced current in a loop: an induced current has a direction such that the magnetic field due to the current opposes the change in the magnetic field that induced the current.Furthermore, the direction of an induced emf is that of the induced current.dBdt, with the minus 6.See above figures.7.Electric Guitars:

21.2 Induction and Energy Transfers 1.See right figure.(1)The magnitude of the emf is dd(BLx)BLv.dtdt(2)The Rmagnitude of the

current.is BLvRinduced i(3)The magnitude of the magnetic force on the loop is B2L2v.FiBLR(4)The rate at which you do work on the loop as you pull it from the magnetic field is

(BLv)2PFvR.(5)The rate at which thermal energy appears in the loop as you pull it along at constant speed is

BLv2(BLv)2PiR()RRR2.(6)Thus the work that you do in pulling the loop through the magnetic field appears as thermal energy in the loop.2.Eddy currents: See right figure.21.3 Induced Electric Fields 1.Let us place a copper ring of radius r in a uniform external magnetic field, as in the figure.Suppose that we increase the strength of this field at a steady rate.The magnetic

flux through the ring will then change at a steady rate, and by Faraday’s law, an induced emf and thus an induced current will appear in the ring.From Lenz’s law we can deduce that the direction of the induced current is counterclockwise in above figure.2.If there is a current in the copper ring, an electric field must be present along the ring;an electric field is needed to do the work of moving the conduction electrons.Moreover, the field must have been produced by changing magnetic flux.This induced electric field is just as real as an electric field produced by static charges.So we are led to a useful and informative restatement of Faraday’s law of induction: A changing magnetic field produces an electric field.3.The striking feature of this statement is that the electric field is induced even if there is no copper ring.To fix this ideas, consider above figure(b), in which the copper ring has been replaced by a hypothetical circular path of radius r.The electric field induced at various points around the circular path must be tangent to the circle, as the figure(b)shows.Hence the circular path is also an electric field line.This is nothing special about the circle of radius r.So the electric field lines produced by the changing magnetic field must be a set of concentric circles as above figure(c).4.As long as the magnetic field is changing with time, the electric field represented by the circular field lines in figure(c)will be present.If the magnetic field remains constant with time, there will be no induced electric field and thus no electric field lines.5.From Faraday’s law, we have the electromotive force is dBEds.dt

21.4 Inductors and Inductance 1.We shall consider a long solenoid, a short length near the middle of a long solenoid, as our basic type of inductor.An inductor can be used to produce a desired magnetic field.2.If we establish a current i in the winding of an inductor, the current produces a magnetic flux through the central region of the inductor.The inductance of the inductor is then LN, in which N is the number of turns.The SI unit of iinductance is the tesla-square meter per ampere.We call this the henry(H), after American physicist Joseph Henry.3.Inductance of a solenoid:

LN(nl)(0ni)(A)0n2lA.So the iiinductance per unit length for a long solenoid near its center is L0n2A.l

4.Self-induction:(1)an induced emf L appears in any coil in which the current is changing.This proce, as shown in

figure,is

called self-induction, and the emf that appears is called a self-induced emf.(2)For any inductor, we haveNLi.Therefore the induced emf is Ld(N)diL.(3)The direction of a self-induced emf dtdtcan be found from Lenz’s law, as shown in figure.5.Mutual induction:(1)we will return to the case of two interacting coils.If the current i changes with time in one coil, an emf will appear in the second coil.We call this proce mutual induction, to suggest the mutual interaction of the two coils and to distinguish it from self-induction, in which only one coil is involved.(2)The mutual inductance can be defined as

M21N221.The emf appearing in coil 2 i1due to the changing current in coil 1 is

2M21di1dt.Similarly, The emf appearing in coil 1 due to the changing current in coil 2 is

21.5 Energy Stored in a Magnetic Fields 1.We consider again the figure.We have the equation

LdiiRdt1M12di2dt.(3)It can be proved that

M21M12M..If the resistance is zero, the work done by the battery will be stored into the magnetic field.So iwe have the magnetic energy is

idi1UBidt(L)idtLidiLi2.00dt22.Energy density of a magnetic field:(1)Consider a length l near the middle of a long solenoid of cro-sectional area A;the volume aociated with this length is Al.The energy per unit volume of the field is

UBLi2Li2i22uB()(0nA)

Al2All2A2AB2(0ni)202012.21.6 Induced Magnetic Fields 1.We know that a changing magnetic flux induces an electric field, and we ended up with Faraday’s law of induction in the form dBEdsdt.Here

E is the electric field induced along a closed loop by the changing magnetic flux through that loop.2.Because symmetry is often so powerful in physics, we should be tempted to ask whether induction can occur in the opposite sense.That is, can a changing electric flux induce a magnetic field? The answer is that it can;furthermore, the equation governing the induction of a magnetic field is almost symmetric with above equation.We often call it Maxwell’s law of induction after James Clerk Maxwell, and we write it as dEBds00dt.The circle on the integral sign indicates that the integral is taken around a closed loop.3.We now consider the charging of a parallel-plate capacitor with circular plates, as shown in figure.We aume that the charge on the capacitor is being increased at a steady rate by a constant current i in the connecting wires.Then the magnitude of the electric field between the plates must also be increasing at a steady

rate.The experiments prove that while the electric field is changing, magnetic fields are induced between the plates, both inside and outside the gap.When the electric field stops changing, these induced magnetic field disappear.4.Combining the Ampere’s law to Maxwell’s law, we have

dEAmpere-Maxwell law to be as:Bds00dt0ienc.21.7 Displacement Current and Maxwell’s Equations 1.Historically, the portion

0dEdt in the ride side of Ampere-Maxwell law has been treated as being a fictitious current called the displacement current: Ampere-Maxwell Bds0id,enc0iencid0dEdt.So as

law.can be rewritten 2.Let us again focus on a charging capacitor with circular plates, as in figure(a).The real current

idqdddE(A)(0EA)0Adtdtdtdtis.On the other hand, the magnitude

of

displacement current between the plates of the id0capacitor dEd(EA)dE00Adtdtdtis

.The real current charging the capacitor and the fictitious displacement current between the plates have the same magnitude.Thus, we can consider the fictitious displacement current to be simply a continuation of the real current from one plate, acro the capacitor gap, to the other plate.3.The induced magnetic field between the plates.We can use the Ampere-Maxwell law to find the induced magnetic field between the plates.It is capacitor and

BiB0d2r2R inside a circular

0id2r outside a circular capacitor.4.Maxwell’s Equations: Table displays Maxwell’s equations.The are the basis for the functioning of such electromagnetic devices as electric motor, cyclotrons, television transmitters and receivers, telephones, fax machines,radar,and microwave ovens.