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- Title: Week9_2
- Type: Notes
- School: Drexel
- Course: PHYS 102
- Term: Spring
Generators AC Electric generators take in energy by work and transfer it out by electrical transmission The AC generator consists of a loop of wire rotated by some external means in a magnetic field 1 Induced emf In an AC Generator The induced emf in the loops is d B = -N dt = NAB sin t This is sinusoidal, with max = N A B 2 Lenz' Law Faraday's Law indicates the induced emf and the change in flux have opposite algebraic signs This has a physical interpretation that has come to be known as Lenz' Law It was developed by a German physicist, Heinrich Lenz 3 Lenz' Law, cont Lenz' Law states the polarity of the induced emf in a loop is such that it produces a current whose magnetic field opposes the change in magnetic flux through the loop The induced current tends to keep the original magnetic flux through the circuit from changing 4 Lenz' Law Example 1 When the magnet is moved toward the stationary loop, a current is induced as shown in a This induced current produces its own magnetic field that is directed as shown in b to counteract the increasing external flux 5 Lenz' Law Example 2 When the magnet is moved away the stationary loop, a current is induced as shown in c This induced current produces its own magnetic field that is directed as shown in d to counteract the decreasing external flux 6 In equal-arm balances from the early 20th century (Fig. 23.18), it is sometimes observed that an aluminum sheet hangs from one of the arms and passes between the poles of a magnet, which causes the oscillations of the equal arm balance to decay rapidly. In the absence of such magnetic braking, the oscillation might continue for a very long time and the experimenter would have to wait to take a reading. Why do the oscillations decay? 1. 2. 3. The aluminum sheet is attracted to the magnet. Currents in the aluminum sheet set up a magnetic field that opposes the oscillations. Aluminum is ferromagnetic. 33% 33% 33% ... m ... ti a. .. um is fe rr om in u m um ur re n ts in th e 10 7 ee sh Th C A lu m in e al al u Problem 23.22. A rectangular coil with resistance R has N turns, each of length and width w as shown in Figure P23.22. The coil moves into a r uniform r magnetic field B with constant velocity v . What are the magnitude and direction of the total magnetic force on the coil (a) as it enters the magnetic field, (b) as it moves within the field, and (c) as it leaves the field? 8 Problem 23.21. A conducting rectangular loop of mass M, resistance R, and dimensions w by falls from rest into a magnetic field as shown in Figure P23.21. During the time interval before the top edge of the loop reaches the field, the loop approaches a terminal speed vT. MgR (a) Show that vT = B 2 w 2 (b) Why is vT proportional to R? (c) Why is it inversely proportional to B2? 9 Induced emf and Electric Fields An electric field is created in the conductor as a result of the changing magnetic flux Even in the absence of a conducting loop, a changing magnetic field will generate an electric field in empty space This induced electric field has different properties than a field produced by stationary charges 10 Induced emf and Electric Fields, cont The emf for any closed path can r r be expressed as the line integral of E ds over the path Faraday's Law can be written in a general form r r d B = E ds = - dt 11 Induced emf and Electric Fields, final The induced electric field is a nonconservative field that is generated by a changing magnetic field The field cannot be an electrostatic field because if the field were electrostatic, and hence conservative, the line integral would be zero and it isn't 12 In a region of space, a magnetic field is uniform over space but increases at a constant rate. This changing magnetic field induces an electric field that 1. 2. 3. 4. ns er se ct io di re re a co th e in ha is s a co ns in c is ta n tm ag increases in time, is conservative, is in the direction of the magnetic field, or has a constant magnitude. 25% 25% 25% 25% e, iv e , tim in va t s n of n. .. ... 10 13 Self-Induction When the switch is closed, the current does not immediately reach its maximum value Faraday's Law can be used to describe the effect 14 Self-Induction, 2 As the current increases with time, the magnetic flux through the circuit loop due to this current also increases with time The corresponding flux due to this current also increases This increasing flux creates an induced emf in the circuit 15 Self-Inductance, 3 The direction of the induced emf is such that it would cause an induced current in the loop which would establish a magnetic field opposing the change in the original magnetic field The direction of the induced emf is opposite the direction of the emf of the battery Sometimes called a back emf This results in a gradual increase in the current to its final equilibrium value 16 Self-Induction, 4 This effect is called self-inductance Because the changing flux through the circuit and the resultant induced emf arise from the circuit itself The emf L is called a self-induced emf 17 Self-Inductance, Equations An induced emf is always proportional to the time rate of change of the current dI L = -L dt L is a constant of proportionality called the inductance of the coil It depends on the geometry of the coil and other physical characteristics 18 Inductance of a Coil A closely spaced coil of N turns carrying current I has an inductance of NB L= =- I dI dt The inductance is a measure of the opposition to a change in current Compared to resistance which was opposition to the current 19 Inductance Units The SI unit of inductance is a Henry (H) V s 1H = 1 A Named for Joseph Henry 20 Joseph Henry 1797 1878 Improved the design of the electromagnet Constructed one of the first motors the Discovered phenomena of selfinductance 21 Inductance of a Solenoid Assume a uniformly wound solenoid having N turns and length l Assume l is much greater than the radius of the solenoid The interior magnetic field is N B = o nI = o I l 22 Inductance of a Solenoid, cont The magnetic flux through each turn is Therefore, the inductance is 2 N B o N A L= I = l NA I B = BA = o l This shows that L depends on the geometry of the object 23 RL Circuit, Introduction A circuit element that has a large selfinductance is called an inductor The circuit symbol is We assume the self-inductance of the rest of the circuit is negligible compared to the inductor However, even without a coil, a circuit will have some self-inductance 24 RL Circuit, Analysis An RL circuit contains an inductor and a resistor When the switch is closed (at time t=0), the current begins to increase At the same time, a back emf is induced in the inductor that opposes the original increasing current 25 RL Circuit, Analysis, cont Applying Kirchhoff's Loop Rule to the previous circuit gives dI - IR - L = 0 dt Looking at the current, we find -Rt L I= (1 - e R ) 26 RL Circuit, Analysis, Final The inductor affects the current exponentially The current does not instantly increase to its final equilibrium value If there is no inductor, the exponential term goes to zero and the current would instantaneously reach its maximum value as expected 27 RL Circuit, Time Constant The expression for the current can also be expressed in terms of the time constant, , of the circuit I (t ) = (1 - e ) R -t where = L / R Physically, is the time required for the current to reach 63.2% of its maximum value 28 RL Circuit, Current-Time Graph, 1 The equilibrium value of the current is /R and is reached as t approaches infinity The current initially increases very rapidly The current then gradually approaches the equilibrium value I (t ) = ( R 1 - e -t ) 29 RL Circuit, Current-Time Graph, 2 The time rate of change of the current is a maximum at t = 0 It falls off exponentially as t approaches infinity In general, dI -t = e dt L 30 The circuit in Figure 23.28 includes a power source that provides a sinusoidal voltage. Thus, the magnetic field in the inductor is constantly changing. The inductor is a simple air-core solenoid. The switch in the circuit is closed and the lightbulb glows steadily. An iron rod is inserted into the interior of the solenoid, which increases the magnitude of the magnetic field in the solenoid. As that happens, the brightness of the lightbulb 1. 2. 3. increases, decreases, or is unaffected. 33% 33% 33% 10 , or se s s, cr ea in de is un af fe c cr ea se te d . 31 Two circuits like the one shown in Active Figure 23.26 are identical except for the value of L. In circuit A, the inductance of the inductor is LA, and in circuit B, it is LB. The switch has been in position b for both circuits for a long time. At t = 0, the switch is thrown to a in both circuits. At t = 10 s, the switch is thrown to b in both circuits. The resulting graphical representation of the current as a function of time is shown in Figure 23.29. Assuming that the time constant of each circuit is much less than 10 s, which of the following is true? 1. 2. 3. LA > LB. LA < LB There is not enough information to determine the relative values. 33% 33% 33% LB . LB > < i.. no te no . 10 32 LA Th er e is ug h LA Problem 23.34. Show that I = Iie t/ is a solution of the differential equation IR + L where = L/R and Ii is the current at t = 0. dI =0 dt 33 Problem 23.39. The switch in Figure P23.39 is open for t < 0 and then closed at time t = 0. Find the current in the inductor and the current in the switch as functions of time thereafter. 34 Energy in a Magnetic Field In a circuit with an inductor, the battery must supply more energy than in a circuit without an inductor Part of the energy supplied by the battery appears as internal energy in the resistor The remaining energy is stored in the magnetic field of the inductor 35 Energy in a Magnetic Field, cont Looking at this energy (in terms of rate) dI 2 I = I R + LI dt I is the rate at which energy is being supplied by the battery I2R is the rate at which the energy is being delivered to the resistor Therefore, LI dI/dt must be the rate at which the energy is being delivered to the inductor 36 Energy in a Magnetic Field, final Let U denote the energy stored in the inductor at any time The rate at which the energy is stored is dUB dI = LI dt dt To find the total energy, integrate and UB = L I2 37 Energy Density of a Magnetic Field Given U = L I2, B 1 B2 U = o n 2 Al Al = 2 2 o o n 2 Since Al is the volume of the solenoid, the magnetic energy density, uB is U B2 uB = = V 2o This applies to any region in which a magnetic field exists not just the solenoid 38 You are performing an experiment that requires the highest possible energy density in the interior of a very long solenoid. Which of the following increases the energy density? (More than one choice may be correct.) 1. 2. 3. th e on ly th e g g g si n si n si n cr ea cr ea cr ea in in in in cr ea si n g th e 4. increasing the number of turns per unit length on the solenoid increasing the crosssectional area of the solenoid increasing only the length of the solenoid while keeping the number of turns per unit length fixed increasing the current in the solenoid 25% 25% 25% 25% s... m be l.. rr en cu t. . ... . 10 39 cr os nu th e Inductance Example Coaxial Cable Calculate L and energy for the cable The total flux is B = BdA = b a o I Il b ldr = o ln 2 r 2 a Therefore, L is L= The total energy is 1 2 o lI 2 b U = LI = ln 2 4 a 40 B o l b = ln 2 a I
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