Presentation on theme: "CHAPTER 32 inductance 32.1 Self-Inductance 32.3 Energy in a Magnetic Field."— Presentation transcript:
1CHAPTER inductanceSelf-InductanceEnergy in a Magnetic Field
2example Current clockwise; force up Current counterclockwise; force up Week 9, Day 2exampleA coil moves up from underneath a magnet with its north pole pointing upward. The current in the coil and the force on the coil:Current clockwise; force upCurrent counterclockwise; force upCurrent clockwise; force downCurrent counterclockwise; force downClass 22
3Answer: 3. Current is clockwise; force is down Week 9, Day 2Answer: 3. Current is clockwise; force is downThe clockwise current creates a self-field downward, trying to offset the increase of magnetic flux through the coil as it moves upward into stronger fields (Lenz’s Law).The I dl x B force on the coil is a force which is trying to keep the flux through the coil from increasing by slowing it down (Lenz’s Law again).Class 22
4example Bout v Current CW, Force Left, No Torque Week 09, Day 1examplevBoutA rectangular wire loop is pulled thru a uniform B field penetrating its top half, as shown. The induced current and the force and torque on the loop are:Current CW, Force Left, No TorqueCurrent CW, No Force, Torque Rotates CCWCurrent CCW, Force Left, No TorqueCurrent CCW, No Force, Torque Rotates CCWNo current, force or torqueClass 20
5Bout v Answer: 5. No current, force or torque Week 09, Day 1vBoutAnswer: 5. No current, force or torqueThe motion does not change the magnetic flux, so Faraday’s Law says there is no induced EMF, or current, or force, or torque.Of course, if we were pulling at all up or down there would be a force to oppose that motion.Class 20
6Self-InductanceWhen the switch is closed, the current does not immediately reach its maximum valueFaraday’s law can be used to describe the effect
7Self-inductance, cont.(a) A current in the coil produces a magnetic field directed to the left. (b) If the current increases, the coil acts as a source of emf directed as shown by the dashed battery. (c) The induced emf in the coil changes its polarity if the current decreases.
8Self-inductance occurs when the changing flux through a circuit arises from the circuit itself As the current increases, the magnetic flux through a loop due to this current also increasesThe increasing flux induces an emf that opposes the currentAs the magnitude of the current increases, the rate of increase lessens and hence the induced emf decreasesThis opposing emf results in a gradual increase in the current
9Self Inductance Define: Self Inductance An inductor is a device that produces a uniform magnetic field when a current passes through it. A solenoid is an inductor.The magnetic flux of an inductor is proportional to the current.For each coil (turn) of the solenoid:Φper coil = A•BΦsol = N(A•B) = NAB = NA(µ0NI/ℓ) = (Au0N2/ℓ)IsolDefine: Self Inductance
10Inductance of a Solenoid The magnetic flux through each turn isTherefore, the inductance isThis shows that L depends on the geometry of the object
12Self-inductance, cont.The self-induced emf is given by Faraday’s law and must be proportional to the time rate of change of the currentL is a proportionality constant called the inductance of the deviceThe negative sign indicates that a changing current induces an emf in opposition to that change
13Inductor has a large inductance (L) and consist of closely wrapped coil of many turns Inductance can be interpreted as a measure of opposition to the rate of change in the currentRemember resistance R is a measure of opposition to the currentAs a circuit is completed, the current begins to increase, but the inductor produces an emf that opposes the increasing currentTherefore, the current doesn’t change from 0 to its maximum instantaneously
15Potential difference across an inductor An inductor is a conducting coil.An inductor in a circuit resists change in current with an induced potential.An inductor stores energy in a magnetic field.The “strength” of an inductor is determined by is “inductance”, represented by the letter L.For the ideal inductor, R = 0, therefore potential difference across the inductor also equals zero, as long as the current is constant.What happens if we increase the current?
16Potential difference across an inductor Increasing the current increases the flux.An induced magnetic field will oppose the increase by pointing to the right.The induced current is opposite the solenoid current.The induced current carries positive charge to the left and establishes a potential difference across the inductor.Induced currentInduced fieldPotential difference
17Potential difference across an inductor The potential difference across the inductor can be found using Faraday’s Law:Where Φm = Φper coilΦsol = N Φper coilWe defined Φ = LIdΦsol/dt = L |dI/dt|Induced currentInduced fieldPotential difference
18Potential difference across an inductor If the inductor current is decreased, the induced magnetic field, the induced current and the potential difference all change direction.Note that whether you increase or decrease the current, the inductor always “resists” the change with an induced current.
19The sign of potential difference across an inductor ∆VL = -L dI/dt∆VL decreases in the direction of current flow if current is increasing.∆VL increases in the direction of current flow if current is decreasing.∆VL is measured in the direction of current in the circuit
20Comparison of R and L in a simple circuit e=-L(DI/Dt)e=-IRR is a measure of opposition to the currentL is a measure of opposition to the rate of change in current
25exampleElectrons are going around a circle in a counterclockwise direction as shown. At the center of the circle they produce a magnetic field that is:eA. into the pageB. out of the pageC. to the leftD. to the rightE. zeroA
26Physics C4/12/2017example: How much current flows through the resistor? How much power is dissipated by the resistor?50 cmB = 0.15 T3 Wv = 2 m/sBertrand
27Physics CInductor, L4/12/2017LeeLiWhen switch is first closed, eL opposes emf of cell.Bertrand
28eL Inductor, L e When switch is opened, eL supports emf of cell. L i Physics CInductor, L4/12/2017LeieLWhen switch is opened, eL supports emf of cell.Bertrand
29Energy Stored in Magnetic Field Energy density = Magnetic Energy per unit volume (J/m3)Energy density = B2 / (2 m0)Example: 1 Tesla fieldEnergy density = (1 T)2 / [8p·10-7T ·m /A]= 3.98 ·105 T·A / m= 3.98 ·105 T·A ·m / m2= 3.98 ·105 N / m2= 3.98 ·105 J / m3
30Physics C4/12/2017example: A coil has an inductance of 3.00 mH and the current changes from A to 1.5 A in a time of s. Find the magnitude of the average induced emf in the coil during this time.Bertrand
32exampleWhat e.m.f. will be induced in a 10 H inductor in which current changes from 10A to 7A in 9x10-2 s ?Solution: L= 10H, I1= 10A, I2= 7A, dt= 9x10-2s 19x10 sε= -L dI/dt,= -L (I2-I1)/dt= -10 (7-10)/ 9x10-2= Volt
33ExampleiThe current in a 10 H inductor is decreasing at a steady rate of 5 A/s.If the current is as shown at some instant in time, what is the magnitude and direction of the induced EMF?Magnitude = (10 H)(5 A/s) = 50 VCurrent is decreasingInduced emf must be in a direction that OPPOSES this change.So, induced emf must be in same direction as current(a) 50 V(b) 50 V
34Example :Find the inductance of a uniformly wound solenoid having N turns and length . Assume that is much longer than the radius of the windings and that the core of the solenoid is air.We can assume that the interior magnetic field due to the source current is uniform and given by Equationwhere n = N/ is the n umber of turns per unit length.The magnetic flux through each turn is :where A is the cross-sectional area of the solenoid.Using this expression and Equation we find that :Because N = n , we can express the result in the form :V = volume of the solenoid = A
35Example : Calculating Inductance and emf Calculate the inductance of an air-core solenoid containing 300 turns if the length of the solenoid is 25.0 cm and its cross-sectional area is 4.00 cm2.Calculate the self-induced emf in the solenoid if the current through it is decreasing at the rate of 50.0A/s.Solution for (a)Using Equation (32.4), we obtain :Solution for (b)Using Equation (32.1) and given that dI/dt = -50.0A/s, we obtain :
36exampleAn inductor is made by tightly winding 0.30 mm diameter wire around a 4.0 mm diameter cylinder. What length cylinder has an inductance of 0.01 mH?
37example A 10.0 A current passes through a 10 mH inductor coil. What potential difference is induced across the coil if the currentdrops to zero in 5 ms?
38Energy in Inductors and Magnetic Fields A magnetic field stores considerable energy. Therefore, an inductor, which operates by creating a magnetic field, stores energy. Let’s consider how much energy UL is stored in an inductor L carryingcurrent I:
39exampleA solenoid of radius 2.5cm has 400 turns and a length of 20 cm. Find(a) its inductance and (b) the rate at which current must changethrough it to produce an emf of 75mV.
42Example: Current I increases uniformly from 0 to 1 A. in 0.1 seconds. Find the induced voltage across a50 mH (milli-Henry) inductance.i+-ELApply:Substitute:Negative result means that induced EMF is opposed to both di/dt and i.
43examplea) At equilibrium (infinite time) how much energy is stored in the coil?E = 12 VL = 53 mH
45exampleWhat is the magnetic energy stored in a3-mH inductor when the current through it is 4 mA?24 × 10-9 jouleor2.4 × 10-8 joule
46exampleWhat happens to the energy stored by an inductor when the current through it is doubled?Its energy is quadrupled(i.e. 4 times of original)
47exampleThe current through a 4-mH inductor is varying as follows: I = 2t What will be the induced emf att = 1 second?ε = -24 mV
48ExampleCalculate the inductance of a solenoid with 100 turns, a length of 5.0 cm, and a cross sectional area of 0.30 cm2. L = m0N2A l L = (4p X 10-7 T m/A)(100)2(3 X 10-5m2) (0.05 m) L = 7.5 X 10-6 H or 7.5 mH
49ExampleThe same solenoid is now filled with an iron cores (m = 4000 m0). Calculate the inductance L = (4000)(7.5 X 10-6H) L = H or 30 mH
50Energy stored in a Magnetic Field Physics C4/12/2017Energy stored in a Magnetic FieldUB = ½ L I2UB : energy stored in magnetic fieldL: inductance in HenrysI: current in amperesBertrand
51Inductance (review)Increasing current in a coil of wire will generate a counter emf which opposes the current.Applying the voltage law allows us to see the effect of this emf on the circuit equation.The fact that the emf always opposes the change in current is an example of Lenz's law.The relation of this counter emf to the current is the origin of the concept of inductance.The inductance of a coil follows from Faraday's law.
52Inductance (review)Inductance of a coil: For a fixed area and changing current, Faraday's law becomesSince the magnetic field of a solenoid isthen for a long coil the emf is approximated by
53Inductance (review)From the definition of inductancewe obtain
54Energy in a Magnetic Field The energy stored in the inductorBattery PowerInductor PowerResistor PowerFor energy density, consider a solenoid