March 2, 2005 Physics for Scientists&Engineers 2 1Physics for Scientists &Physics for Scientists &EngineersEngineers 22Spring Semester 2005Lecture 27March 2, 2005 Physics for Scientists&Engineers 2 2ReviewReview! Lenz’s law states that a current is induced in the loop thattends to oppose the change in magnetic flux! The induced emf due to a changing magnetic field is given by! The unit of inductance is the henry (H)! The inductance of a solenoid of length l and area A with nturns per unit length an is given by !E • d!s"!= "d#Bdt[L] = [!B][i]" 1 H =1 Tm21 AL =µ0n2lAMarch 2, 2005 Physics for Scientists&Engineers 2 3Induced Voltage on a Moving Wire in aInduced Voltage on a Moving Wire in aMagnetic FieldMagnetic Field! Consider a conducting wire of length L moving withconstant speed v perpendicular to a constantmagnetic field B as shown belowMagnetic field isdirected into thescreenNegative chargePositive chargeMarch 2, 2005 Physics for Scientists&Engineers 2 4Induced Voltage on a Moving Wire in aInduced Voltage on a Moving Wire in aMagnetic Field (2)Magnetic Field (2)! The magnetic field will exert a force on the electrons the wire, causingthem to move! A negative charge will build up at one end of the wire and a positivecharge will build up at the opposite end of wire, producing an electricforce that cancels the magnetic force, leaving the electrons atequilibrium! Thus we have an electric field with magnitude! Which produces a voltage between the ends of thewire given byFB= qvB = FE= qEE = vBVL= vB ! V = vLBMarch 2, 2005 Physics for Scientists&Engineers 2 5Self Inductance and Mutual InductionSelf Inductance and Mutual Induction! Consider the situation in which two coils, orinductors, are close to each other! A current in the first coil produces magnetic fluxin the second coil! Changing the current in the first coil will induce anemf in the second coil! However, the changing current in the first coil alsoinduces an emf in itself! This phenomenon is called self-induction! The resulting emf is termed the self-induced emf.March 2, 2005 Physics for Scientists&Engineers 2 6Self InductionSelf Induction! Faraday’s Law of Induction tells us that the self-inducedemf for any inductor is given by! Thus in any inductor, a self-induced emf appears when thecurrent changes with time! This self-induced emf depends on the time rate change ofthe current and the inductance of the device! Lenz’s Law provides the direction of the self-induced emf! The minus sign in provides the clue that the induced emfalways opposes any change in currentVemf , L= !d N"B( )dt= !d Li( )dt= ! LdidtMarch 2, 2005 Physics for Scientists&Engineers 2 7Self Inductance (2)Self Inductance (2)! In the figure below, the current flowing throughan inductor is increasing with time! Thus a self-induced emf arises to oppose theincrease in currentMarch 2, 2005 Physics for Scientists&Engineers 2 8Self Inductance (3)Self Inductance (3)! In the figure below, the current flowing throughan inductor is decreasing with time! Thus a self-induced emf arises to oppose thedecrease in currentMarch 2, 2005 Physics for Scientists&Engineers 2 9RL CircuitsRL Circuits! We have assumed that our inductors have no resistance! Now let’s treat inductors that have resistance! We know that if we place a source of external voltage, Vemf,into a single loop circuit containing a resistor R and acapacitor C, the charge q on the capacitor builds up overtime as! where the time constant of the circuit is given by!C = RC! The same time constant governs the decrease of the initialcharge q in the circuit if the emf is suddenly removedq = CVemf1 ! e! t /"C( )q = q0e! t /"CMarch 2, 2005 Physics for Scientists&Engineers 2 10RL Circuits (2)RL Circuits (2)! If we place an emf in a single loop circuit containing a resistance R andan inductor L, a similar phenomenon occurs! If we had connected only the resistor and not the inductor, the currentwould instantaneously rise to the value given by Ohm’s Law as soon aswe closed the switch! However, in the circuit with both the resistor and the inductor, theincreasing current flowing through the inductor creates a self-inducedemf that tends to oppose the increase in current! As time passes, the change in current decreases and the opposing self-induced emf decreases and after a long time, the current is steadyMarch 2, 2005 Physics for Scientists&Engineers 2 11RL Circuits (3)RL Circuits (3)! We can use Kirchhof’s loop rule to analyze this circuitassuming that the current i at any given time is flowingthrough the circuit in a counterclockwise direction! The emf source represents a gain in potential, +Vemf, andthe resistor represents a drop in potential, -iR! The self-inductance of the inductor represents a drop inpotential because it is opposing the increase in current! The drop in potential due to the inductor is proportional tothe time rate change of the current and is given byVemf , L= ! LdidtMarch 2, 2005 Physics for Scientists&Engineers 2 12RL Circuits (4)RL Circuits (4)! Thus we can write the sum of the potential drops aroundthe circuit as! We can rewrite this equation as! The solution to this differential equation is! We can see that the time constant of this circuit is!L = L/RVemf! iR ! Ldidt= 0Ldidt+ iR = Vemfi(t) =VemfR1 ! e! t / L / R( )( )March 2, 2005 Physics for Scientists&Engineers 2 13RL Circuits (5)RL Circuits (5)! Now consider the case in which an emf source hadbeen connected to the circuit and is suddenlyremoved! We can use our previous equation with Vemf = 0 todescribe the time dependence of this circuitLdidt+ iR = 0March 2, 2005 Physics for Scientists&Engineers 2 14RL Circuits (6)RL Circuits (6)! The solution to this differential equation is! where the initial conditions when the emf wasconnected can be used to determine the initialcurrent, i0 = Vemf/R! This equation describes a single loop circuit with aresistor and an inductor that initially has a currenti0! The current drops with time exponentially with atime constant !L = L/R and after a long time thecurrent in the circuit is zeroi(t ) = i0e! t /"LMarch 2, 2005 Physics for Scientists&Engineers 2 15Energy of a Magnetic FieldEnergy of a Magnetic Field! We can think of an inductor as a device that can storeenergy in a magnetic field in the manner similar to the waywe think of a capacitor as a device that can store energy inan electric field! The energy stored in the electric field of a capacitor isgiven by! Consider the situation in which an
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