Review Lenz s law states that a current is induced in the loop that tends to oppose the change in magnetic flux The induced emf due to a changing magnetic field is given by Physics for Scientists Engineers 2 E ds d B dt The unit of inductance is the henry H Spring Semester 2005 L Lecture 27 B 1 Tm 2 1 H i 1A The inductance of a solenoid of length l and area A with n turns per unit length an is given by L 0 n 2lA March 2 2005 Physics for Scientists Engineers 2 1 March 2 2005 Self Inductance and Mutual Induction Physics for Scientists Engineers 2 Self Induction Consider the situation in which two coils or inductors are close to each other Faraday s Law of Induction tells us that the self induced emf for any inductor is given by A current in the first coil produces magnetic flux in the second coil Vemf L d N B d Li di L dt dt dt Changing the current in the first coil will induce an emf in the second coil Thus in any inductor a self induced emf appears when the current changes with time However the changing current in the first coil also induces an emf in itself This self induced emf depends on the time rate change of the current and the inductance of the device This phenomenon is called self induction Physics for Scientists Engineers 2 Lenz s Law provides the direction of the self induced emf The minus sign in provides the clue that the induced emf always opposes any change in current The resulting emf is termed the self induced emf March 2 2005 2 3 March 2 2005 Physics for Scientists Engineers 2 4 Self Inductance 2 Self Inductance 3 In the figure below the current flowing through an inductor is increasing with time In the figure below the current flowing through an inductor is decreasing with time Thus a self induced emf arises to oppose the increase in current Thus a self induced emf arises to oppose the decrease in current March 2 2005 Physics for Scientists Engineers 2 5 March 2 2005 RL Circuits Physics for Scientists Engineers 2 6 RL Circuits 2 We have assumed that our inductors have no resistance If we place an emf in a single loop circuit containing a resistance R and an inductor L a similar phenomenon occurs 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 a capacitor C the charge q on the capacitor builds up over time as q CVemf 1 e t C If we had connected only the resistor and not the inductor the current would instantaneously rise to the value given by Ohm s Law as soon as we closed the switch where the time constant of the circuit is given by C RC However in the circuit with both the resistor and the inductor the increasing current flowing through the inductor creates a self induced emf that tends to oppose the increase in current The same time constant governs the decrease of the initial charge q in the circuit if the emf is suddenly removed q q0 e t C March 2 2005 Physics for Scientists Engineers 2 As time passes the change in current decreases and the opposing selfinduced emf decreases and after a long time the current is steady 7 March 2 2005 Physics for Scientists Engineers 2 8 RL Circuits 4 RL Circuits 3 Thus we can write the sum of the potential drops around the circuit as di Vemf iR L 0 dt We can rewrite this equation as We can use Kirchhof s loop rule to analyze this circuit assuming that the current i at any given time is flowing through the circuit in a counterclockwise direction The emf source represents a gain in potential Vemf and the resistor represents a drop in potential iR L The self inductance of the inductor represents a drop in potential because it is opposing the increase in current The solution to this differential equation is Vemf i t 1 e t L R R The drop in potential due to the inductor is proportional to the time rate change of the current and is given by Vemf L L March 2 2005 di iR Vemf dt di dt We can see that the time constant of this circuit is L L R Physics for Scientists Engineers 2 9 March 2 2005 RL Circuits 5 10 The solution to this differential equation is i t i0 e t L where the initial conditions when the emf was connected can be used to determine the initial current i0 Vemf R This equation describes a single loop circuit with a resistor and an inductor that initially has a current i0 We can use our previous equation with Vemf 0 to describe the time dependence of this circuit The current drops with time exponentially with a time constant L L R and after a long time the current in the circuit is zero di L iR 0 dt Physics for Scientists Engineers 2 Physics for Scientists Engineers 2 RL Circuits 6 Now consider the case in which an emf source had been connected to the circuit and is suddenly removed March 2 2005 11 March 2 2005 Physics for Scientists Engineers 2 12 Energy of a Magnetic Field Energy of a Magnetic Field 2 We can think of an inductor as a device that can store energy in a magnetic field in the manner similar to the way we think of a capacitor as a device that can store energy in an electric field The energy stored in the electric field of a capacitor is given by 1 q2 UE 2C di P Vemf i L i dt Integrating this power over the time it takes to reach a final current yields the energy stored in the magnetic field of the inductor Consider the situation in which an inductor is connected to a source of emf The current begins to flow through the inductor producing a self induced emf opposing the increase in current March 2 2005 Physics for Scientists Engineers 2 The instantaneous power provided by the emf source is the product of the current and voltage in the circuit UB 13 March 2 2005 Applications 0 Pdt i 0 Li di 1 2 Li 2 Physics for Scientists Engineers 2 14 Retrieval of the information basically reverses the process of information storage As the storage medium passes by the read head which is another coil the magnetization causes a change of the magnetic field inside the coil Examples are computer hard drives videotapes audio tapes and the magnetic strips on credit cards Storage of the information is accomplished by using an electromagnet in the write head A current that varies in time is sent to the electromagnet and creates a magnetic field that magnetizes the ferromagnetic coding of the storage medium as it passes by the magnet Physics for Scientists Engineers …
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