(11.3.4) The above expression is analogous to the electric energy stored in a capacitor, .102 U:=2C (11.3.5) From an energy perspective there is an important distinction between an inductor and a resistor.Whenever a current I goes through a resistor,energy flows into the resistor and dissipates in the form of heat regardless of whether I is steady or time-dependent(recall that power dissipated in a resistor is P=IV=IR).Energy flows into an ideal inductor only when the current is varying with dl/dt >0.The energy is not dissipated but stored there;it is released later when the current decreases with dl/dt<0.If the current that passes through the inductor is steady,then there is no change in energy since P.LI(dI dt)=0. Example 11.5 Energy Stored in a Solenoid A long solenoid with length and a radius R consists of N turns of wire.A current I passes through the coil.Find the energy stored in the system. Solution:Using Egs.(11.2.6)and(11.3.4),we readily obtain V,-jir-wiFal. (11.3.6) The result can be expressed in terms of the magnetic field strength B=uonl, U224 nFπR)= 1 (πR21) (11.3.7) 24 Because R2I is the volume within the solenoid,and the magnetic field inside is uniform, the magnetic energy density,or the energy per unit volume of the magnetic field is given by B2 Ug= (11.3.8) 24 11-11
11-11 UB = 1 2 LI 2 . (11.3.4) The above expression is analogous to the electric energy stored in a capacitor, UE = 1 2 Q2 C . (11.3.5) From an energy perspective there is an important distinction between an inductor and a resistor. Whenever a current I goes through a resistor, energy flows into the resistor and dissipates in the form of heat regardless of whether I is steady or time-dependent (recall that power dissipated in a resistor is PR = IVR = I 2 R ). Energy flows into an ideal inductor only when the current is varying with dI / dt > 0 . The energy is not dissipated but stored there; it is released later when the current decreases with dI / dt < 0 . If the current that passes through the inductor is steady, then there is no change in energy since PL = LI(dI / dt) = 0 . Example 11.5 Energy Stored in a Solenoid A long solenoid with length l and a radius R consists of N turns of wire. A current I passes through the coil. Find the energy stored in the system. Solution: Using Eqs. (11.2.6) and (11.3.4), we readily obtain UB = 1 2 LI 2 = 1 2 µ0n 2 I 2 ! R2 l . (11.3.6) The result can be expressed in terms of the magnetic field strength B = µ0nI , UB = 1 2µ0 (µ0nI) 2 (! R2 l) = B2 2µ0 (! R2 l) . (11.3.7) Because ! R2 l is the volume within the solenoid, and the magnetic field inside is uniform, the magnetic energy density, or the energy per unit volume of the magnetic field is given by uB = B2 2µ0 (11.3.8)
The above expression holds true even when the magnetic field is non-uniform.The result can be compared with the energy density associated with an electric field, 1 (11.3.9) 11.3.1 Creating and Destroying Magnetic Energy Animation Let's consider the process involved in creating magnetic energy.Here we discuss this process qualitatively.A quantitative calculation is given in Section 13.6.2.Figure 11.3.1 shows the process by which an external agent(s)creates magnetic energy.Suppose we have five rings that carry a number of free positive charges that are not moving.Because there is no current,there is no magnetic field.Suppose a set of external agents come along (one for each charge)and simultaneously spin up the charges counterclockwise as seen from above,at the same time and at the same rate,in a manner that has been pre- arranged.Once the charges on the rings start to move,there is a magnetic field in the space between the rings,mostly parallel to their common axis,which is stronger inside the rings than outside.This is the solenoid configuration(see Section 9.4). Figure 11.3.1 Creating (http://youtu.be/GI2Prj4CGZI) and destroying (http://youtu.be/iesoHVflg6I )magnetic field energy As the magnetic flux through the rings grows,Faraday's law of induction tells us that there is an electric field induced by the time-changing magnetic field that is circulating clockwise as seen from above.The force on the charges due to this induced electric field is thus opposite the direction the external agents are trying to spin the rings up (counterclockwise),and thus the agents have to do additional work to spin up the charges because of their charge.This is the source of the energy that is appearing in the magnetic field between the rings-the work done by the agents against the back emf. Over the course of the "create"animation associated with Figure 11.3.1,the agents moving the charges to a higher speed against the induced electric field are continually doing work.The electromagnetic energy is being created at the place where they are 11-12
11-12 The above expression holds true even when the magnetic field is non-uniform. The result can be compared with the energy density associated with an electric field, uE = 1 2 ! 0E2 . (11.3.9) 11.3.1 Creating and Destroying Magnetic Energy Animation Let’s consider the process involved in creating magnetic energy. Here we discuss this process qualitatively. A quantitative calculation is given in Section 13.6.2. Figure 11.3.1 shows the process by which an external agent(s) creates magnetic energy. Suppose we have five rings that carry a number of free positive charges that are not moving. Because there is no current, there is no magnetic field. Suppose a set of external agents come along (one for each charge) and simultaneously spin up the charges counterclockwise as seen from above, at the same time and at the same rate, in a manner that has been prearranged. Once the charges on the rings start to move, there is a magnetic field in the space between the rings, mostly parallel to their common axis, which is stronger inside the rings than outside. This is the solenoid configuration (see Section 9.4). Figure 11.3.1 Creating (http://youtu.be/GI2Prj4CGZI) and destroying (http://youtu.be/iesoHVfIg6I ) magnetic field energy. As the magnetic flux through the rings grows, Faraday’s law of induction tells us that there is an electric field induced by the time-changing magnetic field that is circulating clockwise as seen from above. The force on the charges due to this induced electric field is thus opposite the direction the external agents are trying to spin the rings up (counterclockwise), and thus the agents have to do additional work to spin up the charges because of their charge. This is the source of the energy that is appearing in the magnetic field between the rings — the work done by the agents against the back emf. Over the course of the “create” animation associated with Figure 11.3.1, the agents moving the charges to a higher speed against the induced electric field are continually doing work. The electromagnetic energy is being created at the place where they are
doing work (the path along which the charges move)and that electromagnetic energy flows primarily inward,but also outward.The direction of the flow of this energy is shown by the animated texture patterns in Figure 11.3.1.This is the electromagnetic energy flow that increases the strength of the magnetic field in the space between the rings as each positive charge is accelerated to a higher and higher speed.When the external agents have spun the charges to a pre-determined speed,they stop the acceleration.The charges then move at a constant speed,with a constant field inside the solenoid,and zero induced electric field,in accordance with Faraday's law of induction. We also have an animation of the"destroy"process linked to Figure 11.3.1.This process proceeds as follows.Our set of external agents now simultaneously starts to spin down the moving charges (which are still moving counterclockwise as seen from above),at the same time and at the same rate,in a manner that has been pre-arranged.Once the charges on the rings start to decelerate,the magnetic field in the space between the rings starts to decrease in magnitude.As the magnetic flux through the rings decreases,Faraday's law tells us that there is now an electric field induced by the time-changing magnetic field that is circulating counterclockwise as seen from above.The force on the charges due to this electric field is thus in the same direction as the motion of the charges.In this situation the agents have work done on them as they try to spin the charges down. Over the course of the "destroy"animation associated with Figure 11.3.1,the strength of the magnetic field decreases,and this energy flows from the field back to the path along which the charges move,and is now being provided to the agents trying to spin down the moving charges.The energy provided to those agents as they destroy the magnetic field is exactly the amount of energy that they put into creating the magnetic field in the first place,neglecting radiative losses(such losses are small if we move the charges at speeds small compared to the speed of light).This is a totally reversible process if we neglect such losses.That is,the amount of energy the agents put into creating the magnetic field is exactly returned to the agents as the field is destroyed. There is one final point to be made.Whenever electromagnetic energy is being created, an electric charge is moving (or being moved)against an electric field (gv.E<0) Whenever electromagnetic energy is being destroyed,an electric charge is moving (or being moved)along an electric field (gv.E>0).This is the same rule we saw above when we were creating and destroying electric energy above. 11.4 RL Circuits 11.4.1 Self-Inductance and the Faraday's Law The addition of time-changing magnetic fields to simple circuits means that the closed line integral of the electric field around a circuit is no longer zero (Chapter 10.3). Instead,we have,for any open surface 11-13
11-13 doing work (the path along which the charges move) and that electromagnetic energy flows primarily inward, but also outward. The direction of the flow of this energy is shown by the animated texture patterns in Figure 11.3.1. This is the electromagnetic energy flow that increases the strength of the magnetic field in the space between the rings as each positive charge is accelerated to a higher and higher speed. When the external agents have spun the charges to a pre-determined speed, they stop the acceleration. The charges then move at a constant speed, with a constant field inside the solenoid, and zero induced electric field, in accordance with Faraday’s law of induction. We also have an animation of the “destroy” process linked to Figure 11.3.1. This process proceeds as follows. Our set of external agents now simultaneously starts to spin down the moving charges (which are still moving counterclockwise as seen from above), at the same time and at the same rate, in a manner that has been pre-arranged. Once the charges on the rings start to decelerate, the magnetic field in the space between the rings starts to decrease in magnitude. As the magnetic flux through the rings decreases, Faraday’s law tells us that there is now an electric field induced by the time-changing magnetic field that is circulating counterclockwise as seen from above. The force on the charges due to this electric field is thus in the same direction as the motion of the charges. In this situation the agents have work done on them as they try to spin the charges down. Over the course of the “destroy” animation associated with Figure 11.3.1, the strength of the magnetic field decreases, and this energy flows from the field back to the path along which the charges move, and is now being provided to the agents trying to spin down the moving charges. The energy provided to those agents as they destroy the magnetic field is exactly the amount of energy that they put into creating the magnetic field in the first place, neglecting radiative losses (such losses are small if we move the charges at speeds small compared to the speed of light). This is a totally reversible process if we neglect such losses. That is, the amount of energy the agents put into creating the magnetic field is exactly returned to the agents as the field is destroyed. There is one final point to be made. Whenever electromagnetic energy is being created, an electric charge is moving (or being moved) against an electric field ( q ! v ! ! E < 0 ). Whenever electromagnetic energy is being destroyed, an electric charge is moving (or being moved) along an electric field ( q ! v ! ! E > 0 ). This is the same rule we saw above when we were creating and destroying electric energy above. 11.4 RL Circuits 11.4.1 Self-Inductance and the Faraday’s Law The addition of time-changing magnetic fields to simple circuits means that the closed line integral of the electric field around a circuit is no longer zero (Chapter 10.3). Instead, we have, for any open surface
死ds=-广Ban (11.4.1) Any circuit in which the current changes with time will have time-changing magnetic fields,and therefore associated induced electric fields,which are due to the time changing currents,not to the time changing magnetic field (association is not causation). How do we solve simple circuits taking such effects into account?We discuss here a consistent way to understand the consequences of introducing time-changing magnetic fields into circuit theory--that is,self-inductance. As soon as we introduce time-changing currents,and thus time changing magnetic fields, the electric potential difference between two points in our circuit is no longer well defined.When the line integral of the electric field around a closed loop is no longer zero,the potential difference between any two points a and b,is no longer independent of the path used to get from a to b.That is,the electric field is no longer an electrostatic (conservative)field,and the electric potential is no longer an appropriate concept(that is, E can no longer be written as the negative gradient of a scalar potential).However,we can still write down in a straightforward fashion the differential equation for the current I(t)that determines the time-behavior of the current in the circuit. d R d B Switch S closed at t=0 Figure 11.4.1 One-loop inductor circuit To show how to do this,consider the circuit shown in Figure 11.4.1.We have a battery,a resistor,a switch S that is closed at t=0,and a "one-loop inductor."It will become clear what the consequences of this"inductance"are as we proceed.For t>0,current is in the direction shown(from the positive terminal of the battery to the negative,as usual). What is the equation that governs the behavior of our current I(t)for t>0? To investigate this,apply Faraday's law to the open surface bounded by our circuit,where we take dA=dAn pointing out of the plane of the Figure 11.4.1,and ds is counterclockwise.First,we would like to evaluate the left-hand-side of Eq.(11.4.1),the integral of the electric field around this circuit.There is an electric field in the battery, directed from the positive terminal to the negative terminal,and when we go through the battery in the direction of ds that we have chosen,we are moving against that electric 11-14
11-14 ! E! d ! s = " "# $ $t ! B! d ! ## A . (11.4.1) Any circuit in which the current changes with time will have time-changing magnetic fields, and therefore associated induced electric fields, which are due to the time changing currents, not to the time changing magnetic field (association is not causation). How do we solve simple circuits taking such effects into account? We discuss here a consistent way to understand the consequences of introducing time-changing magnetic fields into circuit theory--that is, self-inductance. As soon as we introduce time-changing currents, and thus time changing magnetic fields, the electric potential difference between two points in our circuit is no longer well defined. When the line integral of the electric field around a closed loop is no longer zero, the potential difference between any two points a and b, is no longer independent of the path used to get from a to b. That is, the electric field is no longer an electrostatic (conservative) field, and the electric potential is no longer an appropriate concept (that is, ! E can no longer be written as the negative gradient of a scalar potential). However, we can still write down in a straightforward fashion the differential equation for the current I(t) that determines the time-behavior of the current in the circuit. Figure 11.4.1 One-loop inductor circuit To show how to do this, consider the circuit shown in Figure 11.4.1. We have a battery, a resistor, a switch S that is closed at t = 0 , and a “one-loop inductor.” It will become clear what the consequences of this “inductance” are as we proceed. For t > 0 , current is in the direction shown (from the positive terminal of the battery to the negative, as usual). What is the equation that governs the behavior of our current I(t) for t > 0? To investigate this, apply Faraday's law to the open surface bounded by our circuit, where we take d ! A = dAnˆ pointing out of the plane of the Figure 11.4.1, and d ! s is counterclockwise. First, we would like to evaluate the left-hand-side of Eq. (11.4.1), the integral of the electric field around this circuit. There is an electric field in the battery, directed from the positive terminal to the negative terminal, and when we go through the battery in the direction of d ! s that we have chosen, we are moving against that electric
field,so the contribution of the battery to our integral is negative and equal to the negative of the emf provided by the battery, 「Eds=-e. battery Then,there is an electric field in the resistor,in the direction of the current,so when we move through the resistor in that direction,the contribution to our integral is positive, E.ds=IR external cIrcwit What about when we move through our one-loop inductor?There is no electric field in this loop if the resistance of the wire making up the loop is zero.Thus,going around the closed loop counterclockwise in the direction of the current,we have E.ds=-e+IR (11.4.2) What is the right-hand-side of Eq.(11.4.1)?Because we have assumed in this section that the circuit is not moving,we can take the partial with respect to time outside of the surface integral and then we simply have the time derivative the magnetic flux through the loop.What is the magnetic flux through the open surface?First of all,we arrange the geometry so that the part of the circuit that includes the battery,the switch,and the resistor makes only a small contribution to e as compared to the (much larger in area) part of the open surface that constitutes our "one-loop inductor".Second,we know that the sign of the magnetic flux is positive in that part of the surface,because current in the counterclockwise direction will produce a magnetic field B pointing out of the plane of Figure 11.4.1,which is the same direction we have assumed for dA,so that B.dA is positive.Note that our magnetic field here is the self-magnetic field-that is the magnetic field produced by the current in the circuit,and not by currents external to this circuit. We also know that at any point in space,B is proportional to the current I,since it can be computed from the Biot-Savart Law,that is, B(c,)=,106×f-) D (11.4.3) 4π (- You may immediately object that the Biot-Savart Law is only good in time-independent situations,but in fact,as long as the current is varying on time scales T long compared to the speed of light travel time across the circuit and we are within a distance cT of the currents,then (11.4.3)is an excellent approximation to the time dependent magnet field. If we look at (11.4.3),although for a general point in space it involves a very complicated 11-15
11-15 field, so the contribution of the battery to our integral is negative and equal to the negative of the emf provided by the battery, ! E! d ! s battery " = #$ . Then, there is an electric field in the resistor, in the direction of the current, so when we move through the resistor in that direction, the contribution to our integral is positive, ! E! d ! s external circuit " = IR . What about when we move through our one-loop inductor? There is no electric field in this loop if the resistance of the wire making up the loop is zero. Thus, going around the closed loop counterclockwise in the direction of the current, we have ! E! d ! s = " "# $ + IR . (11.4.2) What is the right-hand-side of Eq. (11.4.1)? Because we have assumed in this section that the circuit is not moving, we can take the partial with respect to time outside of the surface integral and then we simply have the time derivative the magnetic flux through the loop. What is the magnetic flux through the open surface? First of all, we arrange the geometry so that the part of the circuit that includes the battery, the switch, and the resistor makes only a small contribution to !B as compared to the (much larger in area) part of the open surface that constitutes our “one-loop inductor”. Second, we know that the sign of the magnetic flux is positive in that part of the surface, because current in the counterclockwise direction will produce a magnetic field ! B pointing out of the plane of Figure 11.4.1, which is the same direction we have assumed for d ! A , so that ! B! d ! A is positive. Note that our magnetic field here is the self-magnetic field—that is the magnetic field produced by the current in the circuit, and not by currents external to this circuit. We also know that at any point in space, ! B is proportional to the current I, since it can be computed from the Biot-Savart Law, that is, ! B( ! r,t) = µo I(t) 4! d ! s" # ! r $ " ! ( r ) ! r $ " ! ( r ) "% 3 (11.4.3) You may immediately object that the Biot-Savart Law is only good in time-independent situations, but in fact, as long as the current is varying on time scales T long compared to the speed of light travel time across the circuit and we are within a distance cT of the currents, then (11.4.3) is an excellent approximation to the time dependent magnet field. If we look at (11.4.3), although for a general point in space it involves a very complicated