Generating Electric Fields with Changing Magnetic Fields: Induced Voltages & Electromagnet, Exams of Physics

Download Generating Electric Fields with Changing Magnetic Fields: Induced Voltages & Electromagnet and more Exams Physics in PDF only on Docsity! CH 20 – Induced Voltages In previous sections we have seen how electric fields are produced by charges and magnetic fields are produced by currents. Electricity and magnetism were viewed independently, except for the fact that electric fields drive currents which can produce magnetic fields and forces act on charges moving in a magnetic field. However, it turns out that electric and magnetic fields are intimately related. We can generate an electric field and a current with a time-varying magnetic field. In a later chapter we will also see how a changing electric field can produce a magnetic field. Induced EMF and Magnetic Flux A magnet moved near a coil of wire will induce a potential change around the coil (an emf) and generate a current in the coil. The induced emf and current depend on the rate of the change in the magnetic flux through the coil. The definition of magnetic flux is similar to the definition of electric flux, which was discussed in chapter 15. The flux through a surface depends on the number of field lines that go through the surface. cosBAABB   unit = Tm2 = weber (Wb) B = Bcos is the component of B that is perpendicular to the area A. Faraday’s Law The induced emf in a loop depends on the rate at which the magnetic flux changes through the loop. Faraday’s law gives the emf as t N    B-E 1 where N is the number of turns of wire in the coil and t is the time during which the flux changes. Lenz’s Law The minus sign in Faraday’s law is symbolic and is determined by Lenz’s law. Lenz’s law says that the direction of the induced emf is such that any resulting induced current must be in a direction to generate a magnetic field in the loop which opposes the change in the original flux. It is important to note that it is not the magnitude or direction of the original flux that matters. It is whether this flux is increasing or decreasing and the rate at which it changes. The figure to the right illustrates how the flux in a loop can be changed by moving a bar magnet either toward or away from the loop. If there is no motion, as in (b), then there is no induced current. If you reverse the direction of the magnet, then the induced direction of the current is reversed. Example: A magnetic field points into the page, in which lies a circular coil of wire of radius 2 cm. If the field is increased from 0.5 T to 0.6 T in a time of 0.05 s. What is the average induced emf in the loop during this time? B into page and increasing Solution: Vx s Wbx t WbxmTBAABAB B ifiBfBB 3 4 42 ,, 1051.2 05.0 1026.1 1026.1)02.0()1.0(         |E|  2 Consider the circuit below, in which you have a battery connected in series with a resistor and a coil (e.g., a solenoid). When the switch (S) is closed, the current through the coil starts to increase from zero to some final steady-state value. This changing current produces a changing magnetic field and a changing magnetic flux in the coil. According to Faraday’s law, an emf will be induced in the coil whose direction will oppose the change taking place. The polarity of the emf (a potential difference between the two ends of the coil) will be as shown in the figure. It will oppose the current that the battery is attempting to deliver to the circuit and slow its increase to its final steady-state value. If the flux in each coil is B and there are N turns in the coil, then the total flux is NB. This total flux is proportional to the current in the coil. LIN B  The constant of proportionality, L, is called the ‘inductance’. I N L B   unit = Wb/A = henry (H) According to Fararday’s law and the above definition of L, the emf of the inductor is given by t I L t N BL      E There will only be an emf across the inductor when the current is changing. After it reaches a steady state, there will be no emf and the current in the above circuit will be R I f E  , where E is the battery voltage. 5 It can be shown that after the switch is closed the current increases exponentially in time, somewhat like the voltage across a capacitor as it is charged. )e( R I /t  1 E , where the time constant for energizing the inductor is Note that the time constant for the LR circuit varies inversely with R, whereas it is proportional to R for an RC circuit ( = RC). An inductor carrying a current contains energy in the form of the magnetic field. The energy can be calculated from the power delivered by the current., P = E LI. The work done in increasing the current by  I is ILItI t I LtItPW L     E The total work done to increase the current to I, which is the magnetic potential energy stored in the inductor, is 2 2 1 LIPE  The factor ½ comes from averaging the work as I increases from zero to its final value – somewhat like the ½ in the expression for the electrical energy stored in a capacitor (PE = ½ CV2) and the elastic potential energy stored in a spring (PE = ½ kx2). When you de-energize an inductor, a resistive path must exist for the current to flow to allow the energy to dissipate and the current to go to zero. The current will then decrease exponentially in time according to the equation /teII  0 6 R L  t I E/R 0.63 E/R  = L/R t I  = L/R I0 0.368 I0 Energizing an inductor De-energizing an inductor Inductance of a solenoid The inductance of a solenoid can be calculated using the previously given definition of L, I A)L/NI(N L L/NIB BA I N L B B 0 0        L AN L 2 0 (inductance of a solenoid) Example: A solenoid has 100-turns of wire, a diameter of 2 cm, and a length of 5 cm. What is its inductance? Hx. . ).())(x( L AN L 5 2272 0 1097 050 010100104     If the current in the inductor is 2 A, what is the stored energy? Jx.))(x.(LIPE 425 2 12 2 1 1058121077   If this inductor is de-energized through a 50- resistor, what is the time constant? s.sx. x. R L  611061 50 1097 6 5    7