Induced Electric Fields - Engineering Physics - Lecture Slides, Slides of Engineering Physics

Download Induced Electric Fields - Engineering Physics - Lecture Slides and more Slides Engineering Physics in PDF only on Docsity! Today’s agenda: Induced Electric Fields. You must understand how a changing magnetic flux induces an electric field, and be able to calculate induced electric fields. Eddy Currents. You must understand how induced electric fields give rise to circulating currents called ―eddy currents.‖ Displacement Current and Maxwell’s Equations. Displacement currents explain how current can flow ―through‖ a capacitor, and how a time- varying electric field can induce a magnetic field. Leftovers. Any assorted topics I need to ―cover‖ before finishing magnetism. docsity.com Time-Varying Magnetic Fields and Induced Electric Fields This suggests that a changing magnetic flux produces an electric field. This is true not just in conductors, but any- where in space where there is a changing magnetic flux. A Changing Magnetic Flux Produces an Electric Field? Bd = - N dt ε  b aV=V - V - Ed= V= - Edε= Bd - N = - Ed dt docsity.com             r We can look at work from a different point of view. The electric field exerts a force qE on the charged particle. The instantaneous displacement is always parallel to this force. Thus, the work done by the electric field in moving a charged particle once around the loop is.         W F ds q E ds=qE ds qE 2 r The sign is positive because the particle’s displacement and the force are always parallel. I E E E E ds docsity.com             r Summarizing… I E E E E ds W=qε   W=qE ds 2 r qE Bd = - dt ε Combining and generalizing…  q =qE ds=q E dsε Bd= - dt ε   BdE ds = - dt docsity.com             Generalizing still further… r I E E E E ds The loop of wire was just a convenient way for us to visualize the effect of the changing magnetic field. The electric field exists whether or not the loop is present. Was there anything in this discussion that bothered you?   BdE ds = - dt A changing magnetic flux gives rise to an electric field. docsity.com But the work done is If we tried to define a potential energy, it would not be single- valued:   W=q E ds= qE 2 r Work depends on the path!  F F I I U -U =-q E ds    B F I d U -U =-q E ds = - 0 even if I=F dt U is not single-valued! We can’t define a U for this E! (*%&^#!)             E I and F r docsity.com One or two of you might not have followed the discussion on the previous 9 slides. Did I confuse anybody? You can start taking notes again, if you want. docsity.com Induced Electric Fields: a summary of the key ideas A changing magnetic flux induces an electric field, as given by Faraday’s Law:   BdE ds = - dt This is a different kind of electric field than the one you are familiar with; it is not the electrostatic field caused by the presence of stationary charged particles. Unlike the electrostatic electric field, this ―new‖ electric field is nonconservative. C NCE=E +E ―conservative,‖ or ―Coulomb‖ ―nonconservative‖ docsity.com Example—details to be shown at the board A long thin solenoid has 500 turns per meter and a radius of 3.0 cm. The current is decreasing at a steady rate of 50 A/s. What is the magnitude of the induced electric field near the center of the solenoid 1.0 cm from the axis of the solenoid? 0 r dI E= n 2 dt -4 VE= 1.57x10 m docsity.com Some Revolutionary Applications of Faraday’s Law  Electric Guitar Pickup Coils  Magnetic Tape Readers   Phonograph Cartridges   Ground Fault Interruptors  Alternators  Generators  Transformers  Electric Motors docsity.com Application of Faraday’s Law (MAE Plasma Lab) From Meeks and Rovey, Phys. Plasmas 19, 052505 (2012); doi: 10.1063/1.4717731. Online at http://dx.doi.org/10.1063/1.4717731.T ―The theta-pinch concept is one of the most widely used inductive plasma source designs ever developed. It has established a workhorse reputation within many research circles, including thin films and material surface processing, fusion, high-power space propulsion, and academia, filling the role of not only a simply constructed plasma source but also that of a key component… ―Theta-pinch devices utilize relatively simple coil geometry to induce electromagnetic fields and create plasma… ―This process is illustrated in Figure 1(a), which shows a cut-away of typical theta-pinch operation during an initial current rise. ―FIG. 1. (a) Ideal theta-pinch field topology for an increasing current, I.‖ docsity.com Today’s agenda: Induced Electric Fields. You must understand how a changing magnetic flux induces an electric field, and be able to calculate induced electric fields. Eddy Currents. You must understand how induced electric fields give rise to circulating currents called ―eddy currents.‖ Displacement Current and Maxwell’s Equations. Displacement currents explain how current can flow ―through‖ a capacitor, and how a time- varying electric field can induce a magnetic field. Leftovers. Any assorted topics I need to ―cover‖ before Spring Break. docsity.com Eddy Currents You have seen how a changing magnetic field can induce a ―swirling‖ current in a conductor (the beginning of this lecture). These currents are called ―eddy currents.‖ If a conductor and a magnetic field are in relative motion, the magnetic force on charged particles in the conductor causes circulating currents. Eddy currents give rise to magnetic fields that oppose any external change in the magnetic field. docsity.com Eddy Currents Eddy currents are useful in generators, microphones, and roller coaster brakes (among other things). However, the I2R heating from eddy currents causes energy loss, so if you don’t want energy loss, you probably think eddy currents are ―bad.‖ docsity.com “Quiz” time (maybe for points, maybe not!) docsity.com Today’s agenda: Induced Electric Fields. You must understand how a changing magnetic flux induces an electric field, and be able to calculate induced electric fields. Eddy Currents. You must understand how induced electric fields give rise to circulating currents called ―eddy currents.‖ Displacement Current and Maxwell’s Equations. Displacement currents explain how current can flow ―through‖ a capacitor, and how a time- varying electric field can induce a magnetic field. Leftovers. Any assorted topics I need to ―cover‖ before we leave magnetism. docsity.com Displacement Current + - -q +q IC IC ds Apply Ampere’s Law to a charging capacitor. μ 0 CB ds = I docsity.com + - -q +q IC IC ds As the capacitor charges, the electric field between the plates changes. E  0ε  A q=C V = Ed d 0 0ε ε  E= EA= As the current and electric field change, the electric flux changes.    0 0ε ε   E E dq d d = = dt dt dt This term has units of current. docsity.com + - -q +q IC IC ds We define the displacement current to be E  0ε . D E d I = dt The changing electric flux through the ―bowl‖ surface is equivalent to the current IC through the flat surface. The generalized (―always correct‖) form of Ampere’s Law is then  μ μ μ ε .     E 0 C D 0 encl 0encl d B ds = I I = I dt Magnetic fields are produced by both conduction currents and time varying electric fields. docsity.com The ―stuff‖ inside the gray boxes serves as your official starting equation for the displacement current ID.  μ μ μ ε .     E 0 C D 0 encl 0encl d B ds = I I = I dt  is the relative dielectric constant; not emf. In a vacuum, replace  with 0. ID Why ―displacement?‖ If you put an insulator in between the plates of the capacitor, the atoms of the insulator are ―stretched‖ because the electric field makes the protons ―want‖ to go one way and the electrons the other. The process of ―stretching‖ the atom involves movement of charge, and therefore a current. docsity.com Today’s agenda: Induced Electric Fields. You must understand how a changing magnetic flux induces an electric field, and be able to calculate induced electric fields. Eddy Currents. You must understand how induced electric fields give rise to circulating currents called ―eddy currents.‖ Displacement Current and Maxwell’s Equations. Displacement currents explain how current can flow ―through‖ a capacitor, and how a time- varying electric field can induce a magnetic field. Leftovers. Any assorted topics I need to ―cover‖ before Exam 3. docsity.com A changing magnetic field in wire produces a current. A constant magnetic field does not. An electrical current produces a magnetic field, which by Lenz’s law, opposes the change in flux which produced the current in the first place. http://campus.murraystate.edu/tsm/tsm118/Ch7/Ch7_4/Ch7_4.htm back emf (also known as ―counter emf‖) (if time permits) docsity.com The effect is ―like‖ that of friction. The counter emf is ―like‖ friction that opposes the original change of current. Motors have many coils of wire, and thus generate a large counter emf when they are running. Good—keeps the motor from ―running away.‖ Bad—‖robs‖ you of energy. docsity.com