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PES 1120 Spring 2014, Spendier Lecture 33/Page 1 Today: continue chapter 30 - Induction and energy transfer - Eddy Currents Magnetic flux: B B dA B dA cos( ) units: [Wb] = [T m2] weber (Wb) d B (induced emf in single loop!) dt For a coil that consists of N loops, the total induced emf would be N times as large: dB ind N dt Lenz’s Law The direction of the induced current is determined by Lenz’s law: The induced magnetic field opposes the change in flux that created it Faraday’s Law : ind We now have an equation that relates a change in magnetic flux through a surface to an induced electromotive force around the surface. Thus, we see that an emf may be induced in the following ways: (i) by varying the magnitude of B with time (ii) by varying the magnitude of A , (iii) varying the angle between B and the area vector A with time Generators! Generator: Alternating current generator One of the most important applications of Faraday’s law of induction is to generators and motors. A generator converts mechanical energy into electric energy, while a motor converts electrical energy into mechanical energy. A simple generator consists of an N-turn loop rotating in a magnetic field which is assumed to be uniform. The magnetic flux varies with time, thereby inducing an emf. From Figure at the right, we see that the magnetic flux through the loop may be written as B BA cos q BA cos wt ω angular velocity The rate of change of magnetic flux is d B eind BAw sin(wt ) dt PES 1120 Spring 2014, Spendier Lecture 33/Page 2 For N turns in the loop, the total induced emf across the two ends of the loop is eind NBAw sin(wt ) For a circular coil with radius R=2.5cm and 150 turns of wire, B = 0.060 T, and coil rotating at 440 rev/min, the maximum induced emf is: If we connect the generator to a circuit which has a resistance R, then the current generated in the circuit is given by I e NBAw sin(wt ) R R Motional EMF: Energy transfer Imagine pulling a closed conducting loop out of a magnetic field at a constant velocity v . While the loop is moving, a clockwise current Iind is induced in the loop. This means that in turn the loop segments still within the magnetic field will now experience forces F1 , F2 , and F3 . Force on a wire segment: F1 I L B Pushing the loop into the field therefore requires work to be done by the arem, even to maintain constant velocity PES 1120 Spring 2014, Spendier Lecture 33/Page 3 v = constant a = 0; F Fpush F1 0 Where does this work (mechanical energy) go? It is converted to electrical energy in the loop! When the loop’s motion is stopped, what happens? The induced emf goes to zero and the induced current dies away. In this case electrical energy is converted to thermal energy due to the resistance of the wire. So we have: Mechanical energy electrical energy thermal energy (or heat) The only time we don’t have transfer to thermal energy is for materials with resistance R=0, i.e. superconductors! From the analysis above, in order for the bar to move at a constant speed, an external agent must constantly supply a force. What happens if at t = 0, the speed of the rod is v0, and the external agent stops pushing? In this case, the bar will slow down because of the magnetic force directed to the left. Using Newton’s laws, one can shoe that the speed decreases exponentially in the absence of an external agent doing work. In principle, the bar never stops moving. However, one may verify that the total distance traveled is finite. DEMO: Lenz's Law with Copper Pipe A magnet is dropped down a conducting copper pipe and feels a resistive force. The falling magnet induces a current in the copper pipe and, by Lenz's Law, the current creates a magnetic field that opposes the changing field of the falling magnet. Thus, the magnet is "repelled" and falls more slowly. Example: A metal rod is forced to move with constant velocity along two parallel metal rails, connected with a strip of metal at one end. A magnetic field of magnitude B = 0.350 T points out of the page. a) In which direction is the induced current? PES 1120 Spring 2014, Spendier Lecture 33/Page 4 b) If the rails are separated by L = 25.0 cm and the speed of the rod is 55.0 cm/s, what is the magnitude of the induced emf? c) If the rod has a resistance of 18.0 and the rails and connector have negligible resistance, what is the current in the rod? d) At what rate is energy being transferred to thermal energy? PES 1120 Spring 2014, Spendier Lecture 33/Page 5 Eddy current We have seen that when a conducting loop moves through a magnetic field, current is induced as the result of changing magnetic flux. What if instead of a loop, we move a conducting plate through the uniform magnetic field? The induced current appears to be circulating and is called an eddy current. A typical loop of eddy current is shown. eddy current loop The region of the conducting plate that is not in the field provides return conducting paths for charges displaced in the region experiencing a magnetic field. The result is a circulating eddy current in the conducting plate. By Lenz’s Law - eddy currents flow oppose the motion of metal - eddy currents act as an effective brake to its motion - mechanical work done is converted into heat inside the metal DEMO: The eddy current damping of a swinging plate (one solid metal, one with slits, one nonmetal). Braking effect - swinging a thick copper plate between poles of magnets - changes in magnetic flux linked with the plate induce large eddy current to produce braking effect on plate - plate oscillation is heavily damped There are important applications of eddy currents. For example, the currents can be used to suppress unwanted mechanical oscillations. Another application is the magnetic braking systems in high-speed transit cars. PES 1120 Spring 2014, Spendier Lecture 33/Page 6 When the magnetic flux created at the coil passes through the bottom of the pot placed on the top plate, eddy currents are generated. This eddy current then is transformed into heat by the resistant element of the pan and heats up the pan for cooking. This eddy current then is transformed into heat by the resistant element of the pan and heats up the pan for cooking. Induction cooktops use the high resistance of irom to their advantage, converting a little current into a lot of heat -- highly conductive materials like copper or aluminum would require much higher frequencies to make their small resistances add up to much heat. With this in mind, some manufacturers have begun adding an iron plate to the bottom of their nonferrous cookware to allow them to work with induction cooktops. Induced Electric fields: We will skip this section