week2 - 1C second week notes - Gekelman Physics 1C notes...

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1C second week notes - Gekelman 1 Physics 1C notes© Week 2 Prof. Walter Gekelman Faraday’s Law (1) ! = " d # B dt = " d dt ! B i ˆ ndA $ ! B is the magnetic flux. The first term is the total time derivative of the magnetic flux and the second is the same (the integral is simply the definition of magnetic flux). The flux can change if any or all of three things happen 1) The magnetic field changes in time 2) The angle between the magnetic field and the element of area changes in time 3) The area through which the flux goes through changes in time The EMF or E as you recall is (2) = ! E i d ! l " Faraday’s Law can be written in differential form ! E d ! l " ! = area ! " # ! E ( ) ˆ n dA = $ % % t ! B ! ˆ n dA = EMF then : " # ! E ( ) = $ % B % t Lets examinehow the Emf is formed when we consider item (3) the area changes:
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1C second week notes - Gekelman 2 The top and bottom lines are fixed rails that can conduct electricity. The magnetic field points into the page. A conducting bar is forced to slide along the tracks in the x direction. Let us use Faraday’s law to find the changing flux. ! " " t ! B ˆ n dA area # = ! "$ B " t . Since B is constant d ! B dt = B d dt dA = " Bl dx dt = Bl v . Now what is the current through the resistor and what is the direction of the current. To find the current use Ohms law ! = iR i = Bl v R The current flows in the counterclockwise direction, why? Now let us examine the situation from the point of view (so to speak) of a moving charge in a magnetic field. The charge experiences a force ! F = q ! v ! ! B at right angles to B and v. In the frame of the charge the force could just as well be due to an electric field in which case ! F = q ! E . In either case the force gives rise to an acceleration. If in this though experiment we equate them then the effective electric field the charge thinks its in is ! E = ! v ! ! B . If the charge moves a distance dl then an EMF is generated since = ! E i d ! l " and therefore over a finite path (3) = ! v " ! B ( ) i d ! l # Going back to equation (1) and the production of EMF from changing flux we have several cases: Case (1) the magnetic field changes in time Consider the solenoid shown below with a current in the windings. If we neglect end effects the magnetic field is given by Amperes law as: B = μ 0 ni therefore ! B = BA = 0 niA and, = " d # B dt = " 0 nA di dt B !
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1C second week notes - Gekelman 3 Consider a problem of the evaluation of magnetic flux with a solenoid. This is 29.59. We have a very long wire with a current flowing uniformly distributed across the cross
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This note was uploaded on 01/21/2010 for the course PHYS 1C taught by Professor Whitten during the Winter '07 term at UCLA.

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week2 - 1C second week notes - Gekelman Physics 1C notes...

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