no tagsLecture 016: The Magnetic Field due to Moving ChargeSteveSekula, 29 March 2011 (created 22 March 2011)What is Magnetism?!It was Hans Christian Oersted who is credited with setting physics andchemistry on the path to an understanding of magnetism. While magnetismhas been observed since about 500-600 BC (in the Western world, it wasAristotle who gave credit for the discovery of this phenomenon to a mannamed Thales), it was not at all understood until the 1800s. Oerstedaccidentally observed that an electric current caused a compass needle todeflect. We can easily reproduce this experiment. Magnetism, therefore,seems to have something to do with the MOTION of electric charge. Notlong after Oersted's publicized observation, two French scientists - BaptisteBiot and Felix Savart - performed experiments and determined the exactform of the force law for a steady current. We call this the Biot-Savart Law,and we'll explore it now.The Biot-Savart law considers a steady current moving through a conductor.There are similarities and differences between B-S and Coulomb's Law.Let's write Coulomb's Law for a small piece of a distribution of charge:dE rDraw a picture representing the situation that can be described byCoulomb's Law (a blob of charge, considering the electric field due to apiece of the blob).Now draw a picture representing the situation we want to analyze inmagnetic fields. We want to know the field, , at a point P some distance, ,from a part of the conductor ( ) carrying a steady current . Ourconvention will again be that points from the conductor element to thepoint P.General Physics - E&M (PHY 1308) LectureNotesGeneral Physics - E&M (PHY 1308) LectureNotes~=14ÙÏ0r2dq^B ~r dL ~I r ^General Physics - E&M (PHY 1308) - Lecture Notes file:///home/sekula/Dropbox/Documents/Notebook...1 of 8 03/29/2011 11:35 AMNow let's think about the differences between the static charge situationconsidered by Coulomb's Law and the moving charge situation in B-S Law:In the BSL, we are considering a current element, , which is a vectorquantity (it flows along a direction in the conductor). In CL, we wereconsidering a piece of charge which had no direction. It was just anumber.In the BSL, the source of magnetic field is a VECTOR quantity - currenttimes . In the CL, the source of electric field is a scalar quantity -charge. We have to account for direction of motion in the BSL.In the BSL, the field contribution of depends on the orientiation ofthe conductor to the unit vector - it depends on the sine of the angle,specifically. In CL, we had no such oddity.The B-S Law, which describes all of our observations, is as follows:Here, we have a new constant that has been determined from experimentalmeasurement: , the permeability constant. It's EXACT value is. Equivalent units are often used: .There is one other important distinction between the BSL and CL. The CLgives us the electric field in terms of isolated charge elements. But it'simpossible to talk about an isolated current element, because it necessarilymust be part of a circuit. In order to get the total magnetic field, you have tointegrate around the entire circuit to get the magnetic field at point P.Because magnetic field is a vector, it obeys the superposition principle sowe just have to add up all the current elements:The above is the integral form of the BSL. The magnetic field thus dependson the details of the current distribution. Generally speaking, though, thecross-product in this law tells us that magnetic field lines encircle the pathof the current perpendicular to its direction. Here we have another versionIdL ~dL ~IdL ~dB ~=Ö04Ùr2IdL~Â r^Ö 04Ù 0 N=A Â 172T =A Á mB B ~=Zd~=Ö04ÙZr2IdL~Â r^General Physics - E&M (PHY 1308) - Lecture Notes file:///home/sekula/Dropbox/Documents/Notebook...2 of 8 03/29/2011 11:35 AMof the right-hand rule:To determine the direction of magnetic fields around a conductor, pointyour thumb in the direction of current. The direction your fingers wouldcurl around the conductor indicates the direction of magnetic field lines.Example: magnetic field around a straight conducting wireConsider an infinitely long straight wire carrying a steady current I. Findthe magnetic field at a point P which lies a distance above the wire.Draw the wired and setup the problem. Use the new right-hand rule tonote which direction (into or out of the board) we expect the field topoint.Begin by writing the BSL:Let's choose CARTESIAN coordinates since we have all these handystraight lines in the problem and because whatever the distance ofpoint P above the line, it's fixed no matter where along the conductorwe are considering.What are the unknowns?We need to sort out expressions for , , and the cross-product in termsof the geometry (coordinates) of the problemWe need an expression for in terms of geometry.Let's attack pieces. What are they? They are: (1) the distance, , (2) therelationship between and the geometry of the coordinate system, (3) thedirection of the magnetic field due to , and (4) the magnitude of.y B ~=Ö04ÙZr2IdL~Â r^y r ^ dL ~r 2r dL dL ^Â r^IdL ~Â r^General Physics - E&M (PHY 1308) - Lecture Notes file:///home/sekula/Dropbox/Documents/Notebook...3 of 8 03/29/2011 11:35 AMor is the easiest: it's just Before trying to compute the cross-product from its pieces, let's thinkabout it a bit:Can we figure out the unit vector for ? Both and lie in theplane of the board. Therefore, must point into or out of the board.From the R-H rule, we expect it to point OUT - always perpendicularto both and . There - we've figured out the direction of the cross-product without multiplying a single thing.We can simplify the cross product magnitude as a result of theprevious observation, since we know it points out of the board. Thus where is the angle between and .From the geometry of the problem, we know from trigonometrythat .What about ? Well, the way we've setup the problem, that's just .We have all the pieces. Let's re-write the BSL:We can pull out of the integral since we're not integrating it, and we're leftwith:What have we learned? We have learned that like the electric field froma line of static charge, the magnetic field from a line of moving chargefalls of linearly with distance from the wire. But where the electric fieldlines point OUTWARD from the line, the magnetic field circles AROUNDthe line.Magnetic attraction of two wiresr
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