10.1 The Lorentz force law - MIT

1 Scott Hughes10 March 2005 Massachusetts Institute of TechnologyDepartment of Spring 2004 Lecture 10:Magnetic force ; Magnetic fields; Ampere s The Lorentz force lawUntil now, we have been concerned with electrostatics the forces generated by and actingupon charges at rest. We now begin to consider how things changewhen charges are simple apparatus demonstrates that something wierd happens when charges are inmotion: If we run currents next to one another in parallel, wefind that they areattractedwhen the currents run in the same direction; they arerepulsedwhen the currents run inopposite directions. This is despite the fact the wires arecompletely neutral: if we put astationary test charge near the wires, it feels no 1: Left: parallel currents attract. Right: Anti-parallel currents , experiments show that the force is proportional to the currents double thecurrent inoneof the wires, and you double the force .

2 Double the current in both wires, andyou quadruple the will deviate a bit from Purcell s approach at this point. In particular,we will defer our discussion ofspecial relativity til next all indicates a force that is proportional to the velocity of a moving charge; and,that points in a directionperpendicularto the velocity. These conditions are screaming fora force that depends on a cross we say is that some kind of field~B the magnetic field arises from thecurrent. (We ll talk about this in detail very soon; for the time being, just accept this.) Thedirection of this field is kind of odd: it wraps around the current in a circular fashion, witha direction that is defined by the right-hand rule: We point our right thumb in the directionof the current, and our fingers curl in the same sense as the magnetic this sense of the magnetic field defined, the force that arises when a charge movesthrough this field is given by~F=q~vc ~B ,wherecis the speed of light.

3 The appearance ofcin this force law is a hint that specialrelativity plays an important role in these we have both electric and magnetic fields, thetotalforce that acts on a charge is ofcourse given by~F=q(~E+~vc ~B).This combined force law is known as the Lorentz UnitsThe magnetic force law we ve given is of course in cgs units, in keeping with Purcell s magnetic force equation itself takes a slightly different form in SI units: we do not includethe factor of 1/c, instead writing the force ~F=q~v ~B .90 This is a very important difference! It makes comparing magnetic effects between SI and cgsunits slightly that, in cgs units, the magnetic field has the same overalldimension as the electricfield:~vandcare in the same units, so~Bmust be force /charge. For historical reasons, thiscombination is given a special name: 1 dyne/esu equals 1 Gauss (1G) when the force inquestion is magnetic.

4 (There is no special name for this combination when the force iselectric.)In SI units, the magnetic field doesnothave the same dimension as the electric field:~Bmust be force /(velocity charge). The SI unit of magnetic field is called the Tesla (T): theTesla equals a Newton/(coulomb meter/sec).To convert: 1 T = Consequences of magnetic forceSuppose I shoot a charge into a region filled with a uniform magnetic field:BvThe magnetic field~Bpoints out of the page; the velocity~vinitially points to the right. Whatmotion results from the magnetic force ?At every instant, the magnetic force points perpendicular tothe charge s velocity exactly the force needed to cause circular motion. It is easy tofind the radius of this motion:if the particle has chargeqand massm, thenFmag=FcentripetalqvBc=mv2RR= the chargeqis positive, the particle s trajectory veers to the right, vice versa if its kind of qualitative behavior bending the motion of charges along a curve is typicalof magnetic that magnetic forces do no work on moving charges: if we imagine the chargemoves for a timedt, the work that is done isdW=~F d~s=~F ~v dt=q(~vc ~B) ~v dt= zero follows from the fact that~v ~Bis perpendicular to~ force on a currentSince a current consists of a stream of freely moving charges, a magnetic field will exert aforce upon any flowing current.

5 We can work out this force fromthe general magnetic a currentIthat flows down a wire. This current consists of some linear densityof freely flowing charges, , moving with velocity~v. (The direction of the charges motionis defined by the wire: they are constrained by the wire s geometry to flow in the directionit points.) Look at a little differential lengthdlof this wire (a vector, since the wire definesthe direction of current flow).The amount of charge contained in this differential length isdq= dl. The differentialof force exerted on this piece of the wire is thend~F= ( dl)~vc ~B .There are two equivalent ways to rewrite this in terms of the current. First, because thecurrent is effectively a vector by virtue of the velocity of its constituent charges, we put~I= ~vand findd~F=dl~Ic ~B .Second, we can take the current to be a scalar, and use the geometry of the wire to definethe vector:d~F=Icd~l ~B.

6 These two formulas are completely equivalent to one s focus on the secondversion. The total force is given by integrating:~F=Ic d~l ~B .If we have a long, straight wire whose length isLand is oriented in the ndirection, we find~F=ILc n ~B .This formula is often written in terms of the force per unit length:~F /L= (I/c) n ~ Ampere s lawWe ve talked about the force that a magnetic field exerts on charges and current; but, wehave not yet said anything about where this field comes from. I will now give, withoutanyproof or motivation, a few key results that allow us to determine the magnetic field in main result we need isAmpere s law: C~B d~s=4 words, if we take the line integral of the magnetic field around a closed path, it equals4 /ctimes the current enclosed by the s law plays a role for magnetic fields that is similar to that played by Gauss s lawfor electric fields.

7 In particular, we can use it to calculate the magnetic field in situations thatare sufficiently symmetric. An important example is the magneticfield of a long, straightwire: In this situation, the magnetic field must be constant on any circular path around thewire. The amount of current enclosed by this path is justI, the current flowing in the wire: ~B d~s=B(r)2 r=4 cI B(r) = magnetic field from a current thus falls off as 1/r. Recall that we saw a similar 1/rlawnot so long ago the electric field of a long line charge also falls off as 1/r. As we ll seefairly soon, this is not a direction of this field is in a circulational sense the~Bfield winds around thewire according to the right-hand rule2. This direction is often written , the direction of2In principle, we could have defined it using a left-hand rule.

8 This would give a fully consistentdescription of physics provided we switched the order of all cross products (which is identical to switchingthe sign of all cross products).93increasing polar angle . The full vector magnetic field is thus written~B=2 Icr . Field of a plane of currentThe magnetic field of the long wire can be used to derive one moreimportant result. Supposewe take a whole bunch of wires and lay them next to each other:Lx yThe current in each wire is taken to go into the page. Suppose that the total amount ofcurrent flowing inallof the wires isI, so that the currentper unit lengthisK=I/L. Whatis the magnetic field at a distanceyabove the center of the plane?This is fairly simple to work out using superposition. First, fromthe symmetry, youshould be able to see that only the horizontal component of the magnetic field (pointingto the right) will survive.

9 For the vertical components, therewill be equal and oppositecontributions from wires left and right of the center. To sum what s left, we set up anintegral:~B=2c x L/2 L/2(I/L)dx x2+y2cos .In the numerator under the integral, we are using (I/L)dxas the current carried by a wire of widthdx. Doing the trigonometry, we replace cos with something a little more useful:~B=2c x L/2 L/2(I/L)dx x2+y2y x2+y2=2 Kyc x L/2 L/2dxx2+ integral is doable, but not particularly pretty (you end up with a mess involving arc-tangents). A more tractable form is obtaining by taking the limit ofL : using dxx2+y2= |y|,we find~B= x2 Kyc|y|94= + x2 Kcy >0 (above the plane)= x2 Kcy <0 (below the plane)The most important thing to note here is the change as we cross the sheet of current:| ~B|=4 this remind you of anything? It should! When we cross a sheetofchargewe have| ~E|= 4.

10 The sheet of current plays a role in magnetic fields very similarto that played by the sheetof charge for electric force between two wiresCombining the result for the magnetic field from a wire with currentI1with the force perunit length upon a long wire with currentI2tells us the force per unit length that arisesbetween two wires:|~F|L= right-hand rule, you should be able to convince yourself quite easily that this force isattractive when the currents flow in the same direction, and isrepulsive when they flow inopposite SI unitsIn SI units, Ampere s law takes the form C~B d~S= 0 Ienclwhere the constant 0= 4 10 7 Newtons/amp2is called the magnetic permeability offree space . To convert any cgs formula for magnetic field to SI, multiply by 0 (c/4 ).For example, the magnetic field of a wire becomes~B= 0I2 r .The force between two wires becomes|~F|L= 0I1I22 you try to reproduce this force formula, remember that themagnetic force in SI units doesnot have the factor 1 Divergence of the~BfieldLet s take the divergence of straight wire s magnetic field using Cartesian coordinates.