UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 16 Prof. Steven Errede LECTURE NOTES 16 THE MAGNETIC VECTOR POTENTIAL Ar We saw in electrostatics that E 0 {always} (due to intrinsic / microscopic nature of the electrostatic field) permitted us to introduce a scalar potential Vr such that: Er Vr {n.b. Vr is uniquely defined, up to an (arbitrary) constant.} Analogously, in magnetostatics, the Br 0 (always) {no magnetic charges / no magnetic monopoles} permits us to introduce a magnetic vector potential Ar such that: B rAr S.I. units of the magnetic vector potential Ar = Tesla-meters Teslas 1 m Tesla- Meters Then: Br Ar 0 {always} The divergence of a curl of a vector field Fr is always zero Ampere’s Law: 2 In differential form: BrArArArJr o free Now, just as in the case of electrostatics, where Vr was uniquely defined up to an arbitrary constant Vo , then let: Vr Vr Vo 0 then: Er V r Vr Voo Vr V Vr i.e. Er V r Vr An analogous thing occurs in magnetostatics - we can add / we have the freedom to add to the magnetic vector potential Ar the gradient of any scalar function rrm where 2 m r magnetic scalar potential SI Units of magnetic scalar potential m r Tesla-m Then: Ar Ar r Ar m r Formally known as a Gauge Transformation The curl of the gradient of a scalar field (m r here) automatically/always vanishes, i.e.: 0Always !!! BrArArrArrArr m Ar © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 1 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 16 Prof. Steven Errede Note that the magnetic scalar potential m r has same physical units as magnetic flux m : Tesla-m2 = Weber (Magnetic flux, BrdA !!) eeek!!!! m S Please do NOT confuse the magnetic scalar potential m r (= a scalar point function, whose value can change at each/every point in space, r ) with the magnetic fluxm (which is a constant scalar quantity (i.e. a pure number), independent of position) mmr !!! Thus, like the scalar potential Vr, the magnetic vector potential Ar is (also) uniquely defined, but only up to an (arbitrary) vector function rrm . Ar Ar r Ar m r The definition BrAr specifies the curl of Ar, but in order to fully specify the vector field Ar, we additionally need to specify the divergence of Ar, Ar. We can exploit the freedom of the definition of Ar to eliminate the divergence of Ar - i.e. a specific choice of Ar will make Ar divergenceless: Ar 0 Coulomb Gauge If: Ar Ar r Ar m r 2 Then: A r Ar r Ar mm r Ar r While the original magnetic vector potential, Aris not/may not be divergenceless, we can make Ar Ar r Ar m r divergenceless, i.e. Ar 0 if we chose 2 rrm such that rrArm Coulomb Gauge A Simple Illustrative Example: Suppose a region of space that has a uniform/constant magnetic field, e.g. BrBz o ˆ . Ary Arx Then: BrBzAro ˆˆ z. xy Thus (here): Ar A r xˆˆ A r y A r zArxAryˆ ˆˆ xy z xy 0 1 1 11 If Ax rBy o and AyorBx , then Ar Byxˆˆ Bxy, and thus we see that this 2 2 22oo choice of magnetic vector potential indeed gives us the correct B -field: Ar y Arx 11 BrAr zBBBzˆˆ 22ooo xy 2 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 16 Prof. Steven Errede 1 1 A Ay A By Bx 0 Is A 0 satisfied? A x z 2 o 2 o 0 Yes!!! xyz x y z Note that we could also have instead chosen/used a different magnetic vector potential: Ar Ar r Ar m rwhere e.g. rrAmo , i.e. where Ao is any (arbitrary) constant vector, AAxAyAzooxoyozˆˆˆ . Since (here) rrAmo , then rrAAxAyAzm o oxˆˆ oy oz ˆ means that the gradient of the magnetic scalar Ax Ayoy Az potential (here) is:rxyzAxAyAzAr oxˆˆ oz ˆˆ ˆˆ moxoyozoxyz and thus the magnetic scalar potential itself (here) is: moxoyozrAxxAyyAzzˆˆ ˆ . Thus, here for the case of a constant/uniform magnetic field BrBz o ˆ we see that there is in fact a continuum of allowed magnetic vector potentials Ar Ar Aom Ar r that simultaneously satisfy BrBzArˆ and Ar 0 with the addition of an o (arbitrary) constant magnetic vector potential AAxAyAzooxoyozˆˆˆ contribution with corresponding magnetic scalar potential moxoyozrAxxAyyAzzˆˆ ˆ . Note that this is exactly analogous to the situation in electrostatics where the scalar electric potential Vris unique, up to an arbitrary constant, Vo because there exists no absolute voltage reference in our universe – i.e. absolute measurements of the scalar electric potential are meaningless - only potential differences have physical significance!!! We used this simple example of the constant/uniform magnetic field BrBz o ˆ to elucidate this particular aspect of the magnetic vector potential Ar. Here in this particular example, we found that the addition of an arbitrary constant vector rAAxAyAzooxoyozmˆˆ ˆ rto the magnetic vector potential Ar was allowed, i.e. Ar Ar r Ar Ao , which leaves the magnetic field Br unchanged. In general there are many instances involving more complicated physics situations, where Br constant vector field, where indeed BrAr and Ar 0 are simultaneously satisfied for Ar Ar r, because it is possible to determine/find a 2 corresponding magnetic scalar potential m r for the problem satisfyingm rAr , but it is (very) important to understand that, in general, the allowed rrm (very likely) may not be simply a constant vector field, but indeed one which varies in space (i.e. with position vector, r )! Here again, however, the new Ar Ar rwill also be such that BrAr will be unchanged, exactly analogous to Vr Vr Vo leaving Er V runchanged. © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 3 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 16 Prof. Steven Errede 2 So we see that if m rAr then yes, Ar 0 . It is always possible to find an rrm in order to make Ar 0 . Note however that this situation is then formally mathematically identical to Poisson’s Equation, for the magnetic scalar potential m r because: 2 2 Tot r mmrr Analogous to Vr in electrostatics!!! 0 Equivalent magnetic Physically, m r could e.g. be due to bound effective volume charge densit y magnetic charges associated with a magnetic material… If we assume that the equivalent magnetic volume charge density, m r 0 and we want Ar 0 2 Then: Ar r Ar m r 0 Or: Arm r 0 Arm r Then, the solution to Poisson’s equation for the magnetic scalar potential m r is of the form: 1 r 1 r rdm Analogous to Vr Tot d in electrostatics m v v 4 r 4 0 r with rrr (n.b. these two relations are both valid assuming that m rr and ToT vanish when r !) So then if m rAr , and m rAr vanishes as r , then the magnetic scalar potential m r is given by: 11 rAr rdm d m 44vvrr {Note that if Arm r does not go to zero at infinity, then we’ll have to use some other means in order to obtain an appropriate m r , e.g. in an analogous manner to that which we’ve had to do for the (electric) scalar potential Vrassociated with problems that have electric charge distributions extending out to infinity.} Thus, this choice of m r ensures that indeed we can always make the magnetic vector potential Ardivergenceless, i.e. the conditionAr 0 (Coulomb Gauge) can always be met, for the case of magnetostatics. Note that if m r 0 then m rAr 0 . 4 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 16 Prof. Steven Errede With the choice of the magnetic scalar potential: 1 r rdm and Ar Ar A Ar r, and Ar 0 m 4 v r om Then Ampere’s Law (in differential form) becomes: B rArArArJr 2 0 free 0 2 Which gives: Ar0 Jfree r Vector form of Poisson’s equation for magnetostatics. 2 Arxoxfree J r The three separate/independent scalar forms 2 i.e.: Aryoyfree J r of Poisson's equation are connected by: 2 JrJ rxJryJrzˆˆ ˆ Arzozfree J r free x free y free z free 22 2 2 n.b. in Cartesian coordinates: Ar Arxyz xˆˆ A r y Ar zˆ However, in curvilinear coordinates (i.e. spherical-polar or cylindrical coordinates) rxyzˆˆˆsin cos sin sin cos ˆ e.g. spherical-polar coordinates: ˆ cos cosxˆˆ cos sinyz sin ˆ ˆ ˆˆ sinxy cos Note that the unit vectors rˆ,ˆ ˆ for spherical-polar coordinates are in fact explicit functions of the vector position, r i.e. rrrˆˆ, ˆˆ r and ˆˆ r and therefore rˆ,ˆ ˆ must also be explicitly differentiated in calculating the Laplacian 2 of a vector function (here, Ar) in curvilinear (i.e. either spherical-polar and/or cylindrical) coordinates!!! This is extremely important to keep in mind, for the future… The safest way to calculate the Laplacian of a vector function 2 Ar in terms of curvilinear coordinates, is to NOT use curvilinear coordinates!!! Failing that, then one should use: 2 Ar Ar Ar Ar 0 in the Coulomb Gauge If m r 0 then (automatically) m rAr 0 and we can use Ardirectly.