Lecture 10 Notes, Electromagnetic Theory II Dr. Christopher S. Bair, faculty.uml.eu/cbair University of Massachusetts Lowell 1. Pre-Einstein Relativity - Einstein i not invent the concept of relativity, but instea create a more accurate physical theory to escribe relativity. - The concept of relativity escribes how the values of physical parameters as measure in one frame of reference relate to the values as measure in another frame of reference. - The relativity between two frames that have a fixe translation, a fixe rotation, an a fixe scaling is trivial an will not be aresse here. Furthermore, the relativity that involves accelerating frames requires the introuction of inertial forces that complicate the picture, so we will not look at that type of relativity here. Instea, we look at the case of one frame moving at a constant linear velocity relative to another frame. These are calle inertial frames. - For the purpose of illustration, let us say one frame of reference is always at rest an call it the lab frame or groun frame K while the other frame of reference is moving an we call it the moving frame K'. Frame K' moves at a velocity v relative to the lab frame. Frame K has coorinates (x, y, z, t) an frame K' has coorinates (x', y', z', t'). K K' v - Relativity was first consistently evelope by Galilean-Newtonian mechanics. - Newton's secon law (F = ma) seeme to be vali in all inertial reference frames: F '(x')=m' 2 x' t ' an F (x)=m 2 x 2 t 2 - Galilean-Newtonian mechanics assume universal force (a force is the same in all inertial reference frames), universal time (all clocks run at the same spee in all inertial reference frames), an universal mass (the mass of an object is the same in all inertial reference frames): F '=F, t '=t, m '=m - If the time variables are equal, then incremental time perios in both frames must also be equal. Therefore, the total erivatives with respect to time are also equal: t '=t, t ' = t, t ' = 2 t 2, an t ' t =1
- The fact that the total erivative with respect to time is the same in all frames oes not imply that the partial erivative with respect to time is the same in all frames. The partial erivative is actually ifferent in ifferent frames as we shall shortly euce. (A more formal iscussion is in the Appenix at the en of this ocument.) - Apply these assumptions to Newton's secon law: F '=F m ' 2 x' t ' =m 2 x 2 t 2 m 2 x ' t ' =m 2 x 2 t 2 2 x ' t ' = 2 x 2 t 2 2 x' t = 2 x 2 t 2 - This equation leas irectly to Galilean relativity, or the Galilean concept of how to transform parameters from one frame to the next. - We can simplify the above equation to one imension an integrate both sies of the equation, being careful to keep integration constants: x' t = x t + A x '= x+a t+b - The constant B simply tells us where the spatial origin is set when t = 0. We can easily make B = 0 by aligning the origins of both reference frames at the same point at t = 0, leaing to: x '= x+a t - The constant A must have the imensions of a velocity, be inepenent of time an space, an not epen on any particular reference frame. The only thing we know of in this problem that matches these criteria is the velocity v of the moving frame with respect to the lab frame. Setting A = -v (negative because frame K is moving in the negative x irection relative to frame K'), we have: x '=x v t an t '=t Galilean Relativity - These equations seem so intuitive that there is a anger of thinking they exist on their own an o not nee to be teste.
- We have shown that Galilean relativity is a irect consequence of the following concepts: 1. The laws of mechanics are the same in all inertial frames 2. Newton's secon law is a correct law of mechanics 3. Force is universal 4. Time is universal 5. Mass is universal - The principle of Galilean relativity also leas to other statements. - The length of a moving object as measure in the rest frame is efine as the ifference of two enpoints in the rest frame measure at the same time, which we can transform to a new frame: L '=x 2 ' x 1 ' L '= x 2 vt x 1 v t L '=x 2 x 1 L '= L - Galilean relativity therefore leas to the statement that spatial lengths are universal. - Furthermore, any combination of lengths must therefore be universal in Galilean relativity, such as areas an volumes: A '= A, V '=V - If we take the first time erivative of the Galilean relativity principle, we get: x '=x v t x' t = x t v x' t ' = x t v because t ' = t u '=u v or u=u '+v Galilean Velocity Aition Formula Here u is the velocity of an object as measure in frame K, u' is the velocity of the object as measure in frame K', an v is the velocity of frame K' relative to frame K. The simple linear aition of velocities is one of the most seemingly intuitive parts of pre-einstein relativity. - For example, if I rie in a car at 60 mph an throw a baseball forwar at 100 mph, accoring to Galilean relativity, the ball is traveling 160 mph relative to the groun. - The way that a partial erivative transforms uner Galilean relativity can be foun by expaning the partial erivative: x ' = x x ' x + y x' y + z x' z + t x ' t
which leas to: x ' = x - Doing similar expansions for the other spatial coorinates leas to: y ' = y an z ' = z - Putting these pieces together, we can also show that the graient an Laplacian operator are the same in all inertial reference frames accoring to Galilean relativity: '= an ' 2 = 2 - Make an expansion of the partial erivative with respect to time: t ' = x t ' x + y t ' y + z t ' t ' =v x x +v y y +v z z + t t ' =v + t z + t t ' t This is a very ifferent statement from t ' = t. 2. Galilean Relativity Transformations for Electric an Magnetic Fiels - When you transform an equation to another inertial frame, you must transform everything, incluing the fiels, the sources, an the operators. - We state up front that Maxwell's equations o not obey Galilean relativity, but rather obey Special Relativity. Therefore, it is not immeiately clear how to apply a Galilean transformation to the fiels an sources since the whole approach is wrong from the start. - However, we can get some guiance by requiring Galilean relativity to be the low-spee limit of Special Relativity. Therefore, the Galilean-invariant electromagnetic equations shoul be the low-spee limit of Maxwell's equations. - There are two ifferent, vali, low-spee limits to Maxwell's equations: Electro-Quasi-Statics (EQS) an Magneto-Quasi-Statics (EQS). There are therefore two ifferent, vali sets of Galilean transformations for the electromagnetic fiels an sources. - In the EQS formulation, the charge effects ominate over the current effects. Therefore, the charge ensity is the same in all reference frames. The effect of a frame transformation freezing a current ensity into a charge ensity is consiere small enough in EQS to be ignore. ρ'=ρ (EQS Galilean relativity) - A current ensity J is just a charge ensity ρ moving at some velocity u, J = ρu. Therefore, the
EQS Galilean transformation for current ensity can be foun by using the Galilean velocity aition formula: J '=ρ ' u' J '=ρ u ' J '=ρ(u v) J '=J ρ v (EQS Galilean relativity) - In the MQS formulation, the current effects ominate over the charge effects. Therefore, the current ensity is the same in all reference frames. The effect of a frame transformation turning a charge into a current is consiere small enough in MQS to be ignore. J '=J (MQS Galilean relativity) - Since currents ominate in MQS, we can't ignore the possibility that a frame transformation will catch up with a current ensity an turn part of it into a charge ensity, leaing to: ρ'=ρ 1 c 2 v J (MQS Galilean relativity) - We have therefore foun how the electromagnetic sources transform in the two ifferent Galilean relativity possibilities. - We now nee to fin how the fiels transform in the two cases. Galilean relativity assumes forces are universal, so that the Lorentz force on the sources must be the same in all frames: F '=F (ρ ' E '+J' B ')V '= (ρ E+J B)V - In the EQS framework, the electric fiel effects are much stronger, so the magnetic fiel's contribution to the force rops out, leaving: ρ' E ' V = ρ E V ' - If we trie to keep both the electric an magnetic force, then the equations woul not obey Galilean relativity. - Since the charge ensity an the volume are the same in all frames in the EQS framework, we immeiately see that: E '=E (EQS Galilean relativity) - To see how the magnetic fiel transforms between frames in the EQS Galilean framework, we have to turn to Maxwell's equations. This is vali as long as we stick to the EQS version of Maxwell's equations (i.e. electrostatics plus the no-magnetic-monopole law plus the Ampere-
Maxwell law). - Start with the Maxwell-Ampere law in the prime frame: ' B'= 1 c 2 E ' t +μ 0 J ' - Transform everything possible into the unprime frame accoring to the Galilean rules: B'= 1 2( c v + t) E+μ 0 (J ρ v) B '= 1 c 2 (v )E+ 1 c 2 E t +μ 0 J μ 0 ρ v B '= B+ 1 c 2 (v )E μ 0 ρ v - Now consier the vector ientity: (a b)=a( b) b( a)+(b )a (a )b - Set a = v an b = E in this ientity, recognize that the frame velocity v is constant so that erivatives of it are zero, an use Gauss's law to replace the ivergence of the electric fiel with the charge ensity to fin: (v E)= 1 ϵ 0 ρ v (v )E -Multiply this by (-1/c 2 ): 1 c 2 (v E)= 1 c 2 (v )E μ 0ρ v - Insert this ientity into the transforme Maxwell-Ampere law to fin: B '= B 1 c 2 (v E) - Integrate away the curl operators everywhere to fin: B '=B 1 c 2 v E (EQS Galilean relativity) - Now we nee to know how the fiels transform in the MQS framework. Return to the statement of universal electromagnetic force: (ρ ' E '+J' B ')V '= (ρ E+J B)V - In MQS, the magnetic fiel ominates, so we rop the electric fiel's contribution to the force:
J' B ' V '= J B V - Since the current an the volume are the same in all frames in the MQS framework, this immeiately tells us that the magnetic fiel must also be the same in all frames: B '=B (MQS Galilean relativity) - To see how the electric fiel transforms between frames in the MQS Galilean framework, we have to turn to Maxwell's equations. This is vali as long as we stick to the MQS version of Maxwell's equations (i.e. magnetostatics plus Gauss's law plus Faraay's law). - Start with Faraay's law: ' E'= B ' t ' - Use all of our MQS Galilean transformation rules so far to transform to the unprime frame: E'= ( v + t ) B E'= (v )B B t E'= E (v ) B - Use the same vector ientity again: (a b)=a( b) b( a)+(b )a (a )b - This time set a = v an b = B, apply the fact that the erivative of the frame velocity is zero, an use the no-magnetic-monopole law to rop the term with the ivergence of B to fin: ( v B)= (v )B - Insert this into the transforme Faraay's law to fin: E'= E+ (v B) - Integrate away the curl operators everywhere to fin: E '=E+(v B) (MQS Galilean relativity) - In summary, the electromagnetic fiels an sources transform uner Galilean relativity accoring to the following rules:
Electro-Quasi-Statics ρ'=ρ J '=J ρ v E '=E B '=B 1 c v E 2 Magneto-Quasi-Statics ρ'=ρ 1 c v J 2 J '=J E '=E+(v B) B '=B - Note that some textbooks try to lump these two ifferent cases of electromagnetic Galilean relativity into one case an state that the transformation equations E '=E+(v B) an B '=B 1 c 2 v E can be use at the same time. This is incorrect. These two equations o not simultaneously obey the Galilean relativity requirements. Furthermore, these two equations o not even obey a basic requirement of any relativity system: the group law (i.e. two successive frame transformations shoul be equivalent to a single larger frame transformation). For further information, see M. Le Bellac an J. M. Levy-Leblon, Galilean Electromagnetism, Il Nuovo Cimento. 14, 217-233 (1973). 3. Galilean Relativity Applie to Maxwell's Equations - Now that we know how the fiels, sources, an operators transform uner Galilean relativity, we can see whether Maxwell's equations in their full form obey Galilean relativity. - In the EQS framework, we alreay know that the Ampere-Maxwell law an Gauss's law obey Galilean relativity since we specifically constructe the transformations to obey these laws. - We now test whether Faraay's law obeys EQS Galilean relativity: ' E'= B ' t ' - Apply the EQS Galilean transformation rules to fin: E= (v + t )(B 1 c 2 v E) E= B t (v )B+ 1 c (v )v E+ 1 2 c v E 2 t - This equation oes not have the same form as Faraay's law. Therefore, Faraay's law oes not obey Galilean relativity in the EQS approach. - Now test the no-magnetic-monopole law for EQS Galilean relativity: ' B '=0 (B 1 c 2 v E)=0
B= 1 c 2 (v E) Using a vector ientity on the right-han sie, we fin: B= 1 c 2 v ( E) - Using Faraay's law, we replace the curl of the electric fiel on the right to fin: B= 1 c 2 v ( B t ) - In orer to have the same form as the stanar no-magnetic-monpole law, the term on the right woul have to be zero. Therefore, the no-magnetic-monpole law also oes not obey Galilean relativity in the EQS approach. - The part that breaks Galilean relativity in both Faraay's law an the no-magnetic-monopole law is the time erivative of B. This is exactly the piece that is thrown away to reuce Maxwell's equations to the EQS approximation. - Therefore, whenever the magnetic fiel of a system changes so slowly in time that its time erivative can be approximate to be zero, Special Relativity effects go away an Maxwell's equations reuce own to the equations of electro-quasi-statics, which obey Galilean relativity. - In the MQS framework, we alreay know that Faraay's law an the no-magnetic-monopole law obey Galilean relativity, since we specifically esigne the rules this way. - We now test Gauss's law for MQS Galilean relativity: ' E '= ρ' ϵ 0 (E+(v B))= 1 ϵ 0( ρ 1 c 2 v J ) E= ρ ϵ 0 1 ϵ 0 1 c 2 v J (v B) - Use a vector ientity on the last term to fin: E= ρ ϵ 0 1 ϵ 0 1 c 2 v J+v ( B) - Use the Maxwell-Ampere law to replace the curl of the magnetic fiel an fin: E= ρ ϵ 0 1 ϵ 0 1 c 2 v J+v ( μ 0 J+ 1 c 2 E t )
E= ρ ϵ 0 + 1 c 2 v E t - In orer to have the same form as the stanar Gauss's law, the last term woul have to go away. Therefore, Gauss's law oes not obey Galilean relativity in the MQS approach. - We now test the Ampere-Maxwell law: ' B '=μ 0 J '+ 1 c 2 E ' t ' B=μ 0 J+ 1 c 2( v + t) (E+(v B)) B=μ 0 J+ 1 E c 2 t + 1 c (v )E+ 1 2 c (v )(v B)+ 1 2 c 2 t (v B) - This obviously oes not have the same form as the Ampere-Maxwell law. - Therefore, the Ampere-Maxwell law oes not obey Galilean relativity in the MQS approach. - The part that breaks Galilean relativity in both Gauss's law an the Ampere-Maxwell law is the time erivative of E. This is exactly the piece that is thrown away to reuce Maxwell's equations to the MQS approximation. - Therefore, whenever the electric fiel of a system changes so slowly in time that its time erivative can be approximate to be zero, Special Relativity effects go away an Maxwell's equations reuce own to the equations of magneto-quasi-statics, which obey Galilean relativity. - In summary, Maxwell's equations o not in general obey Galilean relativity. - Since Maxwell's equations o not obey Galilean relativity, which was erive from Newton's laws, we have contraicting escriptions of the physical worl. - There are a few possibilities to resolve this contraiction: 1. Maxwell's equations are wrong. We must fix Maxwell's equations to obey Galilean relativity. 2. Our assumption of inertial frames in electroynamics was wrong. There is a preferre reference frame in which the meium that propagates the waves is at rest. 3. Galilean relativity an Newton's laws are wrong. - Historically speaking, the first option was harly plausible because there was so much experimental evience supporting Maxwell's equations. - The thir option seeme too raical. - Most physicists turne to option two as the best explanation. - They reasone that just like soun is a vibration of air, light must be a vibration of some meium they ubbe the luminiferous ether. - Efforts commence to etect an better unerstan the ether. 4. The Ether - The ether ha to be efine as a substance that is always at rest in some universal frame. - All efforts to etect the ether faile. - Efforts turne to trying to inirectly etect the ether by its effect on light. - Because the earth orbits in a circle an is always changing its velocity, it cannot always be in
this universal ether rest frame. - The ether must be moving linearly with respect to any frame of reference on earth. - Light traveling in the irection of the ether woul travel at a ifferent spee than light traveling in a perpenicular irection. - The Michelson-Morley experiment use an interferometer to split light, sen it in perpenicular irections, recombine it an measure any ifference in their spees as an interference pattern. - This experiment foun no evience of velocity ifference an thus strong evience that there is no ether. - This result was historically assume to be cause by experimental errors. Many more experiments were performe along these lines, each with increasing accuracy, an each emonstrating that no ether exists. 5. Einstein's Postulates - Although the Michelson-Morley experiment an subsequent experiments were isproving the ether concept, Einstein claims he was not motivate by these experiments. - Einstein instea chose to pursue option three, that Galilean relativity an Newton's laws are wrong. He claime to have been motivate to fin a more harmonious an logical theory that matches the relativity obeye by Maxwell's equations. - Replacing Galilean relativity involve removing some of the assumptions inherent in Galilean relativity an using new postulates. - Einstein use as his first postulate of Special Relativity: 1. The laws of physics (incluing electroynamics) are the same in all inertial frames. - This was a safe postulate because all experiments an theories seeme to support this. - This postulate immeiately rule out any special universal reference frame such as the ether. - Einstein iscare the rest of the assumptions of Galilean relativity: that time is universal an that object lengths are universal. - Doing so, he opene up his principle of relativity to the possibility that clocks can run at ifferent spees an that the same object can have ifferent lengths epening on which frame it is measure in. - The challenge before Einstein was now to come up with a secon postulate that woul lea to a relativity principle that woul make Maxwell's equations hol the same in all frames. - Einstein chose as his secon postulate: 2. The spee of light is a constant no matter in what reference frame it is measure. - This postulate is written in its historical form. However, it ultimately is equivalent to the more general postulate: There is a universal spee limit. - This postulate seems counter-intuitive an its consequences are profoun. It took the bolness an genius of Einstein to attempt such a line of thought. - If I am riving in a car close to the spee of light c, an turn on my healights, from my perspective the light will still be emitte an travel away from me at c. - But accoring to Einstein's relativity, the light is also traveling at spee c relative to the groun. This is only possible if time an space are ifferent in ifferent frames. - Since the time when Einstein propose this postulate (1905) there has been consierable experimental evience that the spee of light is constant in all frames.
APPENDIX - We shoul be careful to remember the ifference between a partial an full erivative. - A full erivative operates on all variables in a function: x t = x t î+ y t ĵ+ z t k so that v=v x î+v y ĵ+v z k - A partial erivative only operates on the one variable of interest an hols all others constant: f t = t (2 x+ y3 + xt 2 )=2 x t - There is a ifference between a variable an a function. Partial erivatives only treat variables as constants an not functions. For instance, in the equation x= x'+v x t the symbol x' is a variable an the symbol x is a function. Therefore x' t for the above equation gives zero because x' is a variable, is therefore treate as a constant, an the erivative of a constant is zero. But x t for the above equation gives v x because x is a function in this context an cannot be treate as constant. With this in min, both the full erivative an the partial erivative can be expane into another basis using the chain rule: t ' = x t ' t ' = x t ' x + y t ' x + y t ' y + z t ' y + z t ' z + t t ' z + t t ' t an t Notice that the partial erivative expansion an the full erivative expansion are very similar, but the one uses partial erivatives in the chain rule while the other uses total erivatives. - Let us apply these expansions to Galilean relativity, x=x '+v t ', t '=t. - For the partial erivative, terms like x t ' lea to x =v t ' x, so that: t ' = x t ' x + y t ' y + z t ' t ' =v x x +v y y +v z z + t t ' =v + t z + t t ' t becomes: - For the full erivative, terms like x t ' lea to x t ' = x ' t ' +v x =u x '+v x, so that:
t ' = x t ' x + y t ' y + z t ' z + t t ' t becomes: t ' =(u x '+v x ) x +(u y'+v y ) y +(u z '+v z ) z + t t ' =(u '+v) + t - Now use the velocity aition formula to fin: t ' =u + t - Expan the full erivative with respect to t in the unprime basis as well: t = x t x + y t t =u + t y + z t z + t t - Comparing the two above equations in boxes, we immeiately fin: t ' = t t