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Welcome back to this lecture about the Lawrence Cage, so in the previous lecture, we have transformed



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the Maxwell equations in their most general case to four new equations.



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So these are based on the vector potential and on the electrostatic potential, which is a scalar function.



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Now, you can see two of these equations have the same mathematical shape we have to square operate



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to acting on the potential than we have here, some derivative here, time derivative here.



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It's the gradient, basically the space derivative.



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And then this bracket expression here in both cases and on the right hand side, we have to charge density



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and here we have the current density.



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So these equations rely not on the electric field, but and not on the magnetic field, but they rely



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on the vector potential and on the electrostatic potential and we have already already discussed this



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in electrostatic and in magnetic statics, that these two potentials are not the observables.



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So this means we can use certain gauges and we can change these potentials basically in a certain manner



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without affecting the magnetic and the electric field.



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And maybe you remember this, for example, we have this expression here, this Maxwell equation, where



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the divergence of the magnetic field is zero.



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So this means we have the possibility to express the magnetic field in terms of the rotation of a.



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So this is like an indirect definition of A, we have B is equal to rotation of A, we do not have the



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expression A is equal to something.



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So this means this A is not uniquely defined.



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We can basically add any function to a whose rotation is zero because this leaves B unchanged.



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So this means he is not unique and it can be gauged and the same applies to fire.



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So it would be really good if we could use a gauge where this expression in the brackets would become



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zero, because this would make our equations and our lives much easier.



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And it turns out we can do this.



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There is a certain gauge and it's very famous.



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It's called the Lawrence Gage, which is exactly that this bracket expression becomes zero.



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But of course, we need to verify that this is a valid gauge.



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We cannot do any transformation to a because there are some transformations that change the observables,



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for example, B, in this case, for example, if we would add some function to A whose rotation gives



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some value, then of course this value is added up to B and then B is changed.



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So this would not be a valid gauge.



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So we need to really confirm that as Lawrence is allowed and it's valid that it leaves B and PHI invariant.



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And first of all, I want to tell you how you can gauge these two potentials A and find the answer is



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you can gauge or you can change A in this way that you add a function that is a gradient of a scalar



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so you can take any scalar function.



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It does not matter, can be any scalar dependent on the position R and to time T and then you calculate



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the gradient of this function.



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And in any case you can add it to a this is of course because the rotation of a gradient of some function



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is always zero.



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And then for five you can change your fire to a Newfie by subtracting.



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Or you could also write adding the time derivative of a different function.



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And if you want to use the same function in both cases, then you have to write it down like this.



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And this can easily be verified.



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You can check this for yourself.



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So we must make sure that the electric field and the magnetic fields stay unchanged and that these transformations



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so we can write down that our electric field is this expression from the from the previous lecture.



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So this was the Maxwell equation that we had transferred.



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So we have the time and time derivative of our new vector potential in the gradient of our new scalar



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potential.



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And then we use these two expressions here and set them in.



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So we get here an additional term, which is a time derivative of this gradient.



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And here we get the same thing.



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We get a gradient of the time derivative.



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And since these two derivatives are exchangeable because the position and the time are independent variables,



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these this term here and this term, they compensate.



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So we are left with this term in this term, which is E, so the new E is equal to the old E and then



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for a magnetic field.



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I already mentioned this briefly.



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This is very, very easy.



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You have the new magnetic field is the rotation of the new vector potential, which is of course the



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rotation of the old vector potential, plus the rotation of the verdie and gradient of ky. and a great



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and the rotation of a gradient of any function is always zero.



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So this means it's NewBay is the rotation of the old EI.



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So it's the old B.



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OK, so now we know how we can transform the vector potential and the electrostatic potential.



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And so this means when we take this term here and the brackets and apply this transformation, we get.



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Yeah, this expression.



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But just with this new electrostatic potential and this new vector potential.



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So now let's use these two expressions here for the transformed potentials and put them in and see what



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it looks like.



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So, of course, from this term and from this term, we get the old expression.



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So that's exactly the same as this one, but with the old vector potential and the old scalar potential.



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And we get to new terms from this one and that one here.



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So we get here at a time derivative of the gradient of chi and here we get the rotation of the rotation



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of the gradient of chi.



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So what this means is we need to choose Chi in a certain way so that this whole Brexiteer is equal to



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this Brackett's.



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And we do not really have to say what Chi looks like, we just have to realize that this is, of course,



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possible.



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So we have a certain solution to our problem, a certain calculated fire and a certain calculated A..



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And now we have to choose chi accordingly so that this bracket is the same magnitude, but the opposite



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side to this bracket here.



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And it's possible we will do this.



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And this means this whole expression becomes zero if Chi is chosen accordingly.



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So that means it is indeed possible to gauge this whole term away.



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And so our equations become really simple.



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It's just a scratch.



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Operator Phi is equal to one of absolute zero and the great operator acting on A is equal to zero times



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current density.



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So what we have done in this lecture is we have started from the Maxwell's equation, we have transformed



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them and then we have applied this Lawrence and this led to this to these new Maxwell equations.



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And now you can really see what is the advantage.



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We have now two equations, this first and this fourth one here that are not coupled anymore.



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So you just have to hear the charge density.



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And here you have the I'm sorry, here you have to charge density and here you have to current density.



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And from them you can calculate the electrostatic potential and the vector potential.



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And when you have these, you can just take them and calculate the magnetic field and the electric field.



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And here on the very left, it was much more difficult because here you had a couple of equations,



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for example, here, when you wanted to calculate the magnetic field, you also had to know the electric



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fields.



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And here it's viceversa and you wanted to calculate the electric field.



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You had to know the magnetic fields.



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So this could be an advantage.



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And we will discuss one example in the following.



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So in the next lecture, we will discuss -- potentials, which basically take our knowledge from



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electrostatic and magnetic statics and apply them to time dependent problems based on the solution we



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have just established.