9.10 Maxwell’s Equations Integral Form
Let’s recall the fundamental laws that we have introduced throughout the semester. First, Gauss’s law for the electric field which was E dot dA, integrated over a closed surface S is equal to the net charge enclosed inside of the volume surrounded by this closed surface divided permittivity of free space, ε0. So this was Gauss’s law for electricity or for E field, and basically it gave us the electric flux through this closed surface, S.
We can express a similar type of law for the magnetic field which will be little B dot dA integrated over a closed surface and that will be equal to 0 and recall this as Gauss’s law for B field. The reason that is going to be equal to 0, we have seen this earlier, obviously this expression gives us the magnetic flux. If we calculate the magnetic flux over a closed surface. Since the magnetic flux lines allows close upon themselves, by forming loops, therefore for any closed surface, the number of field lines entering into that surface will be equal to the number of field lines coming out of that surface. Therefore the net flux will be equal to 0 since flux in will be equal to flux out for such a case.
The third fundamental law that we have introduced during the semester was the Faraday’s law of induction and it was in the form of electric field dotted with a displacement vector, dl, integrated over a closed counter or closed loop is equal to minus change in magnetic flux with respect to time. This was Faraday’s law of induction and it simply stated that if we change the magnetic flux through the area, through the surface surrounded by conducting loop then we induce electromagnetic force, hence current along that loop.
The last fundamental law that we studied during the semester was the Ampere’s law and it was in the form of magnetic filled dotted with displacement vector dl integrated over a closed loop is equal to permeable free space, μ0, times the current flowing through the area surrounded by this closed loop, and this was Ampere’s law.
Earlier we have seen how the principle of symmetry permeates physics and how it has often lead to new insights or discoveries. We looked the symmetry between electric field and magnetic field and continuously asked the symmetrical cases as we studied these two fields and try to see the similarities between these two fields. Well, from that point of view, if we look at these four equations, which are the fundamental laws that we have introduced throughout the semester, we see that there is a perfect symmetry on the left-hand side of these equations. First two are the closed surface integrals of electric field and magnetic fields. And the last two are the closed looped integrals of, again, electric field and magnetic fields.
On the right-hand side, of course we don’t see that symmetry. As a matter of fact, there are two basic asymmetries when we look at the right-hand sides of these equations, which we will talk about when we asymmetries in a moment. But in the mean time, one can of course legitimately as that how come we don’t include Coulomb’s law and Biot-Savart law, also these fundamental laws that we have studied throughout the semester. The answer to that question is that those laws are implicitly included in the Gauss’s law for the electric field as well as Ampere’s law for the magnetic field because these two laws simply gives us how to calculate, how to evaluate the electric field and magnetic field from their sources. The other concepts that we have introduced throughout the semester, all those equations mainly deal with special situations and therefore they are not really basic.
Okay. After reminding of that important point, let’s now consider the asymmetries on the right hand sides of these fundamental laws. When we consider the first two equations for the Gauss’s law for the electric field we have q-enclosed, which is the source term for the electric term. In other words this charge generates the corresponding electric field on the left-hand side. Of course we do not have such a term in the case of Gauss’s law for magnetic field and it is because of not having magnetic monopoles. Since we don’t have an isolated north pole by itself or a south pole by itself, then we cannot talk about hose poles as a source of magnetic field.
As you recall, the source of magnetic field was the moving charge or moving charges. And since the magnetic poles are always in the form of dipoles and as a result of that, the magnetic field lines always close upon themselves then the source term on the right hand side of Gauss’s law for the magnetic field becomes 0 over here.
In a similar way, similar asymmetry can be explained again using the same effect of not having a magnetic pole, magnetic monopoles. Since current, by definition, is amount of charge passing through a surface per unit time, or dq over dt in mathematical terms, and therefore not having a single pole by itself, no magnetic monopole implies that there cannot be any magnetic pole current. As a result of that, we don’t have a symmetrical current term over here for the magnetic pole current in Faraday’s law of induction.
Of course the second asymmetry that we observe, now, in these last two equations associated with the flux term. Since magnetic flux is magnetic field dotted with the area vector, therefore this dΦB over dt can loosely be interpreted as the change in magnetic fields. So here in Faraday’s law we say that change in magnetic field generates electric fields.
Well, one can then ask the symmetrical question by hoping that the symmetry exists and saying that does changing electric field generate magnetic fields? So does changing electric fields generate magnetic fields? It turns out to be that the answer to that question is yes, and now we’ll investigate how this happens. Well, just by using direct symmetry we can say that since we cannot find a corresponding term for the current here in the Faraday’s law of induction expression for the magnetic pole current, now going to look at the symmetry in change in flux in Ampere’s law.
Therefore we can say that, well, we can add a term over here as minus change in electric flux with respect to time; change in electric field flux with respect to time. Well, if we directly add this term over here and check the units, what we’ll see is that we’re not going to be able to have a correct unit on the right-hand system. In other words, μ0 i-enclosed will have a different unit than the change in electric field flux term. Well, if we multiply this term by μ0, again, we will not end up with the right unit system. But if we multiply the change in flux with ε0, ε0 times dΦE over dt will have the units or dimensions of current, and therefore μ0 times current will have the same unit with the previous term.
When we test this with the experimental results, we see that, first of all, this term over here, change in electric field flux, case obeys the right hand rule rather than the Lenz law. As you recall this negative sign appears in the Faraday’s due to the Lenz law such that induced current was flowing in such a direction such that it was opposing its course. Whereas in this case, the changing electric field which is generating magnetic field obeys right-hand rule rather than the Lenz law. Therefore this sign becomes positive.
Since this product has the units or dimensions of current, we are going to call this current, displacement current, and well denote that by id. This symmetry analysis first done by Maxwell and by adding this new term to the Ampere’s law, which makes it more complete after this verification called as Ampere-Maxwell’s law.
Now, with this new form of Amperes-Maxwell’s law, these four equations are the fundamental equations for electromagnetic theory. In other words, any electromagnetic phenomena can be explained through these four fundamental laws or equations. Therefore they are commonly called as Maxwell’s equations.
In the next section, we are going to show that, indeed, this quantity is equivalent. In other words ε0 times change in electric flux, with respect to time, is indeed a current and that generates magnetic fields. So in terms with this new term one can express also the Ampere-Maxwell’s law as magnetic field dotted with displacement vector integrated over a closed loop is equal to μ0, permeability of free space, times i-enclosed and that is conduction current, the net current, flowing through the area surrounded by this closed loop, plus id, which is what we call displacement current, and it is arising a result of change in electric flux through the area surrounded by this loop. So the source of magnetic field can either both of these quantities or any one of these currents.