Learning Goal: To show that displacement current is necessary to make Ampère's law consistent for a charging capacitor Ampère's law relates the line integral of the magnetic field around a closed loop to the total current passing through that loop. This law was extended by Maxwell to include a new type of "current" that is due to changing electric fields: B-di-po(Icharge Idisplacement The first term on the right-hand side, Icharge describes the effects of the usual electric current due to moving charge. In this problem, that current is doE designated I (t) as usual. The second term, displacement 0 , is called the displacement current, it was recognized as necessary by Maxwell. His motivation was largely to make Ampère's law symmetric with Faraday's law of induction when the electric fields and magnetic fields are reversed. By calling for the production of a magnetic field due to a change in electric field, this law lays the groundwork for electromagnetic waves in which a changing magnetic field generates an electric field whose change, in turn, sustains the magnetic field. We will discuss these issues later. (Incidentally, a third type of "current," called magnetizing current, should also be added to account for the presence of changing magnetic materials, but it will be neglected, as it has been in the equation above.) The purpose of this problem is to consider a classic illustration of the need for the additional displacement current term in Ampère's law. Consider the problem of finding the magnetic field that loops around just outside the circular plate of a charging capacitor. The cone-shaped surface shown in the figure has a current I (t) passing through it, so Ampère's law indicates a finite value for the field integral around this loop. However, a slightly different surface bordered by the same loop passes through the center of the capacitor, where there is no current due to moving charge. To get the same loop integral independent of the surface it must be true that either a current or an increasing electric field that passes through the Ampèrean surface will generate a looping magnetic field around its edge. The objective of this example is to introduce the displacement current, show how to calculate it, and then to show that the displacement current Idisplacemét is identical to the conduction current Icharg). Assume that the capacitor has plate area A and an electric field E (t) between the plates. Take o to be the permeability of free space and eo to be the permittivity of free space. Figure In ein) 1 of 2 Part B Now find an expression for fRB-di, the same line integral of around the same loop of radius R located just outside the left capacitor plate as before. Use the surface that passes between the plates of the capacitor, where there is no conduction current. This should be found by evaluating the amount of displacement current in the Ampère-Maxwell law above. (Figure 2) Express your answer in terms of the electric field between the plates E (t), dE(t)/dt, the plate area A, and any needed constants given in the introduction. View Available Hint(s) fRB-dl- Submit A necessary consistency check - Part C We now have two quite different expressions for the line integral of the magnetic field around the same loop. The point here is to see that they both are intimately related to the charge (1) on the left capacitor plate. First find the displacement current Idisplacem in terms of g (t) Express your answer in terms of q (f), dg(t)/dt, and any needed constants given in the introduction. View Available Hint(s) Idisplacemetik Submit G: ? Part D Icharg() = Review I Constants (? Now express the normal current Icharge) in terms of the charge on the capacitor plate q (t). Express your answer in terms of g (t), dg(t)/dt, and any needed constants given in the introduction. (3 ?