Second Year Electromagetism: Problem Set 1

 

There are three problem sets for Michaelmas Term. The first two largely cover the first 3 weeks of lectures and are meant to be covered in tutorials during term. The last set covers the last week of lectures and also other topics to study over the Christmas vacation. I would recommend two tutorials on these topics. But it might be reasonable to have one or two this term and then another one very early in Hilary Term.  My own intention is to run 2 tutorials in MT and 3 tutorials in HT, but we will have to see how it goes.

 

In order to keep with long Oxford tradition, there may well be more problems in these sets than one can reasonably set in tutorials and I leave it to the discretion of tutors to decide what they give to students. I have marked the ones I think are particularly difficult or lengthy. I have put many problems in a supplemental problem set that should be useful for further study during the year as well.

 

Note to Students: During the course of your studies I would highly recommend that you, at some point, attempt ALL of these problems.

 

1). Two hollow concentric metal spheres are first isolated from their surroundings and connected electrically. They are then connected to a potential relative to infinity of strength V. Soon after this the connection between the two spheres is broken and the outer sphere connected to the reference potential (ground). Determine the final charge density and potential of the inner sphere.

 

2). An electric dipole p is located at the origin. Show that the electric potential due to this dipole is given by:

                                   

 

3). Calculate the electric potential due to a line of charge with total charge +Q that extends along the z axis from +a/2 to –a/2. Calculate the Electric field for the same configuration. Show that at large enough distances the expression for the potential reduces to that of a point charge.

 

4). Find the electric field at a height z above a uniformly charged square plate with sides of length a and along the central axis of the plate. Check that the result conforms to what you would expect if we allow a to go to infinity or we let z increase to be very much larger than a. (Hint: It appears that, though the voltage integral is easy to set up, it is impossible to solve. Try solving for the z-component of E-field directly instead.)

 

5).!  Suppose an electric field E(x,y,z) has the form:

                                   

where a is a constant. What is the charge density? How do you account for the fact that the field points in a particular direction when the charge density is uniform? [This is a more subtle problem than it looks and worth of careful thought. I am forced to admit that I am not sure I have the correct answer for this one!]

 

6).!  Just as we can imagine spreading a given charge out on a surface to obtain a uniform surface charge density s. So too we could imagine spreading out many small, ideal dipole moments to create a dipole moment per unit area t (which would have to be a vector quantity).

  • Show that the electric potential at a point P from such a surface is given by
       where r is the displacement vector from an infinitesimal dipole element to the point P.
  • Consider a dipole layer of infinite extent lying in the x-y plane with a uniform density . Determine whether the electric potential or the electric field is discontinuous across the layer and find the discontinuity.
  • Consider a positive point charge q located at the centre of a spherical surface of radius a. On this surface there is a uniform dipole layer t and a uniform surface charge density s. Find t and s so that the potential inside the surface will be only that of the point charge, while the electric potential outside will be zero. [1023]

 

7). A sphere in empty space has radius R has ½ of it’s surface held at a potential V0 while the other half is held at ground. The two halves are insulated from each other. Find the potential everywhere due to the sphere. Show the integral relationship needed to find all the constants and work out the first 3 or 4 explicitly. [1070]

 

8). An ideal dipole is located at the origin inside a spherical metal cavity of radius R and held at zero potential. Find the potential in the cavity and show that it reduces to the expected result as R becomes large.

 

9).* We are going to find the potential inside a long pipe with square sides of length a. There is a point charge of size Q located in the centre of the pipe and far away from the edges, indeed, assume the pipe is infinitely long for this problem. First though we need to get some preliminary things out of the way. [1091]

  • Show using Fourier sine and cosine series techniques that over the interval from –a/2 to +a/2 the Dirac delta function d(x) can be expressed by the following series.
  • The Poisson equation describes this situation and then takes the form
    . Assume the Fourier transform of the potential exists in the z direction such that if  is a solution to this problem then
    . Perform a similar integral on the whole Poisson equation to obtain an equation in x, y, and k which does not depend on z at all.
  • Using separation of variables, show that the general solution consists of the multiplication of some function of k only and two cosine terms, one dependent on x and the other dependent on y.
  • Substitute this into the Fourier transformed equation and solve for , then use the result  to transform back and recover the ‘z’ variable.
    With the pipe aligned along the z axis the final result is:

 

! – Problems which might be difficult.

* - Problems which are not in principle difficult, but are quite long and algebra intensive.