Second Year Electromagetism: Problem Set 2

26/10/07

 

Note to Students: During the course of your studies I would highly recommend that you, at some point, attempt ALL of these problems. These problems cover magnetism and also provide a few review problems on magnetic induction and emf, which are good to remind ourselves about before proceeding on to EM waves. This is a long problem set and tutors might want to save questions for HT.

 

 

 

1). Above we see a thin metal sheet which has been bent around a mandrel to form a tube of radius a. The remaining metal sheets extend off to the left for a long distance and the length of the tube itself should be taken as essentially infinite. Let d be the very small gap between the two metal sheets A surface current K0 exists in the plate and it’s direction around the loop and back is shown above.

2).@ A plastic cylinder of equal diameter to the figure above is wound with wire at a density of N turns per unit length in such a way that when a current I is run through the wire that current would run the same direction as shown above. Find the B field everywhere. What current density in problem 1 would be needed to obtain the same B field as in problem 2?

 

3).@ Notice that the vector potential A is related to the magnetic field B the way B is related to the current density J. Meaning curl(A) = B while curl(B) is proportional to J. What mathematical expression about A corresponds to the statement that the line integral of B around any closed path equals m0 times the current enclosed by the path (Ampere’s law)?

 

4).@ Going back to the configuration of problem 1. Calculate the magnetic vector potential, A,  everywhere (again do not worry over-much about what happens near the join). Hint: It is probably not a good idea to use the integral formula that has current density in the integrand to accomplish this because we have a problem where the currents are infinite in extent. So if you skipped question 3 maybe this is a good time to do it anyway.)

 

5).@ In lectures the claim was made that the divergence of the magnetic vector potential could always be set to zero. This allowed us to obtain a set of three Poisson equations relating the vector potential to the three components of the current density…which was very nice and obviously why we wanted to get a divergenceless A in the first place. Prove that this can always be done and satisfy the requirements for static magnetic fields. [Purcell 196]

 

6). A hydrogen atom consists of a proton and an electron which we will think of as describing a circular orbit about the proton at the Bohr radius (0.53 Angstroms).

 

 

7). Show that one can define a magnetic scalar potential of the form  that will satisfy Laplace’s equation. Given a loop of wire with a large radius compared to the diameter of the wire itself in this example under what circumstances would you expect to be able to employ the concept of a scalar magnetic potential and what conditions would this concept be problematic. In particular think about the situation where the wire carries a constant current or where the wire initially carries no current but is placed in an external magnetic field and also about the magnetic field both inside and outside the wire.

 

8). A ring of soft iron (which has a linear magnetic permeability mr) is made from a bar of cross sectional area A and is bent into a ring of radius r. The bar is wound with n turns of wire which carries a constant current I. Sketch the field lines of B and H and deduce the strength of these fields as a function of the given parameters. You can assume that the radius of the ring is much larger than the cross sectional area of the bar.

 

9). A narrow gap of length w is sawn out of the ring in problem 8. Considering the boundary conditions of the fields in the material and the gap and assuming w is very small compared to r show that the magnitude of the magnetic field in the gap B is given by:

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10). The soft iron of the ring in problems 8 and 9 is replaced with a form of iron that is decidedly non-linear. The B-H curve that describes the permeability of this material is shown below.

Show that the magnetic field in the gap may be deduced by finding the intersection of the B-H hysteresis loop with the straight line . Show that the net work expended as heat in magnetising and demagnetising the magnet once around the loop is approximately 104prA Joules. (It might help to think of this loop as mainly rectangular in the figure.)

 

11). A beam of electrons is travelling through space. There is an electric field and a magnetic field present which are perpendicular to each other and both perpendicular to the beam. What velocity would the beam have in order to avoid deflection? If the electric field were then turned off find an expression for the radius of curvature R of the electron beam and then also find the charge to mass ratio of the electron (q/m) in terms of E, B, and R. You must assume the particle is relativistic.

 

12). A proton sits at rest at the origin to start. There is a uniform electric field of strength E in the y direction and a uniform magnetic field B coming out of the page along the z axis. Solve the equations of motion and sketch the trajectory of the particle. You may assume the fields are sufficiently weak relative to the particle mass that the speed is small compared to the speed of light for this problem.

Perform the same problem but with the following sets of initial conditions for the proton’s initial velocity:

 

13). A sphere of radius a is made of a material with an isotropic, uniform, permeability m that responds linearly to the application of a magnetic field. This sphere is placed in an initially uniform magnetic field of strength B0. Calculate the resulting magnetic field (B), H, and magnetization (M).

 

 

 

@ - The vector potential is ‘unexaminable’. I interpret this to mean that you must learn about it anyway even if it doesn’t appear in an exam otherwise no mention of it would have been made on the syllabus or it would have been explicitly been mentioned as off the syllabus.