There are four questions on this exam. All are of equal value but not necessarily of equal difficulty. Answer all four questions

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1 Electricity and Magnetism (PHYS2016) Second Semester Final Exam 2015 There are four questions on this exam. All are of equal value but not necessarily of equal difficulty. Answer all four questions You have a choice in question 4. Do either part A of question 4 or part B of question 4. Page 1 of 11

2 Question 1. The current loop (30 points) A current I flows in a circular current loop of radius R that lies in the xy plane with its centre at the origin as depicted in the figure below. We are interested in the magnetic field at point P positioned at height L directly above the centre of the loop at point (0,0,L). In both parts of this question below ensure that all unit vectors that need to be primed are primed and all coordinates that need to be primed are primed. You will be marked down if you do not do this correctly. Part A (15 Points) i) Write down the Biot Savart Law and write down the vector r that points to the source. ii) Write down the vector Idl that describes the source iii) Write down the vector r that points to point P. iv) Write down the unit vector (r- r )/ r- r and take the appropriate cross product in the Biot Savart law. v) Calculate the magnetic field at point P positioned at z =L. You can use symmetry arguments to avoid calculating any terms that are clearly zero. If there is an appropriate result in your textbook, use it! Page 2 of 11

3 Question 1 (continued) Part B (15 points) You can use your exact result from part A to answer all parts of this question. Although we usually expand before we integrate, there is no need to and here we perturb the exact result. There is no need to do any integrals to answer this question. If you are doing integrals stop and use your result from Part A. i) Our point of observation P in part A of this question is moved a distance epsilon to z=l + ε. Here ε is a displacement that is small compared with L and R. Note that ε has units of length. Based on symmetry arguments, would you expect the first order correction to the magnetic field due the small displacement of P away from z =L to be zero? What about the second order correction? You must state your reasons and be clear to receive credit for this question. A two or three line answer is all that is necessary. Don t write a book but be clear. (5points). ii) In this question, we set L=0 and consider small displacements ε away from z = 0. Based on symmetry arguments, would you expect the first order correction to the field at z = 0 due to a displacement ε to be zero or non- zero? What about the second order correction? You must state your reasons and be clear to receive credit for this question. Again, a two or three line answer is all that is necessary. Don t write a book but be clear. (5 points). iii) Calculate the first and second order corrections in case (i) above. You can assume ε is small compared with R and L (5 points). Hint: Use your solution to part A and collect all contributions to the second order term. The following expansion may be useful: (1 + α)!!/! α α! Page 3 of 11

4 Question 2 (30 Points) In this problem, we study the magnetically clad cable. The inner hollow conductor of radius a carries current I that is uniformly distributed around the conductor. We have a single layer of material that surrounds the inner conductor out to radius b. The magnetic susceptibility of the material varies as a function of s and is given by: Here s is the usual cylindrical radial coordinate. Part A (15 Points) Χ s = Χ! e!!" Using Ampere s law for H, calculate the three vector fields, H, M and B in all three regions (s<a, a<s<b, s>b). Part B (15 Points) Identify all bound currents and again calculate the three vector fields H, M, and B in all three regions but this time using Ampere s law for B. You can use your result for M obtained in part A to calculate the bound currents but ensure that once you have calculated the bound currents, use Ampere s law for B to calculate B, and from B and M calculate H in all three regions. Page 4 of 11

5 Question 3 (30 Points) In this problem we are concerned with the mutual inductance of two loops as shown in the figure below and described here. A square loop of side a is placed directly above a circular loop of radius b as shown below. The planes of the loops are parallel and the centres of the loops are on the z axis. The loops are separated by distance L. You may assume that a <<b and a<<l throughout all parts of this problem and make appropriate approximations. Part A (5 Points) Calculate the mutual inductance of the loops by calculating the magnetic flux through the square loop due to a current of 1 amp flowing in the circular loop. You may make an appropriate approximation because a is the smallest length scale in the system. Part B (10 Points) Calculate the mutual inductance of the two loops by treating the square loop as a perfect dipole and integrating the field over the large circular loop. Part C (15 Points) Calculate the mutual inductance using Neuman s formula: M!" = μ! 4π dl! dl 2 r 1 r 2 Take advantage of the symmetry. All four sides of the square contribute equally so you can compute the integrand for one side of the square, integrate and multiply by 4 to get the result. Step through this solution laying it out clearly as we practiced in homework and workshops! Page 5 of 11

Question 4 (30 Points) DO EITHER PART A OR PART B In this question we are concerned with a magnetized solid exponential cone with constant magnetization M in the z direction.

6 Question 4 (30 Points) DO EITHER PART A OR PART B In this question we are concerned with a magnetized solid exponential cone with constant magnetization M in the z direction. The curved surface of a solid exponential cone (pictured below) is described parametrically by the following three equations. x = R e!!" cos φ y = R e!!" sin φ z = z The top and bottom surface are described by z = 1 and z = 0 respectively. Part A (30 points) Derive the magnetic field at the centre of the upper (top) surface, that is at point (0,0,1) in terms of an integral over the surface current density. Take care on this question to keep primed and unprimed variables clearly labeled. You must express your integral such that all dependences on the integration variables are explicit. For example, if you have an integral over all angles, ensure there is no angular dependence hidden in your choice of basis vectors. You can and should neglect any terms that are clearly zero due to symmetry. I suggest you work in a cylindrical basis initially and move to Cartesian towards the end of the problem. OR Part B (30 points) Derive the magnetic field at (0,0,1) in terms of a volume integral over the field due to the magnetic dipole moment per unit volume. You must express your integral such that all dependences on the integration variables are explicit. Take note of any divergences or delta functions and exclude them/include them as appropriate Page 6 of 11 End of questions Formula sheets follow.

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Physics Education Centre EXAMINATION. PHYS2016_Semester 2 Electromagnetism

Venue Student Number Physics Education Centre EXAMINATION This paper is for ANU students. Examination Duration: Reading Time: 180 minutes 15 minutes Exam Conditions: Central Examination Students must return