Physics Education Centre EXAMINATION. PHYS2016_Semester 2 Electromagnetism

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1 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 the examination paper at the end of the examination This examination paper is NOT available to the ANU Library archives Calculator (non programmable) Materials Permitted In The Exam Venue: Electromagnetism textbook (No electronic aids are permitted e.g. laptops, phones) Calculator (non programmable) Materials To Be Supplied To Students: Writing paper and scribble paper Instructions To Students: You must attempt to answer all four questions. All four questions carry the same mark. All questions to be completed in the script book provided. Page 1 of 6

2 Question 1 (25 marks) Question 1 Part A (15 marks) A square loop (side length a) carries a steady current I, as shown in figure 1. The loop lies in the z = 0 plane with the centre of the loop situated at the origin. Find magnetic field at the point Z situated a distance d above the centre of the loop using the Biot- Savart law. Question 1 Part B (10 marks) (i) (ii) (iii) Find the magnetic moment of the loop. Using the magnetic moment find the dipole term in the multipole expansion of the vector potential on the z-axis. Calculate the magnetic field at the point Z using the vector potential. z Z y d I x Figure 1 Page 2 of 6

3 Question 2 (25 marks) Question 2 Part A (10 marks) A long cylinder of radius R with its axis along the z-axis carries a non-uniform Magnetization M = ks! φ where k is a constant and s is the distance from the z-axis. There is no free current. Find the volume and surface bound currents. Find the magnetic field inside and outside the cylinder using (i) free currents and vector H (ii) bound and free currents, and Ampere s law for the magnetic field vector. Question 2 Part B (15 marks) The space between two infinite planes z = -a/2 and z = a/2 is filled with a magnetic material with susceptibility χ. A free current with surface density: M = Kx flows along the upper plane at z = a/2. A free current with surface density: M = Kx flows along the lower plane at z = -a/2. There is no free volume current. Find all bound currents. Find the magnetic field between the planes and in the outside space using: (i) vector H and free currents. (ii) vector B and bound and free currents. Show that your results for the magnetic field agree with the magnetostatics boundary conditions. Page 3 of 6

4 Question 3 (25 marks) Question 3 Part A (15 marks) Mutual Inductance of Two Concentric Coplanar Loops Consider two single-turn co-planar, concentric coils of radii R 1 and R 2, with R 1 >> R 2, as shown in figure 2. Suppose a current I flows in the big loop. Figure 2. (a) Calculate the mutual inductance of the loops by calculating the magnetic flux through the small loop. (7 Marks) (b) Calculate the mutual inductance using the Neumann formula. Leave your answer in integral form. You may assume that R 2 << R 1 and make appropriate approximations. (8 Marks) M!" = μ! 4π dl 1 dl 2 r 1 r 2 Question 3 Part B (10 marks) Mutual Inductance between two coaxial Solenoids Consider two long thin solenoids, one wound on top of the other. The length of each solenoid is l, and the common radius is r. Suppose that the inner coil has N 1 turns per unit length, and carries a current I 1. The outer coil has N 2 turns per unit length. Calculate the mutual inductance between the two coils. Assume for simplicity that both solenoids are long with respect to the common diameter, in order that end effects may be neglected. The two radii (r) are taken to be approximately equal. Assume that all of the flux produced by one coil passes through the other coil. Page 4 of 6

5 Question 4 (25 marks) A disk of radius R lies in the xy plane centered on the z-axis as shown in figure 3. The disk carries a surface charge density σ = αs! where α is a constant and s is the usual cylindrical coordinate. The disk spins with angular velocity ω = ωz (a) Calculate the surface current density. (2 Marks) (b) Calculate the magnetic field at a point (0, 0, z). (8 Marks) (c) Calculate the magnetic field from the vector potential in a small region around the point (0, 0, z). (15 Marks) Figure 3. Page 5 of 6

6 END OF EXAMINATION Page 6 of 6

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