Question

4. Magnetic Dipole Moment on a PEC Sphere: In HW 4-Problem 5, you found the electric dipole moment for a metal sphere of radius a in a uniform electric field. In this problem you will find the magnetic dipole moment for the same metal sphere but now in a uniform magnetic field, Hext-Hext2. Note:Assume that the sphere is centered at the origin of a spherical coordinate system. Express Hext in spherical coordinates. To find the magnetic field that is induced by presence of the PEC sphere, the total magnetic field can be written as H- Hext Hind. Find an equation for the for the radial component of Hind in terms of Hext ata. Hint What are the boundary conditions for magnetic fields? To find an expression for the induced magnetic field, we first guess that the magnetic field is due to an equivalent z-oriented magnetic dipole (current loop) that is induced on the sphere: a. b. c. Find an expression for magnetic moment m, that satisfies the boundary conditions atr-a, and is thus the unique solution to the problem. Using your answer in (c), find an expression for the total magnetic field H and make a sketch of the magnetic field lines. d. Hext = Hex,2 PEC

Answer 1

Solution:

Given that

Given that inide 0. Howe an the pe

pla quaien (ul) ょ) H.xt - 20...-A D夭 -tば ost

Bn 1-0 rit et

StEm em (V) 2 T1:0 1):-0 (5a) a (5b)

En 60 ft n11 sum equalef) (h) he get (ea) (6b) 6)& 60) 5c) & (c)

il ht e8, 3 0

凝. ind Jt rre

ar Hh a ndr

,于21 liit 21 lead e to maale tt that nne olime та

) 31