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test2-S10 - b Find the net force between the sphere and the...

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Physics 318 R.G. Palmer Electromagnetism 4/5/10 Test 2 1. a. There is a local magnetic field at a magnetic dipole, H local , which is not the same macroscopic field H (like the electric field). In a uniform magnetization M , find H local from H and M . Write down your steps and assumptions. b. A magnetic dipole m is stable if it is the same direction of the local magnetic field. Magnets are generally unstable unless they have use a “finder” (a magnetic material in the gap). Show this for (i) an “anchor ring” (a torus with a gap) and (ii) a bar magnet. 2. Outside a grounded conducting sphere of radius a lies a uniformly charged ring of radius b with total charge q . The sphere lies on the axis of the ring, and the centers of the ring and sphere are a distance d apart. If c = b 2 + d 2 , then c > a since the ring is outside the sphere. a. Find the electrostatic potential everywhere outside the sphere.
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Unformatted text preview: b. Find the net force between the sphere and the ring. You should be able to write the magnitude of this in the form F ∝ q 2 ∞ X l =0 A l P l (cos α ) P l +1 (cos α ) where cos α = d/c and A l depends on the geometrical parameters, by using some recurrence rela-tions. 3. A closed annular tube consists of the region a ≤ ρ ≤ b , 0 ≤ z ≤ c in cylindrical coordinates. The curved surfaces ( ρ = a and ρ = b ) and the bottom ( z = 0) surface are held at 0 potential, while the top ( z = c ) surface is held at potential V . Find an expansion for the potential inside the tube. Notes: For full credit you need to evaluate the V integral, by using a recurrence relation. You do not need to evaluate the normalization integral (although it can be done)....
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