2015 USA Physics Olympiad Exam Part B 21 iii.Evaluate the expression in Part bi in the limit as r0,assuming that the product kIr=p is kept constant,keeping only the lowest non-zero term.Write your answer in terms of k,p,z,and any necessary fundamental constants. Solution B(a)=0I、 2mr2 4r(r2+2p12 2m23 or B)=2器k HoPm iv.Assuming that the two approaches are equivalent,Pm-p Determine the constantk in Part bii. Solution k=m,by inspection. c.Now we try to compare the two approaches if we model a physical magnet as being composed of densely packed microscopic dipoles. L A cylinder of this uniform magnetic material has a radius R and a length L.It is composed of N magnetic dipoles that could be either all Ampere type or all Gilbert type.N is a very large number.The axis of rotation of the cylinder and all of the dipoles are all aligned with the z axis and all point in the same direction as defined above so that the magnetic field outside the cylinder is the same in either dipole case as you previously determined.Below is a picture of the two dipole models;they are cubes of side d<<R and d<<L with volume m=d3. Copyright C2015 American Association of Physics Teachers2015 USA Physics Olympiad Exam Part B 21 iii. Evaluate the expression in Part bi in the limit as r → 0, assuming that the product kIrγ = p 0 m is kept constant, keeping only the lowest non-zero term. Write your answer in terms of k, p 0 m, z, and any necessary fundamental constants. Solution B(z) = µ0 4π I 2πr2 (r 2 + z 2) 3/2 = µ0 2π Iπr2 z 3 or B(z) = µ0 2π π k p 0 m z 3 iv. Assuming that the two approaches are equivalent, pm = p 0 m. Determine the constant k in Part bii. Solution k = π, by inspection. c. Now we try to compare the two approaches if we model a physical magnet as being composed of densely packed microscopic dipoles. R z L A cylinder of this uniform magnetic material has a radius R and a length L. It is composed of N magnetic dipoles that could be either all Amp`ere type or all Gilbert type. N is a very large number. The axis of rotation of the cylinder and all of the dipoles are all aligned with the z axis and all point in the same direction as defined above so that the magnetic field outside the cylinder is the same in either dipole case as you previously determined. Below is a picture of the two dipole models; they are cubes of side d << R and d << L with volume vm = d 3 . Copyright c 2015 American Association of Physics Teachers