2015 USA Physics Olympiad Exam Part B 10 Question B2 The nature of magnetic dipoles. a.A "Gilbert"dipole consists of a pair of magnetic monopoles each with a magnitude gm but opposite magnetic charges separated by a distance d,where d is small.In this case,assume that-gm is located at z=0 and +qm is located at z=d. -9m 9m 00… z=0 z=d Assume that magnetic monopoles behave like electric monopoles according to a coulomb-like force F=Ho 9m19m2 4πr2 and the magnetic field obeys B=F/qm. i.What are the dimensions of the quantity qm? ii.Write an exact expression for the magnetic field strength B(z)along the z axis as a function of z for z>d.Write your answer in terms of gm,d,z,and any necessary fundamental constants. iii.Evaluate this expression in the limit as d-0,assuming that the product qmd=pm is kept constant,keeping only the lowest non-zero term.Write your answer in terms of Pm,z,and any necessary fundamental constants. b.An "Ampere"dipole is a magnetic dipole produced by a current loop I around a circle of radius r,where r is small.Assume the that the z axis is the axis of rotational symmetry for the circular loop,and the loop lies in the ry plane at z=0. i.Write an exact expression for the magnetic field strength B(z)along the z axis as a function of z for >0.Write your answer in terms of I,r,z,and any necessary fundamental constants. ii.Let kIrY have dimensions equal to that of the quantity pm defined above in Part aiii, where k and y are dimensionless constants.Determine the value of y. iii.Evaluate the expression in Part bi in the limit as r>0,assuming that the product kIr=p is kept constant,keeping only the lowest non-zero term.Write your answer in terms of k,pm,z,and any necessary fundamental constants. iv.Assuming that the two approaches are equivalent,pm=pm.Determine the constant k in Part bii Copyright C2015 American Association of Physics Teachers2015 USA Physics Olympiad Exam Part B 10 Question B2 The nature of magnetic dipoles. a. A “Gilbert” dipole consists of a pair of magnetic monopoles each with a magnitude qm but opposite magnetic charges separated by a distance d, where d is small. In this case, assume that −qm is located at z = 0 and +qm is located at z = d. z −qm qm z = 0 z = d Assume that magnetic monopoles behave like electric monopoles according to a coulomb-like force F = µ0 4π qm1qm2 r 2 and the magnetic field obeys B = F/qm. i. What are the dimensions of the quantity qm? ii. Write an exact expression for the magnetic field strength B(z) along the z axis as a function of z for z > d. Write your answer in terms of qm, d, z, and any necessary fundamental constants. iii. Evaluate this expression in the limit as d → 0, assuming that the product qmd = pm is kept constant, keeping only the lowest non-zero term. Write your answer in terms of pm, z, and any necessary fundamental constants. b. An “Amp`ere” dipole is a magnetic dipole produced by a current loop I around a circle of radius r, where r is small. Assume the that the z axis is the axis of rotational symmetry for the circular loop, and the loop lies in the xy plane at z = 0. z I i. Write an exact expression for the magnetic field strength B(z) along the z axis as a function of z for z > 0. Write your answer in terms of I, r, z, and any necessary fundamental constants. ii. Let kIrγ have dimensions equal to that of the quantity pm defined above in Part aiii, where k and γ are dimensionless constants. Determine the value of γ. 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. iv. Assuming that the two approaches are equivalent, pm = p 0 m. Determine the constant k in Part bii. Copyright c 2015 American Association of Physics Teachers