2015 USA Physics Olympiad Exam Part B 19 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-qm is located at z=0 and +qm is located at z=d. -4m 4m 0m-O z=0 z=d Assume that magnetic monopoles behave like electric monopoles according to a coulomb-like force F=H09m19m2 4πr2 and the magnetic field obeys B=F/qm i.What are the dimensions of the quantity qm? Solution By the second expression,gm must be measured in Newtons per Tesla.But since Tesla are also Newtons per Ampere per meter,then gm is also measured in Ampere meters. 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. Solution Add the two terms: B)-绘要+ Simplify,because it is the right thing to do, B到-会n (z+d)2-z2 (z2)(z+d02 or B(a)= (2+d)2-z2\ (z2)(z+d2 or B)=尝nd 2+d (z+d)2 Copyright C2015 American Association of Physics Teachers2015 USA Physics Olympiad Exam Part B 19 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? Solution By the second expression, qm must be measured in Newtons per Tesla. But since Tesla are also Newtons per Ampere per meter, then qm is also measured in Ampere meters. 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. Solution Add the two terms: B(z) = µ0 4π −qm z 2 + µ0 4π qm (z + d) 2 Simplify, because it is the right thing to do, B(z) = µ0 4π qm  (z + d) 2 − z 2 (z 2)(z + d) 2  or B(z) = µ0 4π qm  (z + d) 2 − z 2 (z 2)(z + d) 2  or B(z) = µ0 4π qmd  2 + d z(z + d) 2  Copyright c 2015 American Association of Physics Teachers