2015 USA Physics Olympiad Exam Part B 20 iii.Evaluate this expression in the limit as d0,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. Solution Starting from the last expression, B(e)=9m4= Ho Pm 2T 23 2πz3 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. Solution Apply the law of Biot-Savart,with s the vector from the point on the loop to the point on the z axis: -尝-尝1，n0 in this last case,6 is the angle between the point on the loop and the center of the loop as measured by the point on the z axis. We can rearrange this, 2Tr2 B=丽 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. Solution Pm must have dimensions of Ampere meter square,so y=2. Copyright C2015 American Association of Physics Teachers2015 USA Physics Olympiad Exam Part B 20 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. Solution Starting from the last expression, B(z) = µ0 2π qmd z 3 = µ0 2π pm z 3 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. Solution Apply the law of Biot-Savart, with ~s the vector from the point on the loop to the point on the z axis: B(z) = µ0 4π I I d ~l × ~s s 3 = µ0 4π I 2πr r 2 + z 2 sin θ in this last case, θ is the angle between the point on the loop and the center of the loop as measured by the point on the z axis. We can rearrange this, B(z) = µ0 4π I 2πr2 (r 2 + z 2) 3/2 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. Solution pm must have dimensions of Ampere meter square, so γ = 2. Copyright c 2015 American Association of Physics Teachers