2017 USA Physics Olympiad Exam Part A 8 A pair of iron balls placed in the same horizontal plane as the compass but a distance d away can be used to help correct for the error caused by the induced magnetism of the ship. A binnacle,protecting the ship's compass in the center,with two soft iron spheres to help correct for errors in the compass heading.The use of the spheres was suggested by Lord Kelvin. Just like the ship,the iron balls will become magnetic because of the Earth's field Be.As spheres,the balls will individually act like dipoles.A dipole can be thought of as the field produced by two magnetic monopoles of strength +m at two different points. The magnetic field of a single pole is i=士m2 where the positive sign is for a north pole and the negative for a south pole.The dipole magnetic field is the sum of the two fields:a north pole at y=+a/2 and a south pole at y =-a/2,where the y axis is horizontal and pointing north.a is a small distance much smaller than the radius of the iron balls;in general a =KiBe where Ki is a constant that depends on the size of the iron sphere. North B a iron ball c.Derive an expression for the magnetic field B;from the iron a distance d>a from the center of the ball.Note that there will be a component directed radially away from the ball and a Copyright C2017 American Association of Physics Teachers2017 USA Physics Olympiad Exam Part A 8 A pair of iron balls placed in the same horizontal plane as the compass but a distance d away can be used to help correct for the error caused by the induced magnetism of the ship. A binnacle, protecting the ship’s compass in the center, with two soft iron spheres to help correct for errors in the compass heading. The use of the spheres was suggested by Lord Kelvin. Just like the ship, the iron balls will become magnetic because of the Earth’s field Be. As spheres, the balls will individually act like dipoles. A dipole can be thought of as the field produced by two magnetic monopoles of strength ±m at two different points. The magnetic field of a single pole is B~ = ±m ˆr r 2 where the positive sign is for a north pole and the negative for a south pole. The dipole magnetic field is the sum of the two fields: a north pole at y = +a/2 and a south pole at y = −a/2, where the y axis is horizontal and pointing north. a is a small distance much smaller than the radius of the iron balls; in general a = KiBe where Ki is a constant that depends on the size of the iron sphere. φ North B~ i iron ball a c. Derive an expression for the magnetic field B~ i from the iron a distance d  a from the center of the ball. Note that there will be a component directed radially away from the ball and a Copyright c 2017 American Association of Physics Teachers