2017 USA Physics Olympiad Exam Part A Question A3 A ship can be thought of as a symmetric arrangement of soft iron.In the presence of an external magnetic field,the soft iron will become magnetized,creating a second,weaker magnetic field.We want to examine the effect of the ship's field on the ship's compass,which will be located in the middle of the ship. Let the strength of the Earth's magnetic field near the ship be Be,and the orientation of the field be horizontal,pointing directly toward true north. The Earth's magnetic field Be will magnetize the ship,which will then create a second magnetic field Bs in the vicinity of the ship's compass given by B。=Be(-Kcos06+Ksin0s）》 where Ko and Ks are positive constants,6 is the angle between the heading of the ship and magnetic north,measured clockwise,b and s are unit vectors pointing in the forward direction of the ship (bow)and directly right of the forward direction(starboard),respectively. Because of the ship's magnetic field,the ship's compass will no longer necessarily point North. a.Derive an expression for the deviation of the compass,60,from north as a function of Ko, Ks;and 0. Solution We add the fields to get the local field.The northward component is Bnorth Be-BeKb cos0 cos0-BeKs sin 0 sin 0 while the eastward component is Beast =-BeKb sin 0 cos0+BeKs cos0 sin 0 The deviation is given by sin0 cos0 tan60=(K,-K)1-Kocos20-K sin20 This form is particularly nice,because as we'll see below,Ko and Ks are small enough to ignore in the denominator. b.Assuming that Ko and K's are both much smaller than one,at what heading(s)0 will the deviation 0 be largest? Solution By inspection,0=45°will yield the largest deviation.It's also acceptable to list45°，l35°， 225°，and315°. Copyright C2017 American Association of Physics Teachers2017 USA Physics Olympiad Exam Part A 7 Question A3 A ship can be thought of as a symmetric arrangement of soft iron. In the presence of an external magnetic field, the soft iron will become magnetized, creating a second, weaker magnetic field. We want to examine the effect of the ship’s field on the ship’s compass, which will be located in the middle of the ship. Let the strength of the Earth’s magnetic field near the ship be Be, and the orientation of the field be horizontal, pointing directly toward true north. The Earth’s magnetic field Be will magnetize the ship, which will then create a second magnetic field Bs in the vicinity of the ship’s compass given by B~ s = Be  −Kb cos θ bˆ + Ks sin θ ˆs  where Kb and Ks are positive constants, θ is the angle between the heading of the ship and magnetic north, measured clockwise, bˆ and ˆs are unit vectors pointing in the forward direction of the ship (bow) and directly right of the forward direction (starboard), respectively. Because of the ship’s magnetic field, the ship’s compass will no longer necessarily point North. a. Derive an expression for the deviation of the compass, δθ, from north as a function of Kb, Ks, and θ. Solution We add the fields to get the local field. The northward component is Bnorth = Be − BeKb cos θ cos θ − BeKs sin θ sin θ while the eastward component is Beast = −BeKb sin θ cos θ + BeKs cos θ sin θ The deviation is given by tan δθ = (Ks − Kb) sin θ cos θ 1 − Kb cos2 θ − Ks sin2 θ . This form is particularly nice, because as we’ll see below, Kb and Ks are small enough to ignore in the denominator. b. Assuming that Kb and Ks are both much smaller than one, at what heading(s) θ will the deviation δθ be largest? Solution By inspection, θ = 45◦ will yield the largest deviation. It’s also acceptable to list 45◦ , 135◦ , 225◦ , and 315◦ . Copyright c 2017 American Association of Physics Teachers