b. Assuming that and are both much smaller than one, at what heading(s) will... · OlympiadBench #1438

Physics · electromagnetism

Problem

b. Assuming that KbK_{b} and KsK_{s} are both much smaller than one, at what heading(s) θ\theta will the deviation δθ\delta \theta be largest? Given constants: g=9.8 N/kgg=9.8 \mathrm{~N} / \mathrm{kg} k=1/4πϵ0=8.99×109 N⋅m2/C2k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} c=3.00×108 m/sc=3.00 \times 10^{8} \mathrm{~m} / \mathrm{s} NA=6.02×1023( mol)−1N_{\mathrm{A}}=6.02 \times 10^{23}(\mathrm{~mol})^{-1} σ=5.67×10−8 J/(s⋅m2⋅K4)\sigma=5.67 \times 10^{-8} \mathrm{~J} /\left(\mathrm{s} \cdot \mathrm{m}^{2} \cdot \mathrm{K}^{4}\right) 1eV=1.602×10−19 J1 \mathrm{eV}=1.602 \times 10^{-19} \mathrm{~J} me=9.109×10−31 kg=0.511MeV/c2m_{e}=9.109 \times 10^{-31} \mathrm{~kg}=0.511 \mathrm{MeV} / \mathrm{c}^{2} mp=1.673×10−27 kg=938MeV/c2m_{p}=1.673 \times 10^{-27} \mathrm{~kg}=938 \mathrm{MeV} / \mathrm{c}^{2} sin⁡θ≈θ−16θ3\sin \theta \approx \theta-\frac{1}{6} \theta^{3} for ∣θ∣≪1|\theta| \ll 1 G=6.67×10−11 N⋅m2/kg2km=μ0/4π=10−7 T⋅m/AkB=1.38×10−23 J/KR=NAkB=8.31 J/(mol⋅K)e=1.602×10−19Ch=6.63×10−34 J⋅s=4.14×10−15eV⋅s(1+x)n≈1+nx for ∣x∣≪1ln⁡(1+x)≈x for ∣x∣≪1cos⁡θ≈1−12θ2 for ∣θ∣≪1\begin{aligned} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\ & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} \\ & k_{\mathrm{B}}=1.38 \times 10^{-23} \mathrm{~J} / \mathrm{K} \\ & R=N_{\mathrm{A}} k_{\mathrm{B}}=8.31 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K}) \\ & e=1.602 \times 10^{-19} \mathrm{C} \\ & h=6.63 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}=4.14 \times 10^{-15} \mathrm{eV} \cdot \mathrm{s} \\ & (1+x)^{n} \approx 1+n x \text { for }|x| \ll 1 \\ & \ln (1+x) \approx x \text { for }|x| \ll 1 \\ & \cos \theta \approx 1-\frac{1}{2} \theta^{2} \text { for }|\theta| \ll 1 \end{aligned}  velocity parameter β=vc Lorentz factor γ=11−β2 relativistic momentum p=γβmc relativistic energy E=γmc2 relativistic doppler shift ff0=1−β1+β\begin{array}{lrl} \text { velocity parameter } & \beta & =\frac{v}{c} \\ \text { Lorentz factor } & \gamma & =\frac{1}{\sqrt{1-\beta^{2}}} \\ \text { relativistic momentum } & p & =\gamma \beta m c \\ \text { relativistic energy } & E & =\gamma m c^{2} \\ \text { relativistic doppler shift } & \frac{f}{f_{0}} & =\sqrt{\frac{1-\beta}{1+\beta}} \end{array} Extra Supplementary Reading Materials: 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 BeB_{e}, and the orientation of the field be horizontal, pointing directly toward true north. The Earth's magnetic field BeB_{e} will magnetize the ship, which will then create a second magnetic field BsB_{s} in the vicinity of the ship's compass given by B→s=Be(−Kbcos⁡θb^+Kssin⁡θs^)\overrightarrow{\mathbf{B}}_{s}=B_{e}\left(-K_{b} \cos \theta \hat{\mathbf{b}}+K_{s} \sin \theta \hat{\mathbf{s}}\right) where KbK_{b} and KsK_{s} are positive constants, θ\theta is the angle between the heading of the ship and magnetic north, measured clockwise, b^\hat{\mathbf{b}} and s^\hat{\mathbf{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. Context question: a. Derive an expression for the deviation of the compass, δθ\delta \theta, from north as a function of KbK_{b}, KsK_{s}, and θ\theta. Context answer: δθ=arctan⁡(Ks−Kb)sin⁡θcos⁡θ1−Kbcos⁡2θ−Kssin⁡2θ\delta \theta=\arctan \left(K_{s}-K_{b}\right) \frac{\sin \theta \cos \theta}{1-K_{b} \cos ^{2} \theta-K_{s} \sin ^{2} \theta}

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