“2019年USAAAO决赛第10题”的版本间的差异
Jingsong Guo（讨论 | 贡献） （创建页面，内容为“==英文题目== 10. (25 points) In this problem, we will try to understand the relationship between magnetic moments and angular momenta, first for charged particl…”） |
Jingsong Guo（讨论 | 贡献） |
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v. Write down the angular momentum L of the charge and magnetic moment µ of the effective | v. Write down the angular momentum L of the charge and magnetic moment µ of the effective | ||
current loop. Recall that the magnetic moment of a current loop with current I and radius r is | current loop. Recall that the magnetic moment of a current loop with current I and radius r is | ||
| − | given as µ | + | given as $$µ=IA$$ where A is the area of the loop. |
(b) (3 points) Use the above results to find a relationship between the magnetic moment µ and angular | (b) (3 points) Use the above results to find a relationship between the magnetic moment µ and angular | ||
momentum L in terms of intrinsic properties of the particle (charge,mass). | momentum L in terms of intrinsic properties of the particle (charge,mass). | ||
| − | (c) (2 points) The relationship from part (b) can be expressed as µ | + | (c) (2 points) The relationship from part (b) can be expressed as $$µ=γL$$. γ is usually referred to as the |
classical gyromagnetic ratio of a particle. Evaluate the classical gyromagnetic ratio for an electron | classical gyromagnetic ratio of a particle. Evaluate the classical gyromagnetic ratio for an electron | ||
and for a neutron in SI units. | and for a neutron in SI units. | ||
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whereas the surface magnetic field can be measured. Assuming a magnetic dipole of magnetic | whereas the surface magnetic field can be measured. Assuming a magnetic dipole of magnetic | ||
moment µ located at the center of a sphere of radius r, write down the expression for the surface | moment µ located at the center of a sphere of radius r, write down the expression for the surface | ||
| − | magnetic field | + | magnetic field $$B_{surf}$$ and the surface magnetic moment defined as $$\mathcal{M}_{surf}=B_{surf} r^3$$. You may use |
| − | 3 | ||
| − | . You may use | ||
the value of the angular dependence at the magnetic equator for the following parts. | the value of the angular dependence at the magnetic equator for the following parts. | ||
(e) (3 points) Assuming a gyromagnetic relationship exists between magnetic moment µ and angular | (e) (3 points) Assuming a gyromagnetic relationship exists between magnetic moment µ and angular | ||
momentum L of an extended object, write down the relationship between the surface magnetic | momentum L of an extended object, write down the relationship between the surface magnetic | ||
| − | moment | + | moment $$\mathcal{M}_{surf}$$ and angular momentum L as $$\mathcal{M}_{surf} =\kappa L$$. You will observe that κ depends only |
on fundamental constants and intrinsic properties of the extended object. | on fundamental constants and intrinsic properties of the extended object. | ||
| − | (f) (3 points) The surface magnetic moments for Mercury and Sun are 5 | + | (f) (3 points) The surface magnetic moments for Mercury and Sun are $$5 \times 10^{12} T m^3$$ and $$3 \times 10^{23} Tm^3$$ |
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respectively. Assuming the bodies are perfect spheres, evaluate the constant κ for Mercury and | respectively. Assuming the bodies are perfect spheres, evaluate the constant κ for Mercury and | ||
the Sun. Comment on values obtained and if they fit into the model developed in parts (c) and (d). | the Sun. Comment on values obtained and if they fit into the model developed in parts (c) and (d). | ||
| − | (g) (5 points) The surface magnetic moments | + | [[文件:USAAAO2019Figure3.png|缩略图|Figure 3: Surface magnetic moment vs angular momentum for solar system objects. Figure taken from |
| − | bodies are plotted in the figure 3. Justify that the data implies | + | Vall´ee, Fundamentals of Cosmic Physics, Vol. 19, pp 319-422, 1998. |
| − | + | ]] | |
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| + | (g) (5 points) The surface magnetic moments $$\mathcal{M}_{surf}$$ and angular momenta L of various solar system | ||
| + | bodies are plotted in the figure 3. Justify that the data implies $$\mathcal{M}_{surf} \sim L^{\alpha}$$ and calculate the | ||
constant α. What is the expected value of α from the model developed in parts (c) and (d)? | constant α. What is the expected value of α from the model developed in parts (c) and (d)? | ||
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(h) (2 points) Certain bodies such as Venus, Mars and the Moon are remarkably separated from the | (h) (2 points) Certain bodies such as Venus, Mars and the Moon are remarkably separated from the | ||
trend observed for other bodies. What can you say about magnetism in these bodies when compared | trend observed for other bodies. What can you say about magnetism in these bodies when compared | ||
to the others? | to the others? | ||
2020年3月7日 (六) 18:26的最新版本
英文题目
10. (25 points) In this problem, we will try to understand the relationship between magnetic moments and angular momenta, first for charged particles and how this can be extended to planetary objects.
(a) (5 points) Consider a charge e and mass m moving in circular orbit of radius r with constant speed v. Write down the angular momentum L of the charge and magnetic moment µ of the effective current loop. Recall that the magnetic moment of a current loop with current I and radius r is given as $$µ=IA$$ where A is the area of the loop.
(b) (3 points) Use the above results to find a relationship between the magnetic moment µ and angular momentum L in terms of intrinsic properties of the particle (charge,mass).
(c) (2 points) The relationship from part (b) can be expressed as $$µ=γL$$. γ is usually referred to as the classical gyromagnetic ratio of a particle. Evaluate the classical gyromagnetic ratio for an electron and for a neutron in SI units.
(d) (7 points) For extended objects such as planets, the magnetic dipole moment is not directly accessible whereas the surface magnetic field can be measured. Assuming a magnetic dipole of magnetic moment µ located at the center of a sphere of radius r, write down the expression for the surface magnetic field $$B_{surf}$$ and the surface magnetic moment defined as $$\mathcal{M}_{surf}=B_{surf} r^3$$. You may use the value of the angular dependence at the magnetic equator for the following parts.
(e) (3 points) Assuming a gyromagnetic relationship exists between magnetic moment µ and angular momentum L of an extended object, write down the relationship between the surface magnetic moment $$\mathcal{M}_{surf}$$ and angular momentum L as $$\mathcal{M}_{surf} =\kappa L$$. You will observe that κ depends only on fundamental constants and intrinsic properties of the extended object.
(f) (3 points) The surface magnetic moments for Mercury and Sun are $$5 \times 10^{12} T m^3$$ and $$3 \times 10^{23} Tm^3$$ respectively. Assuming the bodies are perfect spheres, evaluate the constant κ for Mercury and the Sun. Comment on values obtained and if they fit into the model developed in parts (c) and (d).
(g) (5 points) The surface magnetic moments $$\mathcal{M}_{surf}$$ and angular momenta L of various solar system bodies are plotted in the figure 3. Justify that the data implies $$\mathcal{M}_{surf} \sim L^{\alpha}$$ and calculate the constant α. What is the expected value of α from the model developed in parts (c) and (d)?
(h) (2 points) Certain bodies such as Venus, Mars and the Moon are remarkably separated from the trend observed for other bodies. What can you say about magnetism in these bodies when compared to the others?
