2019年USAAAO决赛第10题 - 版本历史

2020年3月7日 (六) 10:26 Jingsong Guo

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2020年3月7日 (六) 10:25 Jingsong Guo

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2020年3月7日 (六) 10:18 Jingsong Guo

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Jingsong Guo：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…”

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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…”

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==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 Bsurf and the surface magnetic moment defined as Msurf “ Bsurf 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 Msurf and angular momentum L as Msurf “ κL. 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 ˆ 1012 T m3 and 3 ˆ 1023 T
m3
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 Msurf and angular momenta L of various solar system
bodies are plotted in the figure 3. Justify that the data implies Msurf „ L
α and calculate the
constant α. What is the expected value of α from the model developed in parts (c) and (d)?

[[文件:USAAAO2019Figure3.png]]
Figure 3: Surface magnetic moment vs angular momentum for solar system objects. Figure taken from
Vall´ee, Fundamentals of Cosmic Physics, Vol. 19, pp 319-422, 1998.

(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?

@@ 第19行： / 第19行： @@
 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
+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
+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 $$3 \times 10^23 Tm^3$$
+(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).
 [[文件:USAAAO2019Figure3.png|缩略图|Figure 3: Surface magnetic moment vs angular momentum for solar system objects. Figure taken from
 Vall´ee, Fundamentals of Cosmic Physics, Vol. 19, pp 319-422, 1998.
 ]]
 (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?

@@ 第7行： / 第7行： @@
 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.
+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
+(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.
@@ 第19行： / 第19行： @@
 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 Bsurf and the surface magnetic moment defined as Msurf “ Bsurf r
+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 Msurf and angular momentum L as Msurf “ κL. You will observe that κ depends only
+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 ˆ 1012 T m3 and 3 ˆ 1023 T
+(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 Msurf and angular momenta L of various solar system
+(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 Msurf „ L
+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)?