課程名稱︰電磁學上
課程性質︰物理系雙班必帶
課程教師︰王立民
開課學院：理學院
開課系所︰物理系
考試日期（年月日）︰100'01'13
考試時限（分鐘）：110分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
列式、答案要求清楚;滿分170分 (後面有(v)的是向量)
1. A very long cylinder of linear dielectrics material is placed in an
otherwise uniform electric field E_0(v). The radius of cylinder is a, the
susceptibility χ_e, and the axis is perpendicular to E_0(v) as shown in
Fig.1.
(a) Find the resulting field inside and outside the cylinder.(10%)
(b) Find the electric displacement inside and outside the cylinder.(10%)
(c) Calculate the bound charges ρ_b andσ_b.(10%)
[Hint: The general solution with cylindrical symmetry:
∞ k
V(s,ψ) = a_0 + b_0 lns + Σ[s (a_k coskψ+b_k sinkψ) +
k=1
-k
s (c_k coskψ+d_k sinkψ)],]
Fig.1: 一個圓柱，中心軸為z軸，E_0方向和正x方向同向，其餘與圓柱座標同
^
2. A point dipole p(v) = pz, is imbedded at the center of a sphere of linear
dielectric material(with radius R and dielectric constant ε_r). The
electric potentials inside and outside the sphere respectively are:
pcosθ r^3 (ε_r -1) pcosθ 3
────(1+2───────),(r≦R); ────(────),(r≧R);
4πεr^2 R^3 (ε_r +2) 4πεr^2 ε_r +2
(a) Find the electric field inside and outside the sphere.(10%)
(b) There is a tiny sphere of the same linear dielectric material(with
radius R_s, R_s<<R, and dielectric constant ε_r) located at point A
(on z-axis, as shown in Fig.2). Find the polarization P(v) induced in the
tiny sphere.[Remenber that the field inside a sphere of linear
dielectric material due to a uniform field E_ext(v) applied is
E(v) = 3/(ε_r+2)‧E_ext(v).](10%)
(c) Determine the force on this tiny sphere due to the sphere of radius R.
[Here the formula F(v) = ∫(P(v)‧▽)E_ext(v)dτ should be used.](10%)
(d) If the tiny sphere is situated at point B(on y-axis), again determine
the force on this tiny sphere.(10%)
Fig.2: 一個以原點為中心半徑為R的球，中心放置一個dipole p(z方向)，A在(0,0,z)
B在(0,y,0)，A、B兩小球的半徑皆為R_s
3. (a) Find the density ρ of mobile charges in a piece of copper(atomic mass
= 64g/mole, density = 9.0g/cm^3), assuming each atom contributes one
free electron.(10%)
(b) Find the Hall voltage for a copper rectangular bar in a uniform
magnetic field 1T with a given current I = 1A (see Fig.3). [Note the
unit equilibrium: m^2AT/C = V](10%)
▕← 3.0cm →▏
I=1A __________________________ __
──→／ ↗ 1.0cm ／▏↑0.2cm
／ ↙ __／ ▏↓
▕￣￣￣￣￣￣￣￣￣￣￣￣▕ ／ ￣
▕________________________▕／
↙↙↙↙ ￣
B=1T
Fig.3
4. A spherical shell, of radius R, carrying a uniform surface charge σ, is
spinning at angular velocity ω. It produces the vector potential
A(v)(r,θ,φ) at point r(v) is:
μ_0 Rωσ ︿ μ_0 R^4ωσ sinθ ︿
─────rsinθψ(r≦R); ────── ───ψ(r≧R).
3 3 r^2
It also produce an inside magnetic field B_in(v) = (2/3)μ_0 σRω(v), a
magnetic dipole moment of m(v) = (4π/3)σωR^4(z-direction), and the
magnetic force of attraction between the northern and southern hemispheres
is -(π/4)μ_0(σ^2)(ω^2)(R^4)(z-direction).
(a) Use these results to find the vector potential and field outside a
uniformly charged sphere, of total charge Q and radius R, which is
rotating at a angular velocity ω.(10%)
(b) Find the magnetic dipole moment of this charged sphere.(10%)
(c) Calculate the magnetic force of attraction between the northern and
southern hemispheres of this spinning charged sphere.(10%)
︿
[For a vector v(v) = v_ψ(r,θ)ψ,
1 δ ^ 1 δ ︿
▽×v(v) = ───[──(sinθ‧v_ψ)]r + ─[-──(rv_ψ)]θ ]
rsinθ δθ r δr
5. (a) Find the magnetic fields(inside and outside) of a uniformly magnetized
sphere(radius R) with M(v) = M(z-direction). [Hint: use the result of
B_in(v) in question 4, and B_dip(v) = (μ_0 m/4πr^3)(2cosθ(r-dir.)+
sinθ(θ-dir.)).](10%)
(b) A tiny sphere of linear magnetic material(with radius R_s, R_s<<R, and
susceptibility χ_m) is situated on the z-axis with a distance z from
the center of the magnetized sphere in (a). Find the magnetization
induced in this tiny sphere. (Use the result:
B_in(v) = [(1+χ_m)/(1+χ_m/3)]B_0(v) for a sphere of linear magnetic
material in an otherwise uniform magnetic field B_0(v).)(10%)
(c) Find the force of attraction between the two spheres. (Remind that the
force between two parallel magnetic moments is -3μ_0(m_1)(m_2)/2πr^4,
where r is the distance between them.(10%)
6. An infinitely long cylinder, of radius R, carries a "frozen-in"
magnetization, parallel to the axis, M(v) = ks(z-direction), where k is a
constant and s is the distance from the axis; there is no free current
anywhere.
(a) Find the magnetic field inside and outside the cylinder.(10%)
(b) Find H(v) and see where ▽‧H(v)≠0 in this case?(10%)
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