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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