課程名稱︰電磁學下 課程性質︰物理系大二必修 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）︰100'06'21 考試時限（分鐘）：120分鐘 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 註: 全形代表向量，單位向量會加註︿ ┌───────────────────────────────────┐ │Some constants: │ │ -12 2 2 -7 2 -19 │ │ε0 = 8.852 ×10 C /Nm , μ0 = 4π ×10 N/A , e = 1.6 ×10 C, │ │ -31 8 │ │m_e = 9.11 ×10 kg, and c = 3 ×10 m/s, = 1/√(ε0μ0). │ └───────────────────────────────────┘ 1. It is known that the scalar potential of a point charge q moving with constant velocity can be written as 1 g V(ｒ,t) = ─── ────────────, and Ａ(ｒ,t) = ｖV(ｒ,t)/c^2, 4πε0 R√(1-(v^2)sin^2θ/c^2) where Ｒ≡ｒ－ｖt is the vector from the present position to the field point ｒ, and θ is the angle between Ｒ and ｖ. Suppose q is constrained to move ︿ along the x axis with ｖ = vｘ. (a) Find the scalar and vector potentials at the point (a,0,0) on the x axis, at the moment the point charge itself is at the origin. (10%) (b) Find the scalar and vector potentials at the point (0,b,0) on the y axis, at the moment the point charge itself is at the origin. (10%) 2. (a) For the same case in 1(a), find the electric and magnetic fields at point (a,0,0) on the x axis. (10%) [Hint: q η Ｅ(ｒ,t) = ──── ────────[(c^2 - v^2)ｕ + η(向量) ×(ｕ ×ａ)], 4πε0 (η(向量)‧ｕ)^3 and ︿ η ︿ Ｂ(ｒ,t) = ─ ×Ｅ(ｒ,t), where ｕ = cη－ｖ.] c (b) For the same case in 1(b), find the electric and magnetic fields at point (0,b,0) on the y axis. (10%) (c) For the same case in 1(a) again, calculate the total power passing through the plane x = a, at the moment the point charge itself is at the origin. (10%) [Remember that the Poynting vector can be written as Ｓ=ε0[E^2ｖ－(ｖ‧Ｅ)Ｅ] P = ∫Ｓ‧dａ, for a point charge moving with constant velocity, ︿ q 1－v^2/c^2 Ｒ Ｅ(ｒ,t) = ──── ────────────── ──, and the integral: 4πε0 (1－(v^2)sin^2θ/c^2)^(3/2) R^2 ∞ rsin^2θ γ^4 1 ∫ ───────────────dr = ───, γ≡ ────────.] 0 R^4 (1－(v^2)sin^2θ/c^2)^3 4a^2 √(1－v^2/c^2) (d) What about the radiation power in (c)? (10%) 3. (a) Find the radiation resistance for an oscillating magnetic dipole with radius of b = 0.05 m, and radiation wavelength λ = 1000 m. (10%) [Hint: μ0 m0^2 ω^4 〈P〉= ──────── for magnetic dipole radiation.] 12πc^3 (b) Calculate the lifespan of Bohr's atom, the hydrogen atom with an electron traveling in a circle of radius r0 = 5 ×10^(-11) m. (10%) [Hint: Larmor formula: P = μ0 q^2 a^2 / (6πc), and at radius of r0, the speed of electron v = 0.0075c.] (c) Using the concept of radiation resistance, I^2R = Prad, estimate the radiation resistance for the Bohr's atom with radius r0 of 5 ×10^(-11) m. [Hint: the current I = e/T, where T is the period of one revolution, and √(μ0/ε0) = 377Ω]. (10%) 4. A particle of mass m = 0.0911 kg and charge q = 0.16 C is attached to a spring with force constant k = 9.11 N/m, hanging from the ceiling. It is pulled down a distance d = 0.1 m below equilibrium and released at time t = 0. (a) Calculate the radiation damping factor given by γ= ω^2τ with τ≡ (μ0 q^2)/(6πmc), referring to τ= 6 ×10^(-24) s for electrons. According to your result, the damping is "small" or not? (10%) (b) It is known that the average energy per unit time striking the entire floor can be written as P = (μ0 q^2 d^2 ω^4)/(24πc). Calculate the fraction of its initial total energy lost to radiation in one cycle. (10%) 5. (a) When an electron (mass m = m_e, charge q = e) approaches a conducting surface, radiation is emitted, associated with the changing electic dipole moment of the charge and its image. Find the total radiation power as a function of its height z above the plane. (10%) [Hint: P = (μ0 p''^2)/(6πc)] (b) Using the formula of the radiation force on a radiation particle: μ0 q^2 γ^4 Frad = ─────── (a' + 3γ^2 a^2 v/c^2), where γ= (1－v^2/c^2)^(-1/2), a 6πc is the acceleration, and v = z', determine reaction reaction force. Express your answer with v, e, m_e, γ, z... (10%) 6. A point charge q is at rest at the origin in system S0. The electric field of this same charge in system S, which moves to the right at speed v relative to S0 can be written as previously seen in Problem 2(a). (a) Using the results that fields in two different frames are simply related by Ｂ = -(ｖ ×Ｅ)/c^2 (if Ｂ0 = 0 in S0), and Ｅ = ｖ ×Ｂ (if Ｅ0 = 0 in S0), determine the magnetic field of a point charge q moving at constant velocity ｖ (You should draw a diagram to show the directions of ｖ and Ｂ that you define) (10%) (b) Check that the Gauss's law is obeyed by the field of a point charge in uniform motion. (10%) [Hint: π sinθ 2 ∫ ───────────────dθ = ──────.] 0 (1－(v^2)sin^2θ/c^2)^(3/2) 1－v^2/c^2 --

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