課程名稱︰電磁學一 課程性質︰必修 課程教師︰林晃巖 開課學院：電機資訊學院 開課系所︰電機工程學系 考試日期（年月日）︰104/4/24 考試時限（分鐘）：110分鐘 試題 : Electromagnetics (I) Midterm Examination April 24, 2015 1. (a) Write down the Maxwell equations in differential form. (b) Show that these equations are consistent with the law of conservation of charge.(15%) 2. (a) What is the Biot-Savart law? State its mathematical expression. (b) Is it exactly validfor computing the magnetic field due to a time-varying current distribution? (14%) 3. A point charge Q moves with velocity V in a vacuum region characterized by electric field of intensity E and magnetic field of flux density B. (a) Find the force acting on charge Q. (b) Find the power delivered by E field to charge Q. (c) Show that the power delivered by B field to charge Q is zero. (d) If the aforementioned electric and magnetic fields are the corresponding fields of a traveling uniform plane wave, show that the ratio of the magnitude of magnetic force to that of electric force, experienced by charge Q, is less than or equal to |v|/c, where c is the speed of light in vacuum. (16%) 4. A magnetic flux density is given in the xz-plane by B = B_0 sin[π(x + v_0 t)]a_y, where v_0 is much smaller than the speed of light. Consider a rigid square loop situated in the xz-plane with its vertices at (x, 0, 3), (x, 0, 4), (x+1, O, 4), and (x+1, 0, 3). (a) Find the expression for the emf induced around the loop in the sense defined by connecting the above points in succession. (b) Let us move (without rotation) the loop with the velocity v to produce zero induced emf on it. How should we choose v? (14%) 5. A current density due to flow of charges is given by J = -(3x a_x + y a_y +z^2 a_z). (a) Byusing surface integrals directly, find the displacement current emanating from the closed surface of the cubical box bounded by the planes x = ±1, y = ±1, and z = ±1. (b) Redo (a) by using an appropriate volume integral. (14%) 6. Consider a vector field A = y sinx a_x - cosx a_y. (a) Evaluate the line integral ∫A．dlfrom point (x1, y1, z1) to point (x2, y2, z2) along the straight line connecting them. (b) Find ▽ x A. (c) Use the results of (a) and (b) to verify Stokes' theorem. (15%) J_0[1-exp(-10^6t)]a_y, t≧ 0 7. An infinite plane sheet of current density J_S={ 0 , t < 0 A/m lies in the z = 0 plane in free space. (a) Find E_y(z, t) as a function of t in the z = 300 m plane.(b) Find the asymptotic result of (a) as t approaches infinity. (c) As t approaches infinity,the surface current density becomes almost constant, producing an almost constant tangential electric field as well as an almost constant tangential magnetic field. This seems contradictory to the fact that a DC current produces a static magnetic field only. Please explain this paradox briefly. (12%) --

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