课程名称︰电磁学 课程性质︰必修 课程教师︰王立民 开课学院：理学院 开课系所︰物理系 考试日期（年月日）︰100. 10. 27 考试时限（分钟）：120分钟 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : notation: a. The arrow of the vector is denoted by "bar"; e.g., the position vector _ is denoted r. If the vector is an unit vector, "bar" will be replaced by "hat", ^. _ b. л denotes the separation vector from the source pointing to the observer. ^ 1. (a) Find the components of the unit vector n perpendicular to the plane shown in Fig.1. (b) Also find the vector area of the plane shown in Fig.1. _ _ _ _ (c) For the vector function v=r, calculate the integral ∮v╳dl, where the integral is around the boundary line of the plane shown in Fig.1. _ (d) Check the Stoke's Theorem for v over the triangular surface with corners at (1,0,0), (0,1,0), and (0,0,1) as shown in Fig.1. Fig.1 The portion of plane x+y+z=1 in the first octant _ 2. Check the gradient theorem of ∫(▽T)dτ=∮Tda, for the scalar function T=r^2, using the volume of a hemisphere with radius of R as shown in Fig.2. Note. The gradient formula in spherical coordinate is given. Fig.2 A hemisphere with radius R 3. A hollow spherical shell carries charge density ρ=kr^(-2) in the region a≦r≦b. (a) Find the total charge Q. (b) Find the electrostatic energy W stored. (c) Find the net force that southern hemispherical shell exerts on the northern hemispherical shell, in terms of a, b and k. (d) If the same charge Q redistributing on a conducting hollow spherical shell with the same inner and outer radii of a and b. Again find the net force that southern hemispherical shell exerts on the northern hemispherical shell. _ ^ [Hint: the formula of the force per unit area f=σ^2/2ε_0n] 4. A solid conical ice-cream cone carries a uniform volume charge density ρ. The height of the cone is h, as is the radius of the top. Find the potential at the vertex point a. 5. If the actual force of interaction between two point charges is found to be _ 1 q_1q_2 л ^ F= ------- --------(1+---) exp[-л/λ]л, 4πε_0 л λ where λ is a huge-number constant. This makes a new scalar potential admitted the electric field. (a) Find the potential of a point charge with using infinity as the reference point. (b) Using the result in (a), find the potential at a distance z above the center of a circular disk with uniform surface charge σ as seen in Fig.3. (c) Show that the slab looks like a point charge for z>>R, using the approximation: exp[x]~1+x and (1+x)^n~1+nx for x<<1. Fig.3 A circular disk of radius R and surface charge density σ. P:(0,0,z), regarding the center of the disk as origin and the disk as x-y plane. 6. Consider two concentric spherical shells, of radii a and b. Suppose the inner one carries a charge 2q, and the outer one a charge -q(both uniformly distributed). Calculate the energy of this configuration in two different ways: (a) Use the energy density ε_0E^2/2, and _ _ (b) the superposition principle: W_{tot}=W_1+W_2+ε∫E_1。E_2dτ. --

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