计算题25道1. and are two vectors at point and , respectively. Determine (a) ,(b) ,(c) , (d) the unit vector perpendicular to and (e) the angle between and 题2.202. If the electric field intensity in space is given as, find and.题2.423. Given, verify the divergence theorem when the region is bounded by a cylinder and the planes at and .题2.444. In electromagnetic fields, we will define the electric field intensityas a negative gradient of a scalar function; i.e. We will also define the volume charge distribution as, where is the permittivity of free space. Determineandif (a) in cylindrical coordinates, whereandare constants, (b) in spherical coordinates, andin spherical coordinates.题2.38解：（a）在柱坐标下5. 3.6 An electron is circling around nucleus of a hydrogen atom at a radius of 0.05nm. Determine the angular velocity and the time period of the electron. 题3.66. Two straight uniformly charged lines are extending along the z axis fromto. One line carries a charge distribution ofand is situated at, and the other line is atand has a charge distribution of. Determine the E field at appoint on the x axis at.题3.137. Three infinite electric sheets carrying uniform charge densities of 2,-3 and 0.5are each separated by a 1-mm air gap, as shown in figure. Find the electric field intensity everywhere in space. 题3.148. 3.16 A finite line extends fromtoand carries a charge distribution of. Determine thefield at a point 2 meters away from the line in theplane.题3.169. A spherical volume of radius b supports a charge distribution of everywhere except at . Determine the electric flux density in the region (a), and (b)题3.2510. A long coaxial cable consists of an inner conductor of radius a and an outer conductor of inner radius b and outer radius c. The inner conductor carries a uniform surface charge distribution s. Find the E field at any point in space when (a) the outer conductor is not grounded, and (b) the outer conductor is grounded.题3.4111. Find the magnetic flux density at point P.题5.512. A metal sphere of radius a has uniform surface charge distribution. The permittivity of the surrounding region varies as. Find (a) D, E, and P everywhere in space, (b) the bound charge densities, and (c) the energy density, and (d) show that the potential in the dielectric region is , where Q is the total charge on the sphere.题3.4513. The magnetic flux density in a region1, is . If marks the boundary between regions 1 and 2, determine the magnetic flux density in region 2. Consider region1 as free space; regions 2 has a relative permeability of 100.题5.3514. A current sheet of separates two regions at. Region 2, has a magnetic field intensity ofand a relative permeability of 200. If region 1, has a permeability of 1000, determine the magnetic field intensity in this region.题5.3615. Capacitance?题3.5816. Capacitance?题3.5917. 3.60 Capacitance?(a=1cm, b=1.5cm)题3.6018. 3.61 Capacitance? (红色为极板)（xoy方向俯视图，z方向高度为h=10cm）题3.6119. 3.62 Capacitance?（球形）题3.6220. Determine the potential distribution in a coaxial cable when the inner conductor of radius 10cm is at a potential of 100v and the outer conductor radius 20cm is grounded. The space between the conductors is filled with two concentric dielectrics. The dielectric constant is 3 for the inner dielectric and 9 for the outer dielectric. The dielectric interface is at a radius of 15cm. Determine (a) the potential distribution, (b) the D and E field in each region, (c) the surface charge density on the inner conductor, and (d) the capacitance of the 100-meter-long cable.题3.6621. Calculate the capacitance of two spherical shells of radii a and b using Laplaces equation. The inner shell is at a potential of V0, and the outer shell is grounded. What is the surface charge density on the inner shell? Obtain the expression for the capacitance of the system.题3.6722. Show that the electric field intensity as given by is a valid solution of the wave equation in a dielectric medium. Also determine (a) the dielectric constant of the medium assuming that, (b) the velocity of propagation, (c) the magnetic field intensity, (d) the intrinsic impedance, (e) the wavelength, and (f) the average power density.题8.423. The electric field intensity of a uniform plane wave in free space is given by. Determine (a) the phase constant, (b) the magnetic field intensity, (c) the wavelength, (d) the average power density in the medium, (e) the average energy density in the electric field, and (f) the average energy density in the magnetic field.题8.624. A uniform plane wave at a frequency of 100MHz propagates in the z direction in a conducting medium characterized by and. Determine,and . If the maximum value of the electric field intensity in the y direction is and occurs at z=0 and t=0, write expressions for the E and H field. Obtain a general