Homework #4  注意：以下題目向量是以粗體字表示，但是書寫作業時，一般向量請寫成 A ，基底向量請寫成 aˆ x 、 aˆ 等。繳交之作業請用A4紙書寫並裝訂妥。 Deadline: 1/10 下午17:00前交至717室 P. 5-3 A long, round wire of radius a and conductivity  is coated with a material of conductivity 0.1. a) What must be the thickness of the coating so that the resistance per unit length of the uncoated wire is reduced by 50%? b) Assuming a total current I in the coated wire, find J and E in both the core and the coating material. Solution: R1 = resistance per unit length of core  1 1  . Assuming thickness of coating = b,  Score  a 2 R2 = resistance per unit length of coating  1 1 1   . 2 2 0.1 Scoat 0.1  (a  b)  a  0.1  2ab  b 2  a) Resistance is reduced by 50%, thus R1 = R2: a 2  0.1 2ab  b 2   11a 2   a  b  I I b) I core  I coat  . J core  , 2 2 a 2 Ecore  J core   2 J coat   b   11  1 a  b  2.3166a. I I   0.1J core . 2Scoat 20 a 2 J I I , Ecoat  coat   Ecore . 2 2 a  0.1 2 a 2 P. 5-6 Lighting strikes a lossy dielectric sphere— = 1.2 0,  = 10 (S/m)—of radius 0.1 (m) at time t = 0, depositing uniformly in the sphere a total charge 1 (mC). Determine, for all t, a) the electric field intensity both inside and outside the sphere, b) the current density in the sphere. Solution: P. 5-10 The space between two parallel conducting plates each having an area S is filled with an inhomogeneous ohmic medium whose conductivity varies linearly from 1 at one plate (y = 0) to 2 at the other plate (y = d). A d-c voltage V0 is applied across the plates as in Fig. 5.11. Determine a) the total resistance between the plates, b) the surface charge densities on the plates, c) the volume charge density and the total amount of charge between the plates, Solution: Fig. 5.11 P. 5-12 Refer again to Example 5-4. Assuming that a voltage V0 is applied across the parallel-plate capacitor with the two layers of different lossy dielectrics at t = 0, a) express the surface charge density si at the dielectric interface as a function of t, b) express the dielectric field intensities E1 and E2 as functions of t. Solution: P. 5-16 Determine the resistance between two concentric spherical surfaces of radii R1 and R2 (R1 < R2), assuming that a material of conductivity  = 0 (1 + k/R) fills the space between them. (Note: Laplace’s equation for V does not apply here.) Solution: P. 5-18 Redo Problem P5.17, assuming that the truncated conical block is composed of an inhomogeneous material with a nonuniform conductivity (R) = 0R1/R, where R1  R  R2 . 註: P5.17的題目: A homogeneous material of uniform conductivity s is shaped like a truncated conical block and defined in spherical coordinates by R1  R  R2 and 0     0 . Determine the resistance between the R = R1 and R = R2 surfaces. Solution: P. 5-20 Justify the statement that the steady-current problem associated with a conductor buried in a poorly conducting medium near a plane boundary with air, as shown in Fig. 5-12(a), can be replaced by that of the conductor and its image, both immersed in the poorly conducting medium as shown in Fig. 5-12(b). Solution: P. 5-22 Assume a rectangular conducting sheet of conductivity , width a, and height b. A potential difference V0 is applied to the side edges, as shown in Fig. 5-14. Find a) the potential distribution b) the current density everywhere within the sheet. (Hint: Solve Laplace’s equation in Cartesian coordinates subject to appropriate boundary conditions.) Solution: Fig. 5-14 A conducting sheet. P. 6-1 A positive point charge q of mass m is injected with a velocity u0 = ayu0 into the y > 0 region where a uniform magnetic field B = axB0 exists. Obtain the equation of motion of the charge, and roughly plot the path that the charge follows. Solution:  Since B is in the x-direction, the magnetic force will be in the y-z plane. u m z  qu y B0 z t u y m  qu z B0 t  2u y q 2 B02   2 uy → t 2 m 2 q 2 B02  uz   uz t 2 m2 Initial velocity at t = 0:   u0  a y u0 x y  qB  u y  u 0 cos 0 t   m   qB  u z  u0 sin  0 t   m  um   qB  t  and z  0 cos  0 t   C C depends on the position injecting into y > 0. qB0  m   We may take C = 0 for simplicity. y u 0 m  qB0 sin  qB0  m y2  z2  u 02 m 2 → a semi-circle. q 2 B02 P. 6-4 A current I flows lengthwise in a very long, thin conducting sheet of width w, as shown in Fig. 6-35. a) Assuming that the current flows into the paper, determine the magnetic flux density B1 at point P1(0, d). b) Use the result in part a) to find the magnetic flux density B2 at point P2(2w/3, d). Fig. 6-35 Solution: P. 6-6 Figure 6-37 shows an infinitely long solenoid with air core having a radius b and n closely wound turns per unit length. The windings are slanted at an angle  and carry a current I. Determine the magnetic flux density both inside and outside the solenoid. Solution: The problem can be decomposed into two sub-problems: a. A cylindrical tube carrying a uniformly distributed longitudinal surface current 2bnIsin. 0, 0  r  b,   → B1   bnI ˆ r  b. a r sin  , b. A solenoid with n turns per unit length carrying a current Icos.  aˆ  nI cos  , 0  r  b, → B2   z 0 0, r  b.     Total field: B  B1  B2 . Fig. 6-37 A long solenoid with closely wound windings carrying a current I.