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Prof. Yen-Chieh Huang Dept of Electrical Engineering National Tsing-Hua University office: Delta 856, HOPE 301 ext: 62340 email:[email protected] EE214002 Electromagnetics, Fall 2012 ___________________________________________________________________________________ Nov. 11, 2012 EE214000 Electromagnetics, Fall 2012 Homework Assignment # 4 (credit points: 150) Due in class Nov.21, 2012 1. (10 points) An infinitely long line charge along z has a line charge density of l z , where is a constant. The line charge is placed in vacuum. (a) (5 points) What is the electric field at ( r , , z 0 )? Give the magnitude and direction of your answer. (b) (5 points) What is the energy required to take a charge of q from ( r a, 0, z 0 ) to ( r b, , z 0 )? Ans: (a) Refer to the following plot. Only the components in the â r direction exit due to cos symmetry. 1 40 z r z2 r work 2 cos dz . r 2 z2 2r E E r aˆ r aˆ r 40 (b) The Er Use z 0 (r z ) necessary 2 2 3/ 2 to dz be , where Therefore aˆ r 20 done is independent of path. b q W qVba q E dl ( a b) a 20 2. (20 points) An infinitesimal point charge of q locates at the x-y plane, and an infinitely long line charge of a line charge density l is arranged in parallel with the z axis. A 90o R1 B y 30o l 45o q x a. (4 points) Assume l = 0 and q 0. What is the work needed to be done for Prof. Yen-Chieh Huang Dept of Electrical Engineering National Tsing-Hua University office: Delta 856, HOPE 301 ext: 62340 email:[email protected] EE214002 Electromagnetics, Fall 2012 ___________________________________________________________________________________ moving a positive unit point charge from A to B on the x-y plane, as shown above? b. (8 points) Assume q = 0 and l 0. What is the work needed to be done for moving a positive unit point charge from A to B on the x-y plane? c. (8 points) Assume l 0 and q 0. What is the work needed to be done for moving a positive unit point charge from A to B on the x-y plane? Ans: (a) E q A 40 R 2 B 1 A 40 R V EdR (1 |R1 3 ) R1 3 ( 3 1) 80 R1 B q (b) ^ E ar l 20 r A A B A A' V E dR E dR 0 l 2 3 ln 40 3 B 30° , A’ 45° ρl q where A and A’ are in a circle of the same radius R1 (c) Wc Wa Wc q ( 3 1) q l 2 3 ln 80 R1 40 3 3. (10 points) The far-field electric potential at a distance R from an electric dipole p aˆ R with a dipole moment = p is given by V , where â R is the unit vector 40 R 2 pointing from the dipole to the field location. Suppose that two dipoles of dipole Prof. Yen-Chieh Huang Dept of Electrical Engineering National Tsing-Hua University office: Delta 856, HOPE 301 ext: 62340 email:[email protected] EE214002 Electromagnetics, Fall 2012 ___________________________________________________________________________________ moments aˆ z 5 nCm and aˆ z 9 nCm are located at points (0, 0, 2 m) and (0, 0, 3 m), respectively, in a Cartesian coordinate system. (hint: 1 nC = 10-9 C, 0 1 10 9 F/m). 36 a. (5 points) Find the electric potential at the origin. b. (5 points) Find the electric field intensity at the origin. Give the magnitude and direction of your answer. p aˆ R R p ( R R ) Ans: Rewrite the dipole potential as V 2 3 40 R R 40 R R ˆ z ; and for the (a) For the dipole with dipole moment aˆ z 5 , R R 2a ˆ z . Taking the linear sum dipole with dipole moment aˆ z 9 , R R 3a of the electric potential from the two dipoles, one has -19.25 V at the origin. (b) E V . Since both the dipole moments are aligned along the z axis, what we are concerned with is E z or ER ( 0) for the dipole at (0, 0, 2 m) and E R ( 180 ) for the dipole at (0, 0, 3 m). E R ( ) V p cos . R 40 R 3 p1 p where aˆ z 23 (aˆ z )) , 3 40 R1 R2 p1 9, R1 2, p2 5, R2 3 . Therefore, at the origin, E E z aˆ z 203 / 24 E1, R ( 0)aˆ1, R E2, R ( 180 )aˆ 2, R 1 ( V/m. 4. (35 points) A sphere consists of four regions, (1) an inner conductor region in R < a, (2) a vacuum region between a < R < 2a, (3) a dielectric region with a dielectric constant of r = 1.5 between 2a < R < 4a, (4) an outer conductor region in R > 4a. The inner conductor is grounded and the outer conductor is maintained at a voltage V0. a. (5 points) What is the capacitance of this sphere? b. (5 points) What is the electric field intensity in region (2)? c. (5 points) What is the electric field intensity in region (3)? d. (5 points) What is the surface charge density on the inner conductor? e. (5 points) What is the surface polarization charge density at R = 2a? f. (5 points) What is the surface polarization charge density at R = 4a on the Prof. Yen-Chieh Huang Dept of Electrical Engineering National Tsing-Hua University office: Delta 856, HOPE 301 ext: 62340 email:[email protected] EE214002 Electromagnetics, Fall 2012 ___________________________________________________________________________________ dielectric? g. (5 points) What is the surface charge density at R = 4a on the outer conductor? (hint: you can verify your answers from conservation of charges) (4) (2) (3) (1) Ans: (a) Assume the inner conductor has a charge Q. The electric field intensity in region (2) is E 2 Q 40 R 2 aˆ R and that in region (3) is E3 Q aˆ R . 40 1.5R 2 The potential between the inner and outer conductor is given by the integration a 2a Q 1 1 Q 1 1 Q V V0 E3 dR E 2 dR ( ) ( ) 4a 2a 60 2a 4a 40 a 2a 60 a The capacitance is C Q 60 a . V (b) From (a), the charge in region (1) can be calculated to be Q 60 aV0 . E2 Q 40 R 2 aˆ R 3aV0 aˆ R . 2R 2 aV0 (c) Substitute the charge calculated in (b), E3 aˆ R R2 (d) The surface charge density on the inner conductor is equal to s aˆ R D2 3 0V0 2a a 0V0 1 (e) P D3 0 E3 0 E3 aˆ R . The surface polarization charge 2 2R 2 density at R = 2a is given by ps aˆ R P R 2 a a 0V0 2R 2 R 2 a 0V0 8a Prof. Yen-Chieh Huang Dept of Electrical Engineering National Tsing-Hua University office: Delta 856, HOPE 301 ext: 62340 email:[email protected] EE214002 Electromagnetics, Fall 2012 ___________________________________________________________________________________ (f) Continued from (e), the surface polarization charge density at R = 4a is given by ps aˆ R P R 4a a 0V0 2R 2 R 4 a 0V0 32a (g) Similar to (d) The surface charge density on the outer conductor is given by s aˆ R D3 1.5 0 aV0 3 0V0 . The result in (c) is used. 32a ( 4a ) 2 5. (15 points) A large-area parallel-plate capacitor is charged to store an amount of charge Q and then disconnected from the power supply, as shown in (a) in the following. The parallel-plate capacitor has an area A and an electrode separation d in vacuum. If one inserts a perfect dielectric of area A, thickness d/2, and relative permittivity r = 2 into the capacitor, as shown in (b), (1) what is the ratio Vb / Va , where Va, Vb are the voltages across the conducting electrodes before and after inserting the perfect dielectric, respectively. (5 points) Ans: C a 0 A d , Cb 4 A 4 1 0 C a independent of y0 d / 2 y0 d /2 3d 3 0 A 0 A 0 r A y0. Since V Q C with a constant Q, Vb / Va C a C b 3 . 4 (2) what is the ratio Fb / Fa , where Fa , Fb are the forces holding the conducting electrodes before and after inserting the perfect dielectric, respectively. (5 points) Ans: F 1 / Cb ( y) dWe dy Q y 1 Q2 1 2 d (1 / C ) 1 / C a ( y) , Fa ; Q 0 A 2 dC a 2 dy C 3y 1 Q2 3 , Fb Fb / Fa a . 4 0 A 2 dCb Cb 4 (3) what is the polarization charge density induced on the surface of the dielectric facing electrode 1? (5 points) Ans: The voltage across the two conducting plates for capacitor (b) is Q 3Qd D d D d Q Vb . D . According to A C b 4 0 A 2 0 2 0 2 Prof. Yen-Chieh Huang Dept of Electrical Engineering National Tsing-Hua University office: Delta 856, HOPE 301 ext: 62340 email:[email protected] EE214002 Electromagnetics, Fall 2012 ___________________________________________________________________________________ P D 0 E (1 1 r )D 1 Q D . The surface polarization charge density 2 2A Q on the dielectric facing electrode 1 is there P aˆ s . 2A d d d/2 vacuum 1 2 Q -Q r =2 1 2 -Q Q y0 (a) (b) 6. (20 points) Find the work to assemble a sphere of charges with its charge density varying with b between the radius distance 0 R R0 , where 0 and b 0 R are constants. The sphere of charges is in a vacuum. Present at least TWO methods to calculate and cross check the answer. Ans: Method 1 2 b 2 R sin dRd d 2bR 2 0 0 0 0 R Rb 0 QR VR 40 R 2 0 QR R 0 dWe VR dQR R0 We VR dQR 0 Method 2 R0 0 202 b 2 R 2 0 202 bR0 dR 3 0 3 Prof. Yen-Chieh Huang Dept of Electrical Engineering National Tsing-Hua University office: Delta 856, HOPE 301 ext: 62340 email:[email protected] EE214002 Electromagnetics, Fall 2012 ___________________________________________________________________________________ R R0 0bR02 40 R 2 2 0 R 2 b QR ER 0 2 40 R 2 0 QR0 ER 0 R R0 R R0 V EdR R b 0 bR02 bR bR dR 0 dR 0 0 0 2 R 2 0 R 2 0 0 2 0 0 We 20 b V 20 b b 0 0 R0 0 20 b 2 R03 R03 202 bR0 R )dR ( ) 2 0 2 6 3 0 2 ( R0 3 Method 3 R R0 0bR02 40 R 2 2 0 R 2 b QR ER 0 2 40 R 2 0 ER 0 R R0 We QR0 R0 b 0 2 0bR02 2 2 1 2 E ( ( ( ) R dR ( 0 ) 2 R 2 dR ) sin dd ) 0 2 0 2V 2 0 0 R0 2 0 R 2 0 2 3 4 R R b 2 b R3 202 bR0 0 0 4 ( 02 dR R 2 dR ) 0 ( R03 0 ) R R 0 2 2 0 2 0 3 3 0 0 0 7. (40 points) Visit a few electronics stores to find out various kinds of capacitors. Use your cell phone camera to take photographs for those capacitors with your photo ID in photographs. Explain the functioning principles of those capacitors that you found in the stores.