BEng (Hons) Telecommunications. Examinations for / Semester 2

Transcription

1 BEng (Hons) Telecommunicaions Cohor: BTEL/14/FT Examinaions for / Semeser 2 MODULE: ELECTROMAGNETIC THEORY MODULE CODE: ASE2103 Duraion: 2 ½ Hours Insrucions o Candidaes: 1. Answer ALL 4 (FOUR) quesions. 2. Quesions may be answered in any order bu your answers mus show he quesion number clearly. 3. Always sar a new quesion on a fresh page. 4. Quesions carry unequal marks. 5. Toal marks 100. This quesion paper conains 4 quesions and 7 pages. Page 1 of 7

2 ANSWER ALL 4 (FOUR) QUESTIONS QUESTION 1: (25 MARKS) a) Uni vecors are used exensively in Elecromagneic Theory. i) Briefly explain he imporance he uni vecor in Elecromagneic Theory. ii) Assuming ha A is a vecor in a 3-dimensional recangular world, expand he vecor in erms of is componens and uni vecors. iii) Provide an expression for he magniude of he vecor A in erms of is componens. ( marks) b) The recangular coordinae sysem is he mos commonly used sysem. i) Consider Fig 1.1 and complee in your answer book he hird missing axis for each diagram. x x y z Fig 1.1: Incomplee 3D recangular coordinae axes. ii) Consider wo vecors A and B lying on a wo-dimensional plane wih axes x and y and boh spanning from he origin in differen direcions. Skech a parallelogram subending he wo vecors and define he area of such parallelogram. Page 2 of 7

3 iii) Briefly explain how he area of he parallelogram skeched in ii) above links o he definiion of he cross produc. iv) To complee he definiion of he cross-produc, a uni vecor is required. Wha is he uni vecor ha you will assign for A Bin secion iii) and why? ( marks) c) Show by aking he cross-produc of he individual elemens of wo vecors A and B ha: a a a x y z A B A A A x y z B B B x y z (8 marks) d) If wo vecors are parallel, wha would be he magniude of heir cross-produc? Wha if hey were perpendicular? (1 + 1 marks) QUESTION 2: (25 MARKS) a) Consider wo vecors A and B, given as: A 4ax 2 ay az and B ax 4ay 4az i) Wha is he do produc of he wo vecors? ii) Wha is he cross produc of he wo vecors? iii) Assuming ha θ is he smaller angle beween he wo vecors, derive an equaion for sin θ and cos θ in erms of he cross produc and do produc respecively. iv) Use boh equaions derived in iii) o deermine he smaller angle beween he wo vecors. ( marks) b) Coordinae sysems apar from he recangular one are also very imporan in Elecromagneic heory. i) Consider Fig 2.1. Page 3 of 7

4 Fig 2.1: Defining a Poin P in he cylindrical and spherical coordinae sysem Provide he range of values spanned by each elemen in each coordinae sysem. (3 marks) ii) If he poin P was exended o ( r dr, d, z dz ) and ( r dr, d, d ) in Fig 2.1, a differenial volume dv would be obained in each case. Skech he wo volumes in each case along wih he lengh of each side. (4 + 4 marks) iii) Provide an expression for he wo differenial volumes of secion ii) above. (1 + 1 marks) QUESTION 3: (40 MARKS) a) Assume ha here are wo posiive charges in free space, namely Q andq, and ha Q is locaed a he origin of a coordinae sysem. i) Wha is he definiion of Coulomb s law, in words? ii) Skech a diagram of he wo charges Q andq. Assume ha Q is locaed a an arbirary Poin P locaed a a disance from Q (wih Q a he origin). Show he force F acing on Q due o Q. iii) Wrie down an expression for he force F. iv) Which coordinae sysem will you use for he problem specified and why? ( marks) Page 4 of 7

5 b) The Elecric Field Inensiy of a charge is a very imporan noion in elecromagneic heory. i) Define in words he Elecric Field Inensiy using he wo charges of par a), namely Q andq, and assuming ha Q is a es charge. ii) Provide an expression herefore for he elecric field inensiy of Q a an arbirary poin P locaed a disance r from he origin. iii) Wha are he unis of he elecric field inensiy? ( marks) c) In he previous secion, he elecric field inensiy a a poin was considered. Consider he case however where we would like o deermine an expression for he elecric field inensiy due o an infinie line wih consan charge densiy l a an arbirary poin P, as shown in Fig z dq z r P -z Fig 3.1: Calculaing he elecric field inensiy a poin P fixed a a disance r from he line. i) Wha is he mos appropriae coordinae sysem o deal wih he problem a hand? Why? ii) Show ha he differenial elecric field inensiy a he poin P is given by: Page 5 of 7

6 ldz de ra 3 r za r z 0 z iii) Assuming ha he elemen in he below: az direcion vanishes, and using he resuls dx 4ax 2b R 4ac b R, 2 R ax bx c where 3 2 l Show ha E can be derived as E a r. 2 r 0 ( marks) iv) On a line described by x 2m, y 4m, here is a uniform charge disribuion of densiy 20nC/m. Deermine he elecric field inensiy a (-2, -1, 4) m. You l may use Fig 3.2 as suppor o derive he answer. Assume F/m 36. Fig 3.2: Finding he elecric Field inensiy a a poin P due o a given line charge (8 marks) Page 6 of 7

7 QUESTION 4: (10 MARKS) a) Sae Gauss s Law and wrie down is defining equaion. (2 + 1 marks) b) Apply Gauss s Law o he problem of finding he flux inensiy due o an infiniely line wih consan charge densiy l a an arbirary poin P. Use your answer o infer he elecric field inensiy a he poin P. You may consider Fig 4.1 as suppor o derive your answer. Fig 4.1: Finding he Elecric Flux densiy due o an infiniely long line of consan charge densiy. (5 + 1 marks) c) Wha is your observaion when comparing he derivaions in Q4 (b) and Q3 (c) since boh seeks he same answer? (1 mark) ***END OF QUESTION PAPER*** Page 7 of 7