[5 points] (c) Find the charge enclosed by the cylindrical surface of radius ρ 0 = 9 mm and length L = 1 m. [2
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1 STUDENT NAME: STUDENT ID: ELEC ENG FH3: MIDTERM EXAMINATION QUESTION SHEET This emitio is TWO HOURS log. Oe double-sided cib sheet is llowed. You c use the McMste ppoved clculto Csio f99. You c tke y mteil out of the emitio oom fte the em is ove. Attempt ALL questios. Mimum mk is. Ptil mks will be give. of 5 Febuy 5, 6 Wite you solutios i the em booklet. Eos must be cossed out clely. Sloppy witig will be igoed. WRITE YOUR NAME AND STUDENT ID ON TOP OF EACH SUMITTED PAPER SHEET! This emitio ppe icludes: pge of questios d 4 pges of mthemticl fomuls.. [4 poits] The poit C (,,3 m is give with its ectgul coodites whees the poit D (5m,9, 9 is give with its spheicl coodites. The vecto field A= eists t C whees the vecto field = eist t D. Fid the dot poduct A.. [4 poits] A light souce shies oto hemispheicl dome of dius = m, d mkes oud spot 3 m i dimete, d = 3 m. Fid the e A of the dome lit up by the bem. 3. [6 poits] A lie chge of desity ( pbol y l ( y, = + y C/m is distibuted log the = ( = fom A(,, m to (,, m. Fid the totl chge Q. 4. [ poits] A ifiite uifom lie chge of desity l = C/m lies log the is. Fid the potetil diffeece V A betwee A( A =, A =, A = d ( = 4, = 8, = 4. All lie coodites e i metes. Medium is vcuum. 5. [6 poits] A ifiite cylide of dius = mm is uifomly chged with chge desity v = 5 C/m 3. The is of the cylide is log the is. ( Fid the flu desity vecto D t the poit P with coodites P = 9 mm d P = 9. [9 poits] (b Fid the totl flu Ψ though the cylidicl sufce of dius = 9 mm d legth L = m. [5 poits] (c Fid the chge eclosed by the cylidicl sufce of dius = 9 mm d legth L = m. [ poits] 6. [ poits] The flu desity is give s 6 D(, = cos(, C/m. Fid the chge desity s fuctio of positio. v 7. [9 poits] The electic field vecto is give s E( y,, = 6 + 6yy + 4 V/m. Fid the bsolute potetil V M t the poit M (,, m if the potetil t the oigi (,, is V = 5 V. 8. [9 poits] Two ifiite ples e uifomly chged d plced pllel to ech othe t 9 septio distce of d = mm. Ple # is t = mm d it hs sufce chge desity s = C/m 9. Ple # is t = mm d it hs sufce chge desity s = C/m. Medium is vcuum. ( Fid the E-field vecto t the oigi (,,. [5 poits] (b Fid the vecto of the foce pe uit e eeted o plte #. [4 poits] Totl Scoe: poits END OF EXAMINATION QUESTIONS (see ove fo Fomul Sheets
2 FORMULA SHEET ElecEg FH3 TRIGONOMETRIC IDENTITIES si( A± = si Acos ± cos Asi cos( A± = cos Acos si Asi si Asi = cos( A cos( A+ si Acos = si( A+ + si( A cos Acos = cos( A+ + cos( A si A+ si = si[( A+ / ]cos[( A / ] si A si = cos[( A+ / ]si[( A / ] cos A+ cos = cos[( A+ / ]cos[( A / ] cos A cos = si[( A+ / ]si[( A / ] si A= ( cos A / cos A= ( + cos A / si A= si Acos A= t A/ ( + t A cos A= cos A si A= si A= cos A DERIVATIVES OF ELEMENTARY FUNCTIONS ( cost. = (ct = ( = + k k ( = k (c cot = ( e = e + (sih = cosh ( = l (cosh = sih (l = / (log =,, l (th = = th > cosh (si = cos (coth = = coth sih (cos = si (csih = (t =, (k + + cos (ccosh =±, > (cot =, k si (csi (cth =, < =, < (ccoth =, > (ccos =, < pge of 5 ASIC INTEGRALS OF ELEMENTARY FUNCTIONS + d d =, l + = d e d = e, d = t l cos = si d = cos, cot d = l si cos d = si d = l t cos + 4 d cot si = d = l t t d = l cos si SOME INDEFINITE INTEGRALS d = ± 3/ ( ± ± d = 3/ ( + + d = + l 3/ ( + + ( d = + + 3/ ( + + d = ct + d = l ( + + l = cth, < + d = ccoth, > d = + + cotiued
3 FORMULA SHEET ElecEg FH3 d ( + b =, = 4c b 3/ ( + b + c + b + c d ( c + b =, = 4c b 3/ ( + b + c + b + c d= + l ( d = l( / ( l + + d = + d=.5 + csi( / ( + d= l + + d =.5 + csi( / d = b l + b + b b d = l b + + ( + b + b si( d = [si( cos( ] / cos( d = [cos( + si( ] / sih( d = cosh(, = csih( / d= csih( / + cosh( d sih( > ccosh( / d = + < ccosh( /, if ccosh( / ccosh( /, if ccosh( / d g bf = ct,( g > bf ( + b f + g b g bf b f + g SOME DEFINITE INTGERALS pge 3 of 5, m sim si d =, m=, m cosm cos d =, m= sim cos d =, m sim si d = /, m=, m cosm cos d = /, m=, m+ = eve umbe sim cos d = m, m odd umbe + = m ( bcos /, > b> d = + b b, b> > ( cos VECTOR IDENTITIES A ( C = C ( A = ( C A A ( C = A ( C CA ( ( Φ + Ψ = Φ + Ψ ( A+ = A+ ( A+ = A+ ( ΦΨ = Φ Ψ + Ψ Φ Φ Ψ Φ Φ Ψ = Ψ Ψ Φ = Φ Φ ( Φ A = A Φ+Φ A ( A = A A ( Φ A = Φ A+Φ A ( A = A A+ ( A ( A cotiued
4 FORMULA SHEET ElecEg FH3 ( A = A ( + ( A + ( A + ( A Note: A = A + Ay + A y Φ = Φ A = Φ = A= A A DIFFERENTIAL ELEMENTS Ctesi coodites dl = d + dy + d y ds= dyd + ddy + ddy ; dv = ddyd Cylidicl coodites dl= d + d + d ds= dd + dd + dd ; dv = d dd Spheicl coodites dl = d + d + sid ds= sidd + sidd + dd ; dv = siddd COORDINATE TRANSFORMATIONS Rectgul Cylidicl / = cos ( y y = si = = ct( + y / = = Rectgul Spheicl / = sicos = ( + y + y = / sisi = ccos[ / ( + y + ] = cos = ct( y/ Cylidicl Spheicl / = si = ( + = = = cos / = ccos[ / ( + ] Rectgul Compoets Cylidicl Compoets A = A cos Asi A = Acos+ Aysi Ay = Asi + Acos A = Asi + Aycos Rectgul Compoets Spheicl Compoets A = Asi cos+ A coscos Asi Ay = Asisi+ A cossi+ Acos cos A si sicos+ Aysisi+ Acos A = Acoscos+ Aycossi Asi A = A si+ A cos y Note: d e the positio gles of the obsevtio poit. Cylidicl Compoets Spheicl Compoets A = A si + A cos A = A si + Acos A = A A = A cos A si cos A si Note: is the positio gle of the poit t which the vecto eists. SOME CONSTANTS ε F/m 7 µ = 4 H/m; g 9.8 m/s (Eth s cceletio q.6 9 e C; m e kg (electo chge/mss ELECTROMAGNETIC EQUATIONS Mwell s equtios (diffeetil fom D i E= ; H= + σ E+ J ; D= ; = t t Coil lie ε µ b µ C =, F/m; L = l +, H/m l( b/ 8 Twi-led lie VECTOR TRANSFORMATIONS pge 4 of 5 cotiued
5 FORMULA SHEET ElecEg FH3 ε µ h h C = F/m; L = l + H/m h h l + w h Pllel-plte Lie C = ε, L = h µ w DIFFERENTIAL OPERATORS Rectgul Coodites Φ Φ Φ Φ = + y + y F Fy F F = + + y F F F F F F F= y + + y Φ Φ Φ ( Φ Φ Φ= + + y y y y F= F + F + F y y Cylidicl Coodites Φ Φ Φ Φ = + + F F F = ( F + + F F F F ( F F F = + + Φ Φ Φ ( Φ Φ Φ= + + F F F F F F F F F F F F + F F F F F= Spheicl Coodites Φ Φ Φ Φ = + + si F F = ( F ( si + F + si si F F F= ( F si ( F si + si F + ( F Φ si Φ Φ= + + Φ si si F F F cot F F = + F F F cot F F + si si F F F F cot F si F F cot F + + si si F F F F cot F si F F cot F + + si si si THE END pge 5 of 5 cotiued
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