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106-20082-MT1

# 106-20082-MT1 - PROBLEM(1(25 points A cube with side length...

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Unformatted text preview: PROBLEM (1) (25 points) A cube with side length L is placed with one corner at the 52 (top) 36 (back) origin as shown. The electric ﬁeld is not uniform, but is given by 3': Ax2+6§+26223 where A , B and C are positive SI (left face) S3 (right face) constants. S5 (front) 3’ (a) (15 pts) Find the electric ﬂux through each of the six cube faces, i.e. CD1, (152, (153, CD4, CD5, and 456. x EL— E 0141*) A A :f ng—k AAA) :’jE%AA 5 ‘0m 31,; {A =+IL j E AA 5 555 3 ”AA = = , gAL €23 551-?‘0‘ ) 52, 2 E5 (AL) {C’k (b) (5 pts) Find the total electric charge inside the cube ...é.......... . § éFAA sé+§m§+§q+§s+\$l 3 : ~\$LH+ZCL MAL +0+AL *0 : (QC-(AME .1 Qwe/€c (c) (5 pts) Find the electric potential difference Vab = Va —- Vb between point a with coordinates (0, O, 0) and point b with A :m + t coordinates (L, L, 0). L- L ‘— f Ax Ax + {93" 0 o -, ALL ﬁt, 74' Phys 106 First Midterm Examination Page 3 Saturday, April 04, 2009 PROBLEM (2) (25 points) An insulating sphere of radius R has nonuniform volume charge density p = Arz, where A is a positive constant and r is measured from the center. The sphere has total charge Q. Take V = O at infinity. (a) (5 pts) Find A in terms of Q and R. R Q: I r-W'” ; at A a” 5 (b) (10 pts) An electron with mass m and charge —e is released from rest at r = 2R. Find its speed at the instant it reaches the surface of the sphere. Ignore gravitational forces. E— fwu Misti. at, @an Aisiomce r Pole“ ini ewe/J1 avgdeciron '. U: ('dVN) .éE "AK: @ér "z ' 15(76): L ‘9 K14“. a L&+ Ufw ' CM? 7' 36° —a 6” Lana? (aerial) Palm hag at r (Mercia) o + (—e)\1(u)~§ + Vtr)— V190) ; -— frE'.JZ= 4—]?454 if; (Qe (Wm-V ":(21131) . W 00 00 / ° g 3 Ar 1:31,: - ,.‘ 3 AL_. _(%’;;:'Tf; 3 quﬂm (c) (10 pts) Find the electric potential at r: R/2. t Fﬁ’gﬁx AL ima iwar Vlr);V(E)-r jg“ >Ar éo ‘( ti Emm‘-L ijv& ;l g + Q L,&i— men a we a5 'Li Phys 106 First Midterm Examination Page 4 Saturday, April 04, 2009 PROBLEM (3) (25 points) Positive electric char e is distributed uniforml g Q y |<———L ——>i<—L —>l P (A = constant) along a thin rod of length 2L. Take the —'r—é——l—‘— X potential to be zero at infinity. |<—x ——>i I (a) (10 pts) Find the electric potential at point P. I 9i : % PZX-X (c) (10 pts) Find the electric field at P from Coulomb’s law. ............................. l ‘ A'E_ _\_ AC) -1. aax’ X “6° r” “meg 0-01 ‘5’ V r L1 “—60 X'X ' L/ 3 ﬂ ( \ _\_> Aireo _><~L— x+u Phys 106 First Midterm Examination Page 5 Saturday, April 04, 2009 PROBLEM (4) (25 points) Two identical parallel-plate capacitors (with the same plate area and separation) having capacitances C = 2 nF are connected as shown. A » Q1 Q2 material with dielectric constant K = 3 is inserted between the plates of the capacitor on the right. The capacitors are charged until the charge of the capacitor on the left reaches Q 1 = ‘6 nC. The charging battery is then disconnected from the capacitors. (a) (5 pts) Find the charge Q2 on the capacitor with the dielectric. ..................................................................... Caghdie'rs \nave SON/“3 FAQAHOJ a ®L=KQ| : a / C’L Qta’igmc ;Q- Q: C ' Kc (b) (5 pts) Let E 1 and E2 be the electric—ﬁeld magnitudes in the capacitors, respectively. Find the ratio E2/E1. .................. [6 A a 56qu HOM kiwem Fiat-‘63 (c) (10 pts) Find t e wor required to pull the dielectric halfway out of the capacitor. ............................................................. We in Ml clerﬂc C 06 = C(- 91+ Kc/z QM=Q+C01 = 6+l2 =1Lm‘ ca Aces moi change, (Lamina We _, I'Dfocess- mm and I Eﬂgivquﬂ' cqﬁacf (1441:: 1 ’leorcz M a lag} — QM “04!) 4| . _ ’_ ,_’- ’— A chory, Céﬁ =2 C‘fC’L: Cf KC. - :1+3.2:8AF 1’56Km'ﬁ/3— :ggn'j- :irwii VOW \‘V'ﬂ \$6004“ 9:" We, quqci‘i'bfs aceﬁew zan_/[{V Phys 106 First Midterm Examination Page 6 Saturday, April 04, 2009 ...
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106-20082-MT1 - PROBLEM(1(25 points A cube with side length...

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