P Physics d Quate Test Review KEY k 8.99 x 9 Nm /C 8.85 x - C /Nm e.6 x -9 C milli - mico -6 nano -9 pico - Mega 6 Electostatics. Chage, Field, and Potential a. poton is placed in a unifom electic field that has a value of (.i.5j) Newtons/Coulomb. What is the foce on the poton? F E (.6 x -9 C) (.i.5j) (4.8 x -9 i 4. x -9 j) Newtons b. n electon is placed at (.,., -.5) m. The electic field in that aea is given by the function E (.(N/Cm )x i.5(n/cm )y k) Newtons/Coulomb. What is the foce on the paticle? E() i.5() k (7i k) N/C F E (-.6 x -9 C) (7i k) N/C (-4. x -8 i. x -8 k) Newtons c. dipole has a moment of 6.7 x - j Coulomb-metes. It is placed in a unifom field (.i.5j) Newtons/Coulomb. Find the initial foce, toue, & potential enegy on the dipole. ΣF τ p x E (6.7 x - j) x (.i) -. k Nm U - p E -(6.7 x - j) (-.5j).68 J d. Indicate on the diagam below whee the electic field is stongest and whee it is the weakest. weak stong What is the diffeence between electic field lines and electic field vectos? Field lines stat at the chage and use density to show magnitude. Vectos stat at the point whee field is evaluated & use length to show magnitude. Daw field lines fo the non-conducting sheet of chage below. _ e. Calculate the initial acceleation of the electon in #b. -8-8 F (-4.x.x k) Newtons a (-4.74x.5x )m/s - m i 9.x kg i k
f. poton is initially taveling at a speed of.x 5 m/s when it entes a unifom field of 5. V/m. What is the final speed of the poton if it tavels. mm though this field? V Ed (5 V / m)(. m).5v 9 W V (.6x C)(.5 V ).4x J W K m v v v f ( f i ) ( x J) 7 5 W mv.4 (.67x kg)(x m / s) i, 5 m / s 7 m (.67x kg) g. Electic potentials ae dawn fo an electic field in the diagam below. Is the field unifom? yes o no (cicle one) Explain you answe. The euipotential sufaces ae evenly spaced. What is the diection of the field? left h. If a. C chage has. kj of kinetic enegy at point, how much kinetic enegy will the. C chage have when it eaches point. W K K W K V K ( C)( V ), J 9, 4J f i i i. If the function fo electic field is given by.x.x 5., find a function fo the potential at any point in the field. V E dx x x 5dx x x 5x C j. If the function fo potential is given by 5.xy.z, find the electic field stength at (.,. -.) m. E E x y V 5 V V 5 V V 5 V V 5yi 5i x V 5xj j y V Ez 6zk 6k z E ( 5i j 6 k) V / m
. Coulomb s aw and Field and Potential of Point Chages a. Find the foce on the bottom ight chage. Each chage is -. C. P. m tanθ 4 θ 6.6 F k( C) ( m) ( (4 m) ( m) ).,7 x N k( C) F x N 9 4.5, 6.6 k( C) 9 F4 5.6x N, (4 m) i j 8 9 F.7x N, 68.5 (8.68x.x ) N b. Point P is located at the cente of the ectangle above. Detemine the potential at point P due to all 4 chages. n ki 4 k( C) V 4.8 i i ( m) ( m) x V c. How much wok must be applied to bing a 5. Coulomb chage fom infinity to point P in the ectangle above? Wapplied V (5 C)(-4.8 x V V) -.4 x J d. Calculate the total electical potential enegy of the five chage system, afte the 5. Coulomb chage is placed at point P. U Wapplied -.4 x J o you could add up each combination of U U U 4. m Uf.4x J k( C) 8.7x J U k 4 4 f i.4x J k( C)
z.5 m. Fields and Potentials of Othe Chage Distibutions a. Find the potential at point P (diectly above the left end of the line of chage) due to the unifom line of chage. The line of chage has a total chage of 7. C and a length of. m. Its thickness is negligible. P θ x dx x kd V d Q dx Q d dx kq dx V x z kq V x x x tanθ z x z tanθ dx z sec θdθ dx x z x z secθ z kq z sec θdθ kq z sec θdθ kq sec θdθ kq V z tan θ z z (tan θ ) secθ kq V u secθ tanθ du θ x x x x x x x x secθ tanθ kq x sec θ secθ tanθ secdθ dθ secθ tanθ x secθ tanθ x x θ θ dθ (sec sec tan ) secθdθ x x kq x du kq x kq x z x kq ln secθ tanθ ln ln( x z x) lnz x x u z z x x V kq V kq ln( z ) lnz lnz lnz ln( z ) lnz k(7 C) ln( ( ) (.5 ) ( )) ln(.5 ) 5. V m m m m x V ( m) 4
decosθ b. Find the electic field and potential at point P that is.5 m above a ing of chage. Point P lies on the cental axis of the ing of chage. The total chage of the ing is. micocoulombs. The adius of the ing is. m. desinθ d k E k d cosθ de R z Θ z P cosθ R z k z kz Qkz E d d R z R z Θ z ( µ C) k(.5 m) E,95 V / m ((. m) (.5 m) ) ( R z ) ( R z ) d k k( µ C) V k 7, 6V R z (. m) (.5 m) R R 5
c. P Find the electic field and potential at point P that is.5 m above a unifom disk of chage. Point P lies on the cental axis of the disk of chage. The total chage of the ing is 5. micocoulombs. The adius of the ing is.m. d z R ( z R z) Q (5 µ C) σ.59x C / m π π (. m) d σd σ πd 4 zd zσ πd de 4 π ( ) 4 ( ) z R π z R σz R d E 4 ( z ) u z du d σz R du σz u E 4 4 / ( u) R E (.5m) (.m) fom electic field fo single ing σz σ z z R z z R 4.59x C / m.5m 5.75 / d σd σ πr ' dr ' σ R' R πr ' dr ' 4π ' 4π 4π 4 π R z R ' V u z R ' du R ' dr ' R' R σ σ σ σ V u u 4 4 ' R R R R' Rdu ' z R ' R' u R' σ.59x C / m V 4 R' ( ) (.5 ) (. ) (.5 ) 8.9 x V m 4 m m m x V d. Daw field lines fo a set of paallel plates. - - - - - - e. Daw field lines fo a long unifomly chaged wie o thin cylindical shell. Use a coss-sectional view. 6
f. Daw field lines fo a thin spheical shell. Use a coss-sectional view. 4. Gauss aw a. non-unifom electic field given by E (.x)i (.5y y)j (.z )k pieces the Gaussian cube shown below. The back bottom left vetex of the cube is located at the oigin. The cube has a side length of. m. Detemine the electic flux though each face and the total flux though the cube. y m font Φ E Φ k k Φ back ight Φ left top Φ bottom ( ). / x Φ i i.(). / Φ j j Φ (.5( ) ).5 / 7.7 / Nm C Nm C Nm C Nm C z b. egin with Gauss law and deive an expession fo the electic field at a point a distance of fom a unifomly chaged infinitely long thin wie. The wie has a linea chage density of λ. Include a elevant sketch. Eπl λl E d enclosed λ E π 7
c. The diagam below shows a spheical shell made of pocelain, i.e., it is non-conducting. Chage is unifomly distibuted thoughout the shell. Find an expession fo the electic field fo <a, fo a<<b, and fo >b. Make a plot of vs. b. ssume the total chage on the shell is Q. a b E a b fo <a E E d fo a<<b ρ 4 ( b a ) π E π E 4 π ( a ) fo >b E d Q ( a ) 4 π ( b a ) E4 Q π Q E 4π Q E d enclosed enclosed Q 4 4 π ( b a ) enclosed d. Find the suface chage density of the conducting sheet of chage below. ssume that the sheet is vey lage compaed to the small -. micocoulomb chage suspended with a non-conducting sting. The -. micocoulomb chage has a mass of. milligams. ssume eveything is at est. E mg T F y T cosθ mg mg T cosθ F x T sinθ E σ mg tanθ mg tanθ σ 6 (x kg) g tan σ 5 pc / m ( µ C) Note: mg -6 kg 8
Conductos, Capacitos, and Dielectics 5. Electostatics with Conductos a. Give at least two possible explanations fo why thee is no chage inside a conducto. Chage edistibutes to make like chages fa away fom each othe. If thee was chage on the inside, thee would be field. This would cause the chages to be in pepetual motion. b. Why is a conducto an euipotential suface? Othewise, chage would constantly be in motion on the suface. c. In the diagam below, the sphee on the left,, initially has a adius of.5 mm and a chage of. nc. The sphee on the ight, has a adius of. mm and a chage of 5. nc. The sphees ae. metes apat. Find the net chage on each sphee, afte they ae connected with a conducting wie. d. 5. pf chage is bought nea a lage non-conducting sheet of chage that has a chage density of 5. µc/m. If the 5. pf chage has a mass of. µg, calculate its acceleation. σ F ma E σ (5 pf)(5 µ C / m ) a 47 m / s m (x kg) 9 Note: µ 9 g x kg V V k k.5 mm.mm 5nC.5. 5 e. On the gaph below, make a sketch of V vs. fo a conducting sphee with a adius R and chage Q. abel impotant points. fo <R: VkQ/R fo >R: VkQ/ V kq/r nc nc R 9
6. Capacitos a. paallel plate capacito has an aea. x -4 m and a plate sepaation d. mm. Find its capacitance. 4 (x m ) C.77pF d (x m) b. If the capacito above is then connected to a. V battey, what is the chage on the capacito? CV (.77 pf)( V ).pc c. How much enegy is stoed in the above capacito when it is fully chaged? (.77 )( ).7 U CV pf V nj d. The battey is disconnected and a.8 mm slab of pocelain is inseted between the plates of the capacito. Calculate the electic field in the slab of pocelain. σ.pc E 846 V / m κ κ 6.5(x m ) 4 e. Calculate the electic field in the ai gaps between the capacito plates. E σ κ(846 V / m), V / m f. Calculate the potential diffeence acoss the capacito. V Ed (846 V / m)(.8x m) (, V / m)(.x m).88v g. Calculate the new capacitance..pc c 5.5pF V.88V h. If the battey had not been disconnected, the chage stoed on the capacito would have changed. What would the new chage be? CV (5.5 pf)( V ) 66pC i. Calculate the wok applied when the dielectic was inseted. (5.5 )( ).7.69 Wapplied U pf V nj nj j. The oute adius of a cylindical capacito is inceased while the capacito is connected to a battey. How will the chage change? C π and CV ln( b / a) b C
k. The oute adius of a spheical capacito is inceased while the capacito is connected to a battey. How will the chage change? ab a C 4 π 4 π and CV b a a / b b C i. Find the euivalent capacitance of the cicuit below. C. µf, C 4. µf, and C 5. µf. CC ( µ F)(4 µ F) Ce C 5µ F 6.7µ F C C µ F 4µ F j. In the cicuit below, the esisto R 6Ω, the emf is V, and the capacito C. nf. Find the cuent in the cicuit when the switch is fist placed at a. V V i. R 6Ω k. Find the cuent in the cicuit above, a long time afte the switch is in position a. i l. What is the maximum chage on the capacito above? CV ( nf)( V ) 6nC
m. How long does it take fo the capacito above to each half its maximum chage? Q CV e t / RC ( ) CV CV e t / RC e t / RC e ln t / RC t / RC ( ) t RC Ω nf x 5 ln ln (6 )( ). sec n. fte a long time, the switch is flipped to position b. Calculate the initial cuent. i-. (opposite diection of the initial cuent when the switch is flipped to a). o. What is the cuent a long time afte the switch is flipped to position b. i p. What is the time constant of the cicuit? RC Ω nf x Electic Cicuits 5 (6 )( ).8 sec 7. Cuent, Resistance, Powe a. If thee is a cuent of 5. in a esisto, how much chage passes though the esisto in. hous? 5C 6s i t h( )( ) 6C s h b. What is the potential diffeence acoss the above esisto, if it has a esistance of 5. Ω? V ir (5 )(5 Ω ) 5V c. conducto of a unifom adius. cm caies a cuent of. poduced by an electic field of V/m. What is the esistivity of the mateial? ρ ( m) E E Eπ ( V / m) π. J i / i.8x Ω d. Detemine the esistance of a cylindical esisto at K that is made of silve and has a adius of 5. mm. The esisto is. cm long. fom p. 669, ρ.6x Ωm and α 4.x K R 8 - ρ ρ ρα Ω Ω Ω 8 8-8 ( T T ).6x m (.6x m)(4.x K )(K 9 K).67x m x Ωm m π (.5 m) 8 ρ (.67 )(. ) 6.x Ω m
e. n electic heate is constucted by applying a potential diffeence of V to a nichome wie of total esistance 8. Ω. Find the cuent caied by the wie and the powe ating of the heate. V ( V ) P 5W R 8Ω V V i.75 R 8Ω f. On the cicuit below, indicate how you would connect an ammete and a voltmete to ead the cuent and voltage espectively, in R. i i V i g. Given the values of E. V, E 4. V, E 9. V, R 6. Ω, and R. Ω, detemine the cuent in each banch of the cicuit. i i i i i i ξ ir ir ξ ir i i 4 i i ξ ir ir ξ ir 4 i i 9 i i 5 i i i i i i 5.7.69ampees.97
h. Find the euivalent esistance of the netwok shown below. Re R R R R R R RRR 4 RR 4 RR 4 RR RR 4 RR 4 RR 4 4 RRR i. Explain why the esistance of an ammete must be low. If it was high, it would dastically educe the cicuit cuent since ammetes ae connected in seies. k. Explain why the esistance of a voltmete must be high. If it was low, moe of the cuent would be diveted fom the tested banch. This would alte the value of the potential dop. 4
6.7Ω i E i C i i D 6.7Ω.7Ω i F.7Ω l. In the cicuit above, R. Ω, R. Ω, R. Ω, R4 4. Ω, R5 5. Ω, R6 6. Ω, R7 7. Ω. R8 8. Ω, E. V, and E. V. Find the value i and i. loop : V 9iD V id to calculate i, detemine the euivalent esistance fo all esistos to the left of E (see diagam) V i.6 6.7Ω junction : i ic id ic.6 loop : 6i ic-v i (-(.6))/6.86 junction: ie ic- i.6-.86.76 loop: 8i 7iF junction : ie i if.76 i(8/7)i i.6 5
R m. battey has an emf of 5. V. The teminal voltage of the battey is.6 V when it is deliveing. W of powe to an extenal load esisto R. What is the value of R? V P R V (.6 V ) R 6.7Ω P W n. In the above poblem, what is the intenal esistance of the battey? V.6V i.7 R 6.7Ω V ξ i.6v 5 V (.7 ).97Ω Magnetostatics 8. Foces on Moving Chages in Magnetic Fields a. Explain why a magnetic field cannot do wok. Magnetic fields cannot do wok because the foce they exet is always pependicula to the displacement. b. n electon moving along the positive x axis pependicula to a magnetic field expeiences a magnetic deflection in the negative y diection. What is the diection of the magnetic field? -z c. n electon is pojected into a unifom magnetic field (.4i.j) T. Find the vecto expession fo the foce on the electon when its velocity v.7 x 5 j m/s. Fvx (-.6x -9 C)(-.7x 5 jm/s x.4i) -8.9x -4 k Newtons d. poton is moving in a cicula obit of adius 4 cm in a unifom magnetic field of magnitude.5 T diected pependicula to the velocity of the poton. Find the obital speed of the poton. F ma v v m 9 (.6x C)(.5 T )(.4 m) v 7 m (.67x kg) 6 4.69 / x m s 6
e. cossed field velocity selecto has a magnetic field of magnitude. T. What electic field stength is euied if. kev electons ae to pass though undeflected? K mv 9.6x J (, ev ) (9.x kg) v ev v E 7 5.9 / F F E v x m s 7 5 E v (5.9x m / s)(. T ) 5.9x V / m 9. Foces on Cuent Caying Wies in Magnetic Fields a. wie having a mass pe unit length of.5 g/cm caies a. cuent hoizontally to the south. What ae the diection and magnitude of the minimum magnetic field needed to lift this wie vetically upwad? West.5g kg cm.5kg cm g m m F mg il mg mg.5 kg (9.8 m / s ).45T li m b. cuent of 7. m is maintained in a single cicula loop of. m cicumfeence. unifom magnetic field of.8 T is diected paallel to the plane of the loop. Calculate the magnetic moment of the loop. πd.m d. m / π.64m π (.64 m) µ Ni ()(7x ) 5.4x m 4 c. What is the magnitude of the toue exeted on the loop in the above poblem by the magnetic field when the plane of the loop is paallel to the magnetic field? τ µx (5.4x m )(.8 T ) 4.x Tm 7