( ) ( )()4 x 10-6 C) ( ) = 3.6 N ( ) = "0.9 N. ( )ˆ i ' ( ) 2 ( ) 2. q 1 = 4 µc q 2 = -4 µc q 3 = 4 µc. q 1 q 2 q 3
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1 3 Emple : Three chrges re fed long strght lne s shown n the fgure boe wth 4 µc, -4 µc, nd 3 4 µc. The dstnce between nd s. m nd the dstnce between nd 3 s lso. m. Fnd the net force on ech chrge due to the other chrges. Emple :. m. m k 9. 9 N m 4 o C 4-6 C 3.6 N. m 9. 9 N m & 4-6 C C ˆ 4 µc -4 µc 3 4 µc F, F 3, 4 o, ˆ Emple :. m. m 4 µc -4 µc 3 4 µc 3 r r ˆ r 3, 4-6 C.9 N.4 m 9. 9 N m & 4-6 C C ˆ 3.6 Nˆ nd F 3,.9 Nˆ F, F 3, 3.6 Nˆ.9 Nˆ.7 Nˆ ˆ 3 Emple :, 3 r r ˆ. m. m, F,3 4 µc -4 µc 3 4 µc F F, F 3, F, 3.6 Nˆ, 3 r r ˆ r 3, 9. 9 N m & 4-6 C C ˆ, 3.6 Nˆ 4-6 C. m F F, F 3, 3.6 Nˆ 3.6 Nˆ F 4 Emple :. m. m, F,3 4 µc -4 µc 3 4 µc F,3 F,3 F 3,.9 Nˆ F,3 F 3, -3.6 Nˆ F,3 F,3.9 Nˆ 3.6 Nˆ.7 Nˆ Emple : Two poste pont chrges re plced on the -s t nd -. A negte pont chrge - s locted t some pont on the -s. Fnd the net force tht the two poste chrges eert on
2 Emple : - There re three cses tht must be consdered becuse the force s nfnte t the locton of ech chrge. I. < - II. - < < III. < 7 Emple : - 3 r - - I. < r r ˆ ˆ ˆ F,3 3 r r ˆ ˆ ˆ F 3, F 3, ˆ k ˆ 8 Emple : I. < - Emple : II. - < < ˆ k ˆ ˆ k ˆ ˆ ˆ < 9-3 r ˆ r r ˆ ˆ F,3 3 r r ˆ ˆ ˆ F 3, F 3, ˆ k ˆ Emple : II. - < < Emple : III. < ˆ k ˆ ˆ k ˆ ˆ 4k ˆ < < - 3 r ˆ r r ˆ ˆ F,3 3 r r ˆ ˆ ˆ F 3, F 3, ˆ k ˆ
3 Emple : III. < ˆ k ˆ ˆ k ˆ ˆ k ˆ > Emple : F < 4k < < k > Emple 3: 3 Emple 3: Three chrges re fed long the corners of n eulterl trngle wth sdes eul to. m s shown n the fgure boe wth 6 µc, 6 µc, nd 3-6 µc. Fnd the net force on ech chrge due to the other chrges. 5 6 Emple 3: Emple 3:
4 Emple 3: 3 r r ˆ Emple 3: r 3, - 3, C 9. 9 N m & C 6-6 C ˆ. m 8. Nˆ r. m 3, 3 r ˆ r r.3 m 3, r 3,. m 6-6 C 9. 9 N m & C 6-6 C.ˆ.3ˆ &. m. 4.5 Nˆ 7. Nˆ 6-6 C 9. 9 N m & C 6-6 C ˆ. m 8. Nˆ 8. N r ˆ 6 Emple 3: r 3, - 3 cos 3, 4.5 N, 8. Ncos6 F 3, sn 3, 8. Nsn6 7. N Emple 3: 8. Nˆ 4.5 Nˆ 7. Nˆ F 3, F, -4.5 Nˆ 7. Nˆ -4.5 N 7. N 8. N tn F & tn 7. & N Emple 3: Emple 3: 3 3 F,3, 3 4
5 Emple 3: Emple 3: 3 3, F, F, 5 6 F, F, 8. Nˆ Emple 3: - 3 r 3,.3 m r 3,. m r 3,. m, 3 r ˆ, 9. 9 N m & C 6-6 C 6-6 C.ˆ.3ˆ &. m., 4.5 Nˆ 7. Nˆ F F, F 3, 4.5 Nˆ 7. Nˆ F 8. N6 7 F, Emple 3: F, 8. Nˆ, 8. N, F 3 6, F F, F 3,,, cos 3,, 4.5 N 8. Ncos F, 3, sn 3, 8. Nsn, 7. N F 8. N6, 4.5 Nˆ 7. Nˆ 4.5 Nˆ 7. Nˆ 8 Emple 3: Emple 3: 3 3 F,3, 9 3
6 Emple 3: Emple 3: 3 F,3 F, F,3 Emple 3: F 3, 4.5 Nˆ 7. Nˆ F,3 F 3, 4.5 Nˆ 7. Nˆ F,3 F,3-4.3 Nˆ 4.3 N 9 8. N Emple 3: N 9 F 8. N Emple 4: 3 3 Emple 4: Three chrges re fed long the corners of n eulterl trngle wth sdes eul to. m s shown n the fgure boe wth 6 µc, 6 µc, nd 3 6 µc. Fnd the net force on ech chrge due to the other chrges
7 Emple 4: Emple 4: F,3 3 3, Emple 4: Emple 4: F,3 3 F,3 3, 6-6 C. m F,3 9 9 N m & 6-6 C C 8. N F, 39 4 Emple 4: Emple 4: F, N9 sn 8. Nsn6 4.3 N 4.3 N N F 4.3 N 3 3 4
8 Emple 5: d d 3 - Emple 5: Two dentcl conductng spheres re chrged to 3 nd, respectel, nd re seprted b dstnce d much greter thn the rd of the spheres s shown boe. The mgntude of the force of ttrcton on the left sphere s. After the spheres re mde to touch nd then re reseprted b dstnce d, the mgntude of the force on the left sphere s F. Wht s the reltonshp between nd F? d d Totl 3 d F d d X Y Emple 6 X Y Emple 6: Two metl spheres tht re ntll unchrged re mounted on nsultng stnds, s shown boe. A postel chrged rubber rod s brought close to, but does not mke contct wth sphere X. Sphere Y s then brought close to X on the sde opposte to the rubber rod. Y s llowed to touch X nd then s remoed some dstnce w. The rubber rod s then moed fr w from X nd Y. Wht re the fnl chrges on the spheres? X Y X Y Emple 7:,. m 3. m,. m -, r. mˆ - Emple 7: Two chrges re fed long the -s t -. m nd. m s shown n the fgure boe wth 6 µc nd -6 µc. Fnd the electrc feld due to both chrges t,, nd 3. r ˆ ˆ E r r ˆ 9. 9 N m & C 6-6 C ˆ m N C& ˆ 47 48
9 Emple 7: Emple 7:. mˆ - E E E - r ˆ ˆ E r r ˆ 9. 9 N m & C 6-6 C ˆ N. m C& ˆ E E E N N C E.8 6 N C& ˆ C& ˆ 49 5 Emple 7: Emple 7: E - E E E - E E ˆ 5 5 Emple 7: E E E E ˆ Emple 7: E E r E 9. 9 N m & C 6-6 C N. m C r. m r. m -. m E E ˆ E.8 6 N ˆ C& N ˆ C& r r r ˆ E r r ˆ r. mˆ. mˆ. mˆ. mˆ. m. m ˆ ˆ 53 54
10 Emple 7: E ˆ r r ˆ. m E -. m. m. mˆ. mˆ. mˆ. mˆ. m. m ˆ ˆ 55 Emple 7: ˆ 9. 9 N m & C 6-6 C E. m r r r ˆ.7 6 N r ˆ C& E.7 6 N C ˆ ˆ.9 6 N & C& ˆ.9 6 N C& ˆ ˆ 9. 9 N m & C 6-6 C E. m r r ˆ.7 6 N& ˆ C E.7 6 E N C ˆ ˆ &.9 6 N C& ˆ.9 6 N C& ˆ E E E E E N C & ˆ N C 56 Emple 7: Emple 7: E E E E E E E E ˆ E. m E E E E Emple 7: E E E E r E 9. 9 N m & C E E cos ˆ E E ˆ 6-6 C. m E cos ˆ.7 N 6 C.7 6 N C& m ˆ &. m. Emple 7: 3 3 E - E. m E N ˆ C& 59 6
11 Emple 7: 3 Emple 7: 3 3 E 3 E 3 E E - E E r r ˆ r.5 m r. m. m E r r ˆ r. mˆ. mˆ. m ˆ. m. mˆ. mˆ ˆ r. m. m.5 m&.5 m& r ˆ cos ˆ sn ˆ 6 6 Emple 7: 3 Emple 7: 3 E ˆ 3 E. m -. mˆ r ˆ ˆ r ˆ 9. 9 N m & C 6-6 C E r r ˆ.5 m E.8 6 N. m ˆ. m ˆ C&.5 m&.5 m& & E N ˆ N ˆ C& C& E ˆ N.8 6 r ˆ C& 9. 9 N m & C 6-6 C ˆ. m E N& ˆ C Emple 7: 3 3 E E, - E 3 E E E N ˆ N ˆ C& C& E N& ˆ C E 3 E E N ˆ N ˆ C& C&, - Emple 8: Three chrges re fed long the -s s shown n the fgure boe. Fnd the electrc feld t pont on the - s t dstnce of from the orgn., E N C
12 Emple 8:, - r ˆ ˆ r r ˆ ˆ ˆ Emple 8:, -, - 3 E r r ˆ ˆ ˆ & E, ˆ ˆ 3, - 3, r ˆ 3 ˆ ˆ r 3 ˆ ˆ r 3 E 3 r ˆ ˆ ˆ 3 3 & E 3 ˆ ˆ Emple 8:, -, - 3 ˆ r ˆ ˆ E r r ˆ ˆ E ˆ, 69 Emple 8:, -, - 3 E 3 ˆ ˆ E E E E E 3 k ˆ, E ˆ k k E 3 ˆ 3 E ˆ 3 ˆ 7 - Emple 9: oste electrc chrge s dstrbuted unforml long lne wth length, lng long the -s between - nd. Fnd the electrc feld t pont on the -s t dstnce of from the orgn. 7 Emple 9 :, - d d d r ˆ ˆ r, de d r r ˆ de d ˆ ˆ de d ˆ ˆ 3 r ˆ ˆ ˆ 7
13 Emple 9 : de d ˆ ˆ 3 d de 3 d de 3 Emple 9 : From ntegrl tbles: k E de k d 3 3 d d E k 3 d C & 3 k k Emple 9 : k E & k k E k k Emple 9 : de k d E 3 d 3 d E Es ntegrl 3 k E o k & 4 o 4 o & E & & E k Emple 9 :, E 4 o d E, de ˆ Emple 9 : E 4 o E ˆ - Chrge dstrbuton s smmetrcl wth respect to the -s
14 Emple 9 :, - de decos d d d r d E k d!, de d 3 d d 3 79 Emple 9 :, - d d d r!, de d d d de desn 3 d d E k 3 3 d 8 Emple : r d ds d d * & Emple : A negte chrge - s unforml dstrbuted round semcrcle of rdus. Fnd the electrc feld t the center of curture. 8 8 Emple : cos sn r - r d ds d * - & de d r r ˆ d r ˆ r ˆ cosˆ snˆ cosˆ snˆ de d -cosˆ - snˆ 83 Emple : de d -cosˆ - snˆ E E E k d -cosˆ - snˆ k d -cos k d -sn 84
15 Emple : E k d -cos k & cosd Es ntegrl Emple : - E k sn k sn sn & k E - d E d E Once gn we he smmetr Emple : Emple : E k d -sn k & snd Es ntegrl E cos & E o cos cos k & 4 o E * o & - - E o ˆ O Emple : d O r Emple : A rng-shped conductor wth rdus n the -z plne crres totl chrge unforml dstrbuted round t. Fnd the electrc feld t pont tht les on the s of the rng t dstnce from the orgn. E becuse of smmetr. E z becuse of smmetr. E s onl component tht s nonzero 89 9
16 Emple : d O de decos r! de d d! d 3 Emple : d de d de k E k E 3 d o Emple : O E 4 o 3 ˆ Emple : E 4 o E 3 ˆ Emple : d da rdr rdr R O R O r d!! Emple : A dsk of rdus R n the -z plne hs poste chrge unforml dstrbuted on ts surfce. Fnd the electrc feld t pont long the s of the dsk dstnce from ts center. 95 Usng the result for rng of chrge E 3 de rdr 3 96
17 Emple : de rdr 3 R E rdr R rdr 3 R 3 E E rdr R R & 3 97 Emple : E R E & R E o & R E & R * * * * * * o R & R k & & 4 o R * * 98 Emple : E E * o R * ˆ & 99
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