HO 40 Solutions ( ) ˆ. j, and B v. F m x 10-3 kg = i + ( 4.19 x 10 4 m/s)ˆ. (( )ˆ i + ( 4.19 x 10 4 m/s )ˆ j ) ( 1.40 T )ˆ k.

Transcription

1 .) m.8 x -3 g, q. x -8 C, ( 3. x 5 m/)ˆ, and (.85 T)ˆ The magnetc force : F q (. x -8 C) ( 3. x 5 m/)ˆ (.85 T)ˆ F.98 x -3 N F ma ( ˆ ˆ ) (.98 x -3 N) ˆ o a HO 4 Soluton F m (.98 x -3 N)ˆ.8 x -3 g.65 m.98 x -3 N ˆ ˆ.) q -.48 x -8 C and ( 3.85 x 4 m/)ˆ + ( 4.9 x 4 m/)ˆ (.4 T)ˆ F q.48 x -8 C (ˆ + ( 4.9 x 4 m/ )ˆ ) (.4 T )ˆ 3.85 x 4 m/ F q.34 x -3 N ( ˆ ˆ ) + (.45 x -3 N) ( ˆ ˆ ).45 x -3 N ˆ b.) (.4 T)ˆ F q.48 x -8 C F.34 x -3 N ˆ (ˆ + ( 4.9 x 4 m/ )ˆ ) (.4 T )ˆ 3.85 x 4 m/ ( ˆ ) +.45 x -3 N ˆ F (.45 x -3 N)ˆ (.34 x -3 N)ˆ ( ˆ ) (.34 x -3 N)ˆ (.45 x -3 N)ˆ 3.) F 4.6 x -5 N, 3.5 x -3 T, θ 4, and q -.6 x -9 C F q F qnθ o qnθ 4.6 x -5 N.6 x -9 C n x -3 T.8 x m 7 4.) crcular area R.374 m n the x-y plane (.6 T)ˆ φ m A (.6 T)ˆ ( π(.374 m) )ˆ.5 Wb b.) (.6 T)ˆ φ m A (.6 T)ˆ ( π(.374 m) )ˆ

2 5.) 4. cm b y 3. cm HO 4 Soluton a c e 3. cm (.385 T)ˆ z d 5. cm f x urface abcd A abcd (.4 m) (.3 m) ˆ. m φ m A (.385 T)ˆ (. m )ˆ.46 Wb ˆ b.) urface befc A befc (.3 m) (.3 m) ˆ φ m A (.385 T)ˆ (.9 m )ˆ.9 m ˆ c.) urface aefd A aefd (.3 m) (.5 m)ˆ a (.5 m )ˆ a The drecton the ame a the cro-product of two ector that le n the plane of the urface. (.3 m)ˆ ˆ a and and (.3 m)ˆ + (.4 m)ˆ (.3 m)ˆ (.3 m)ˆ + (.4 m)ˆ.9 m ˆ.9 m + (. m )( ˆ ). m (. m ) + (.9 m ).5 m ˆ a ( ˆ ˆ ) + (. m ) ˆ ˆ + (.9 m )ˆ. m ˆ + (.9 m )ˆ.8ˆ +.6ˆ.5 m ( ˆ ) φ m A (.385 T)ˆ (.5 m )ˆ a (.385 T)ˆ.5 m.8ˆ ( +.6ˆ ).46 Wb d.) No flux pae through urface abe and dcf o the net flux : φ net φ abcd +φ befc +φ aefd +φ abe +φ dcf φ net.46 Wb Wb + + Th agree wth Gau Law for Magnetm whch tate that: da

3 HO 4 Soluton 6.) q 4.8 x -9 C trael n crcular orbt R.468 m due to force from.65 T perpendcular to t orbt F q ma m R a ˆ nce and are perpendcular q m R and qr m p p qr ( 4.8 x -9 C) (.65 T) (.468 m) 3.7 x -9 g m chec unt: p qr [ ] ( C) ( T) ( m) [ ] ( C) N m A m [ ] ( C) g m C [ ] g m b.) L r p o L R p.468 m 3.7 x -9 g m.73 x -9 g m 7.) o _ A. cm o.94 x 6 m ˆ and R.5 m for an electron q -.6 x -9 C and m e 9. x -3 g F q ma m R a ˆ and q m R The magntude of : m qr ( 9. x -3 g).94 x 6 m 3.35 x -4 T.6 x -9 C.5 m The drecton of determned ung Rght-Hand-Rule. At pont A the force to the rght and the elocty upward. So the magnetc feld mut pont outward for pote charge and nward for negate charge. Therefore: ( 3.35 x -4 T)ˆ 8.) o _ A. cm o.94 x 6 m ˆ and R.5 m for a proton q.6 x -9 C and m p.67 x -7 g F q ma m R a ˆ and q m R The magntude of : m qr (.67 x -7 g).94 x 6 m.64 T.6 x -9 C.5 m The drecton of determned ung Rght-Hand-Rule. At pont A the force to the rght and the elocty upward. So the magnetc feld mut pont outward for pote charge and nward for negate charge. Therefore: (.64 T)ˆ

4 HO 4 Soluton 9.) m.6 x -6 g, q.6 x -9 C, ΔV 45 V,.73 T (perpendcular to ) Ue energy coneraton to get the peed of the partcle when t enter the magnetc feld. ΔK ΔU qδv o m qδv and qδv m 45 V.6 x -9 C.6 x -6 g. x 5 m F q ma m R a ˆ o q m R and R m q R m q (.6 x -6 g). x 5 m. m.6 x -9 C.73 T chec unt: R m q ( g) m [ ] ( C) ( T) [ ] ( g ) m N C A m [ ] ( g ) m C C m g m [ ] m.) m 9. x -3 g, q -.6 x -9 C, ΔV, V Ue energy coneraton to get the peed of the electron when t enter the magnetc feld. ΔK ΔU qδv o m qδv and qδv m, V.6 x -9 C 9. x -3 g 8.38 x 7 m Electron enter magnetc feld n a crcular arc R.3 m. F q ma m R a ˆ o q m R and m qr m qr ( 9. x -3 g) 8.38 x 7 m 3.67 x -3 T.6 x -9 C.3 m

5 HO 4 Soluton.) m 9. x -3 g, q -.6 x -9 C, 4 m ˆ + 35 m ˆ, F ( 4. fn)ˆ + ( 4.8 fn)ˆ F q q ˆ ˆ ˆ x y z o F ( 4. fn)ˆ + ( 4.8 fn)ˆ.6 x -9 C ˆ ˆ ˆ 4 m 35 m Expandng determnant: F ( 4. fn)ˆ + ( 4.8 fn)ˆ.6 x -9 C ˆ 35 m ˆ 4 m + ˆ 4 m 35 m ˆ ( 4. fn)ˆ.6 x -9 C 35 m.6 x -9 C 35 m ˆ ( 4. x -5 N) (.6 x -9 C) 35 x 3 m.75 T ˆ ( 4.8 fn)ˆ.6 x -9 C 35 m.6 x -9 C 35 m ˆ ( 4.8 x -5 N).6 x -9 C 4 x 3 m.75 T (.6 x -9 C ) ˆ 4 m 35 m.6 x -9 C 4 m o Therefore: (.75 T)ˆ

6 HO 4 Soluton.) l. m,.87 T, F. N F l l o F l.) 7. A, l (. m)ˆ b.) c.) d.) (.65 T)ˆ F l 7 A (.56 T)ˆ F l 7 A (.3 T)ˆ F l 7 A. m. N. m (.87 T) 3 A (ˆ (.65 T )ˆ ).455 N. m ˆ (ˆ (.56 T )ˆ ).39 N. m (.74 T)ˆ (.36 T)ˆ F l 7 A ˆ (ˆ (.3 T )ˆ ).7 N. m ˆ ( ˆ ) (.455 N)ˆ ( ˆ ) (.39 N)ˆ ( ˆ ) (ˆ ((.74 T)ˆ (.36 T )ˆ )).58 N F (.58 N)ˆ + (.5 N)ˆ (.5 N)ˆ + (.58 N)ˆ ˆ ( ˆ ).5 N ˆ ( ˆ ) 3.) W N l (. m)ˆ E 8. A (out of the page) 6.7 T, eat F l 8 A ( 6.7 T)ˆ (ˆ ( 6.7 T )ˆ ). m b.) S 6.7 T, outh ( 6.7 T)ˆ F.538 N ˆ ( ˆ ) (.538 N)ˆ (north) F l 8 A (ˆ ( 6.7 T )ˆ ).538 N. m ˆ ( ˆ ) (.538 N)ˆ (eat) c.) 6.7 T, 3 outh of wet d.) 6.7 T, 6 north of eat N N 8. A (out of the page) F W E W E 8. A (out of the page) F S S F.538 N (6 outh of eat) F.538 N (6 wet of north)

7 HO 4 Soluton 4.) D 6.5 cm, N,.7 A,.56 T Maxmum torque occur when magnetc feld perpendcular to the ax of the col. τ NAnφ (.7 A) (.56 T) π 3.5 x - m n 9.6 N m b.) Occur when nφ.5 or when φ 3 5.) 5 cm x cm rectangular col, N 6,.63 A,.67 T n 9 τ NAnφ 6(.63 A) (.67 T) (.5 m) (. m).589 N m 6.) q 4.97 nc, when 3.57 x 4 m 45 force F n the ˆ drecton when.6 x 4 m ˆ force F 4. x -5 N along the x-ax 3.57 x 4 m x m 4 co( 45 )ˆ x 4 m n( 45 )ˆ.5 x 4 m ˆ +.5 x 4 m ˆ F q q ˆ ˆ ˆ x y z o F F ˆ z 4.97 x -9 C ˆ.5 x 4 m ˆ.5 x 4 m ˆ.5 x m 4 F z ˆ 4.97 x -9 C ˆ.5 x m 4 ˆ.5 x 4 m.5 x 4 m + ˆ F ha no x or y-component whch mean frt two term are zero and.5 x m 4 F z 4.97 x -9 C.5 x m 4 F z 4.97 x -9 C.5 x 4 m 4.97 x -9 C for th component to be negate ( ) < F ±F xˆ 4.97 x -9 C.5 x 4 m.5 x 4 m.5 x -4 C m z ˆ ˆ ˆ.6 x 4 m.5 x m y 4 x ± ( 4. x -5 N)ˆ ( 4.97 x -9.6 x C) 4 m ˆ.6 x m 4 ˆ + ˆ z F ha no y-component o mddle term zero and

8 HO 4 Soluton For the x-component: ± ( 4. x -5 N) ( 4.97 x -9 C).6 x 4 m and ( 4. x -5 N) ± ±.498 T ( 4.97 x -9 C).6 x 4 m nce ( ) < and t follow that.498 T Therefore: (.498 N)ˆ 7.) q 35 nc, 5.89 x 5 m ˆ n a unform magnetc feld wth. T,.5 T, and.3 T F q q ˆ F 35 x -9 C F 35 x -9 C ˆ ˆ ˆ x y z 35 x -9 C ˆ 5.89 x 5 m ˆ ˆ. T -.5 T.3 T ˆ 5.89 x m 5 ˆ 5.89 x 5 m + ˆ. T.3 T. T.3 T. T.5 T 5.89 x 5 m.3 T ˆ x 5 m.5 T F (.644 N)ˆ + (.8 N)ˆ and F x, F y.664 N, and F z.8 N ˆ 8.) l. m, 8. A n the +y-drecton n a unform magnetc feld wth.7 T,.3 T, and.538 T F l ˆ ˆ ˆ l x l y l z 8 A ˆ ˆ ˆ. m.7 T.3 T.538 T F 8 A F 8 A. m ˆ.3 T.538 T.7 T.538 T (ˆ ˆ + (. m(.7 T) )ˆ ). m(.538 T) ˆ. m +.7 T.3 T F (.6456 N)ˆ (.84 N)ˆ and F x.6456 N, F y, and F z -.84 N ˆ b.) F F x + F z (.6456 N) + (.84 N).658 N θ tan - F z F x tan -.84 N.6456 N.5

9 HO 4 Soluton 9.) q -.6 x -9 C,, V 35 V, mt Coneraton of Energy K +U K +U o K ΔU qδv or m qδv Therefore: qδv m -35 V -.6 x -9 C 9. x -3 g. x 7 m b.) F q ma m R o R m q ( 9. x -3 g). x 7 m 3.6 x -4 m.6 x -9 C ( x -3 T).) l.8 m, 3. A, θ 35,.5 T F l o F lnθ ( 3 A) (.8 m) (.5 T)n( 35 ). N

10 HO 4 Soluton y.) q +.3 m.4 m _ q x q 5. µc q 3. µc r.3 m r.4 m r ˆ -ˆ r ˆ -ˆ 6. x 5 m ˆ 8. x 5 m ˆ µ oq ˆ r -7 N 5 x -6 C 4πr C.3 m 6 x 5 m µ oq ˆ r -7 N -3 x -6 C 4πr C.4 m ˆ -ˆ ˆ 3.33 x -6 T 8 x 5 m ˆ -ˆ ˆ.5 x -6 T + ( 4.83 x -6 T)ˆ.) q + d d + q q 4. µc q 6. µc r d.5 m r d.5 m r ˆ -ˆ ˆ r ˆ 7.5 x 5 m ˆ.5 x 5 m ˆ µ oq ˆ r -7 N 4 x -6 C 4πr C.5 m 7.5 x 5 m µ oq ˆ r -7 N 6 x -6 C 4πr C.5 m ˆ -ˆ ˆ.33 x -5 T.5 x 5 m (-ˆ ) ( ˆ ) ˆ 6.66 x -6 T + (. x -5 T)ˆ 3.) long wre along x-ax wth current 8. A n the x-drecton For mall egment of wre the equaton for an nfntemal current element can be ued wth d l (. mm)ˆ. The magnetc feld of a current element : d µ o 4π d l ˆ r r ˆ r ˆ x 3. m, y, z y d l r ( 3 m)ˆ ˆ r ˆ x d µ o d l ˆ r -7 N ( 8 A) ( x -3 m)ˆ ˆ 4π r C 3 m

11 3.) (cont d) HO 4 Soluton b.) x, y 3., z y r ( 3 m) ) ) r ˆ d l x d µ o d 4π x -7 N l ˆ r C ( 8 A) ( x -3 m)ˆ ˆ 4π r 4π 3 m ˆ.78 x - T c.) x 3. m, y 3., z y d l x r ( 3 m)ˆ + ( 3 m)ˆ r ( 3 m) + 3 m r ˆ r r ( 3 m )ˆ + ( 3 m)ˆ 8 m 8 m (.77)ˆ + (.77)ˆ d µ o d l ˆ r 4π r 4π x -7 N C 4π ˆ (.77)ˆ + (.77 )ˆ ( 8 A) x -3 m 6.9 x ( 8 m) - T ˆ 4.) 7.5 x -4 T, r.5 m for a long wre: µ o r o r µ o 7.5 x -4 T.5 m 4π x -7 N C 88 A b.) r. m, 88 A µ 4π x -7 N o r C 88 A. m 3.76 x -4 T 5.) long wre 9 A from eat to wet r 5 m aboe the ground drectly under the wre r 5 m µ 4π x -7 N o r C 9 A 5 m Ung Rght-hand Rule the drecton outh. 3.6 x -5 T 5.) (cont d)

12 b.) walng away from the wre 5 m r HO 4 Soluton 5 m r ( 5 m) + ( 5 m) 5.5 m µ o r 4π x -7 N C ( 9 A) 5.5 m 3.58 x -6 T 6.) 5. A.4 m. A The magnetc feld on wre by wre : µ o 4π x -7 N r C ( 5 A).4 m.5 x -6 T Ung Rght-hand Rule the drecton nto the page or ˆ below the wre. The magnetc force on wre : F ( A) (. m) ˆ F l and the force on a. m length : (.5 x -6 T) ( ˆ ). x -6 N ˆ The magnetc feld on wre by wre : µ o r 4π x -7 N C ( A).4 m. x -6 T Ung Rght-hand Rule the drecton nto the page or ˆ aboe the wre. 6.) (cont d) The magnetc force on wre : F ( 5 A) (. m) ˆ F l and the force on a. m length : (. x -6 T) ( ˆ ). x -6 N So the wre are repelled by a force of F. x -6 N ˆ

13 HO 4 Soluton b.) when 5 A and 6 A the force tll repule and on wre : µ 4π x -7 N o r C ( 5 A).4 m F l ( 6 A) (. m) ˆ 7.5 x -6 T ( 7.5 x -6 T) ( ˆ ) 9. x -6 N ˆ on wre : µ o r 4π x -7 N C F l ( 5 A) (. m) ˆ ( 6 A).4 m 3. x -6 T ( 3. x -6 T) ( ˆ ) 9. x -6 N ˆ So the wre are repelled by a force of F 9. x -6 N 7.) a a y θ x r r θ θ x 9. A, a.3 m, x.4 m at pont µ o r 4π x -7 N C ( 9 A).5 m 3.6 x -6 T y-component cancel and +.6 x x x ( -6 T) 4.3 x -6 T and.3 m x x coθ 3.6 x -6 T ( 4.3 x -6 T)ˆ (.5 m).6 x -6 T 9. A nto the page or ˆ F l and F ˆ l ˆ 9 A 4.3 x -6 T ˆ 3.89 x -5 N ˆ m b.) 9. A out of the page or ˆ F l and F ˆ l ( 9 A)ˆ ( 4.3 x -6 T)ˆ 3.89 x -5 N ˆ m

14 HO 43 Soluton.) q -.6 x -9 C.5 A r.8 m 4. x 4 m The magnetc feld from the current n the wre µ 4π x -7 N o r C (.5 A).8 m 3.75 x -6 T The magnetc feld nto the page or n the ˆ drecton. The force on the electron F q (.6 x -9 C) 4 x 4 m ˆ ( 3.75 x -6 T) ( ˆ ) (.4 x - N)ˆ Note that th force away from the wre and would be drected n the upward ˆ drecton f the electron were traelng aboe the wre..) 6. A. m Q.5 m.6 m S.8 m.5 m At pont the magnetc feld due to wre r.5 m µ o r 4π x -7 N C Rght-hand Rule: Thumb pont nto the page and fnger wrap around clocwe. ( 6 A).5 m 8. x -7 T For the no net feld at pont the current n wre mut be out of the page to create a magnetc feld to the rght and equal to. r.5 m µ o o r r µ o.5 m 8. x -7 T 4π x -7 N C A out of the page Rght-hand Rule: Thumb pont out of the page and fnger wrap around counterclocwe. b.) Q r.5 m At pont Q the magnetc feld due to wre µ 4π x -7 N o C ( 6 A) r.5 m.4 x -6 T Rght-hand Rule: Thumb pont nto the page and fnger wrap around clocwe. Q r.5 m At pont Q the magnetc feld due to wre µ 4π x -7 N o C ( A) r.5 m.67 x -7 T Rght-hand Rule: Thumb pont out of the page and fnger wrap around counterclocwe. The total magnetc feld at pont Q Q + (.4 x -6 T)ˆ (.67 x -7 T)ˆ (.3 x -6 T)ˆ

15 .) (cont d) c.) 6. A. m.6 m S S.8 m. A HO 43 Soluton µ o r µ o r 4π x -7 N C 4π x -7 N C ( 6 A).6 m. x -6 T ( A).8 m 5. x -7 T θ y. m θ.6 m.8 m coθ a h.8 m. m.8 and nθ o h (.6 m) (. m).6 x x nθ (. x -6 T) (.6). x -6 T and y coθ (. x -6 T) (.8).6 x -6 T.6 m y θ x. m θ.8 m coθ a h.8 m. m.8 and nθ o h (.6 m) (. m).6 x coθ ( 5. x -7 T) (.8) 4. x -7 T and y nθ ( 5. x -7 T) (.6) 3. x -7 T Sx x + x. x -6 T Sy y + y.6 x -6 T + ( 4. x -7 T).6 x -6 T + ( 3. x -7 T).3 x -6 T S Sx + Sy (.6 x -6 T) + (.3 x -6 T).6 x -6 T θ tan Sy Sx tan (.3 x -6 T) x -6 T S.6 x -6 T 9. 3.) R d µ o 4π d l ˆ r r For the traght ecton of wre d l ˆ r and do not contrbute to feld at. d l dθ r Z π µ o 4π d l r Rdθ R For crcular ecton of the wre d l and r are alway perpendcular and by the Rght-hand Rule d l ˆ r pont nto the plane of the page o d l ˆ r Rdθ ˆ µ o 4πR θ π µ o 4πR ( π ) µ o 4R. o µ o ˆ 4R

16 4.). cm A. cm 5. A.6 cm 4. A HO 43 Soluton For left de of the loop the current gong up ˆ and the magnetc feld pont nto the plane of the page ˆ o the force to the left. The drecton of the force determned by drecton of l ˆ ˆ ˆ For rght de of the loop the current gong down ˆ and the magnetc feld pont nto the plane of the page ˆ o the force to the rght. The drecton of the force determned by drecton of l ˆ. ( ˆ ) ˆ. The force on the top of the loop F l loop l µ o wre r Thee force cancel each other out nce they are the ame dtance from the wre. ˆ ( 5 A) (. m)ˆ The force on the bottom of the loop F l loop l µ o wre r The total force on the loop ˆ 5 A. m ˆ 4π x -7 N C 4π x -7 N C ( 4 A). m ˆ ( 4 A).6 m.8 x -5 N ˆ ˆ.8 x -4 N ˆ F top + F bottom (.8 x -5 N)ˆ (.8 x -4 N)ˆ ( 8. x -5 N)ˆ 5.) Solenod N 5, L. cm, R 3. cm, and 6. A Near the center of a olenod nµ o where n number of turn length N L Therefore 5 turn 4π x -7 N 6. m C A.89 T turn The formula for the magnetc feld near the center of long olenod can be found ung Ampere Law. a l b d l µ o encloed d l + d l + ab bc d l + cd da d l µ o encloed l µ o ln µ o n d l dl l becaue and dl are parallel ab d c bc cd d l and da d l becaue and dl are perpendcular d l becaue ery far from the loop

17 HO 43 Soluton 6.) a b c - For nner conductor the current denty J nner A πa Ung Ampere Law and a crcular path of radu r < a d l µ o encloed r µ o J nner A µ o πa πr µ o r (r < a) a b.) For a < r < b the encloed current. Ung Ampere Law d l µ o encloed r µ o µ o r (a < r < b) c.) For the outer conductor the current denty J A ( πc πb ) For b < r < c the encloed current nclude that on the nner conductor and a porton of the current on the outer conductor. Ung Ampere Law d l µ o encloed µ o + r µ o + J outer A µ o r r b c b µ o r c r (b < r < c) c b ( ( πc πb ) πr πb ) µ r b o c b d.) When r > c the encloed current encloed and ung Ampere Law d l µ o encloed o (c < r)

18 HO 43 Soluton 7.) Long wre radu R and current wth J αr where α a contant. R R JdA αrrdr α r r 3 dr dr α r 3 R αr 3 3 da rdr Therefore α 3 R 3 b.) r.) for r R encloed J da r r αrrdr α r dr α r 3 3 r α 3 r 3 ung the reult from ( encloed R 3 r 3 Ung Ampere Law d l µ o encloed r µ o µ o r R 3 r 3 R 3 r 3 µ o R r (r R) 3.) for r > R encloed d l µ o encloed r µ o µ o r (r > R) 8.) y.4 m y r xˆ + yˆ xd l dx 3 A ˆ x x.3 m ˆ r xˆ + yˆ x + y d µ o d l ˆ r µ o ( dx)ˆ 4π r 4π ( x + y ) xˆ + yˆ x + y d µ o 4π µ o 4π d l ˆ r r ydx µ o 4π ( x + y ) 3 ˆ ydx ( x + y ) 3 ˆ from ntegral table dx ( x + y ) 3 x + C o µ o y x + y 4π yxdx y x + y ˆ + C µ o 4π x y x + y x.3 m x ˆ -7 T m A.4 m ( 3 A) (.3 m) ˆ + (.4 m).3 m ( 4.5 x -5 T)ˆ