MAGNETIC FIELD AROUND CURRENT-CARRYING WIRES. point in space due to the current in a small segment ds. a for field around long wire

Transcription

1 MAGNETC FELD AROUND CURRENT-CARRYNG WRES How will we tckle this? Pln: 1 st : Will look t contibution d to the totl mgnetic field t some point in spce due to the cuent in smll segment of wie iot-svt Lw nd : Will use iot-svt Lw to find long the xis of cuent loop. 3 d : Will use iot-svt Lw to get esult fo field ound long wie 7 π di. by RH-ule, 1 T m/a 4 th : Will look t foce between pllel cuent-cying wies 5 th : Ampee s Lw: nothe wy to get π fo field ound long wie d

2 OT-SAVART LAW (.7) P Expession fo contibution d to mgnetic field t P due to cuent in smll segment of wie ˆ is UNT VECTOR pointing P θ is DSTANCE fom P θ is the ANGLE between nd ˆ Then contibution to t P fom ˆ is d 7 1 T m/a is the pemittivity of fee spce Diection of d is fom d s ˆ (cn lso use RH-ule with thumb cuent)

3 ˆ OT-SAVART LAW: undestnding d P Fo dwing, diection of d s ˆ is out of sceen/pge o Finges sweep ˆ ; thumb shows diection of d s ˆ So d t P due to points out of sceen/pge Mgnitude d s ˆ sinθ θ o Fo given, contibutions d fom e mximum fo points on plne pependicul to o Cuent in mkes NO contibution to d t points long diection Fo TOTAL FELD t P, must sum (integte) contibutions fom ll segments of wie (don t pnic. We will only do specil cses)

4 EXAMPLE:.6 in text Mgnetic field on xis of cicul cuent loop Look t loop of dius locted in the yz plne cying cuent. Wht is the mgnetic field distnce x fom the cente of the loop long its xis? y R x d R θ x d θ z P

5 MAGNETC FELD AROUND A LONG (NFNTE) WRE iot-svt Lw gives contibution d t P due to cuent in segment ˆ d P d o d points out of sceen/pge t P by RH-ule To get t P due to WHOLE wie, must sum contibutions fom ll ngles Angle between nd ˆ chnges with position long wie! o Will stte esult nd then sketch deivtion

6 MAGNETC FELD AROUND A LONG (NFNTE) WRE Result of using iot-svt Lw: o Mgnetic field lines cicle wie no component of pllel to wie out Mgnitude of invesely popotionl to pependicul distnce fom wie π (MPORTANT RESULT) Diection of mgnetic field lines: o Anothe Right-Hnd Rule: thumb long ; finges cul in diection of

7 Sketch of how we get π fom the iot-svt Lw Looking fo t point P locted pependicul distnce fom wie Stt with contibution to field t P due to segment locted distnce s long the wie fom the point closest to P : d ˆ o Note tht ˆ points out of the sceen/pge nd tht d s ˆ sinθ To get totl t P fom infinite wie, could sum d fom s to s +. Helps to expess nd in tems of ngle θ between nd ˆ nd then sum fom θ to θ π

8 Sketch of how we get π fom the iot-svt Lw (continued) Chnging Vibles: Fom sin θ get sinθ Ticky point: fo θ s shown, s is negtive so tht tn θ s o s cotθ Use deivtive to elte to d θ o Fom d csc θ sin θ we get θ dθ sin θ

9 Sketch of how we get π fom the iot-svt Lw (continued) Doing the integl. So f, we hve: sinθ dθ sin θ so d s ˆ sinθ dθ sinθ So contibution to field t P fom is d ˆ sinθ dθ Mgnitude of totl field is: π sinθ dθ cosθ π π RESULT: field t distnce fom wie with cuent : π

10 USE π TO FND MAGNETC FORCE ETWEEN PARALLEL WRES 1 F F Field t 1 due to is π o Exets foce on length l of 1 : F1 1l o Mgnitude of foce on length l of 1 is F l 1 1 l1 π

11 MAGNETC FORCE ETWEEN PARALLEL WRES Foce on 1 pe unit length is y Newton s 3 d lw, Foce on pe unit length is F l F l 1 1 π 1 π towd (fo cuents in sme di.) towd 1 (fo cuents in sme di.) n genel: Fo pllel conductos, cuent in sme diection: F 1 o Wies ATTRACT with foce pe unit length l π Fo pllel conductos, cuent in opposite diection: F 1 o Wies REPEL with foce pe unit length l π Povides definition of AMPERE: 7 Fo wies, 1 m pt, 1A foce pe unit length is 1 N/m 1