This immediately suggests an inversesquare law for a "piece" of current along the line.


 Regina Benson
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1 Electomgnetic Theoy (EMT) Pof Rui, UNC Asheville, doctophys on YouTube Chpte T Notes The iotsvt Lw T nvesesque Lw fo Mgnetism Compe the mgnitude of the electic field t distnce wy fom n infinite line of chge to the mgnitude of the mgnetic field t distnce wy fom n infinite line of cuent The fomuls e vey simil You see the behvio, souce pmete λ o, constnt, nd even the π This immeditely suggests n invesesque lw fo "piece" of cuent long the line We did the integtion fo the electic field elie in ode to clculte the esult fo n infinite line of chge Of couse, Guss's Lw is quicke Note the invesesque lw fo the chge element distnce wy nd the need fo the cosine to obtin the component tht sums up Note: dl d Thee is n impotnt diffeence The mgnetic field lwys points into the pge Howeve, it is weke s the ngle ppoches 9 We know tht the cosine tem hs to be thee vi ou nlogy in ode to get the ight integted esult So we wite dl dl d cos sinθ nd PT (Pctice Poblem) Show tht cos sinθ d dl ˆ Michel J Rui, Cetive Commons AttibutionNonCommecilSheAlike 3 Unpoted License
2 T The iotsvt Lw Ou fomul fo the mgnetic field due to n element of cuent is known s the iot Svt Lw We cn wite it s diffeentil o n integl d dl ˆ dl ˆ Let's integte fo the infinite line of cuent to double check things sinθ dl cosdl cos 3 d We will use tn cos PT (Pctice Poblem) Show tht d fom, ie, tn d d d 3 3/ 4 π ( + ) d sinθ dθ cosθ PT3 (Pctice Poblem) We did this integl in Chpte C Use tht esult to finish it We will do it nothe wy 4 ( tn ) cos π / π / 3/ π + θ d PT4 (Pctice Poblem) Show tht tn + Michel J Rui, Cetive Commons AttibutionNonCommecilSheAlike 3 Unpoted License cos
3 Fist fcto out the s 3 in the denominto due to the 3/ powe Now use 4 ( tn ) cos π / 3 π / 3/ π + θ tn cos + to get d 4 π (/ cos ) cos π / 3 d π / 3 cos 3 π / cos d π / Well, we hve the coect fcto /, which is nice Remembe tht we know the nswe: π We e just double checking hee π / π cos d / Since the cosine is n even function nd we hve symmetic nge, we cn wite π π / cos d The integl bette evlute to π / π sin (sin sin ) ( ) π π π π ˆ θ π Michel J Rui, Cetive Commons AttibutionNonCommecilSheAlike 3 Unpoted License
4 The Finite Wie We cn now esily do wht initilly might ppe s difficult poblem cos d ntegte fom some initil ngle to some finl one, both smlle thn 9 nd we hve moe genel solution sin (sin sin ) The Sque Wie Fom this we cn do n even scie poblem: the mgnetic field t the cente of sque wie loop We pply the finite eqution on the left veticl wie whee we multiply by 4 4 (sin sin ) π π sin sin( ) π 4 4 π π sin sin π sin π 4 π π π Michel J Rui, Cetive Commons AttibutionNonCommecilSheAlike 3 Unpoted License
5 T3 Ring of Cuent A nice bsic poblem is to clculte the mgnetic field t the cente of cicul loop of cuent in the xy plne We stt with the iot Svt Lw dl ˆ kdl ˆ dl ˆ This is tue fo ny dl long the loop The mgnetic field points long the xis kˆ π π π dl dl Now we poceed to point long the xis Note tht the coss poduct still includes n ngle of 9, but now the mgnetic field is no longe long the xis Howeve, we wnt since the x component will cncel out The best news is tht the mgnitude of the mgnetic field is constnt, which mens the integl will be tivil Wtch! π π sinθ π + + dl dl 4 ( ) π ( π ) π dl 3/ 3/ 3/ π ( ) 4 π ( ) π ( ) Michel J Rui, Cetive Commons AttibutionNonCommecilSheAlike 3 Unpoted License
6 T4 Solenoids Figue Coutesy HypePhysics, R Nve A solenoid is cicuit element with wie windings See the figue t the left fo the windings, cuent, nd sketch of mgnetic field lines nfinitely Long Solenoid Fo cses of symmety we use Ampèe's Lw insted of the iotsvt Lw We will do this below fo the infinitely long solenoid The windings pe unit length is n, sometimes clled the tun density Figue Coutesy MT Open Cousewe dl enc Refe to the figue fo the Ampèin pth nd pth components though 4 dl dl + dl + dl + dl enc 3 4 dl l nl, whee n is the tun density Then, dl enc leds to l nl nd n The self inductnce L is defined s Φ L, whee Φ is the mgnetic flux Then, the voltge ε dφ dt E dl leds to d ε E dl L dt Fo ou solenoid section l : L l Φ nla nl( n) A n Al Michel J Rui, Cetive Commons AttibutionNonCommecilSheAlike 3 Unpoted License
7 π N Figue Coutesy MT Open Cousewe The Tooidl Solenoid See the figue t the left to the geomety of the tooidl solenoid Tke the Ampèin pth to be cicle inside the tooid whee < < b dl π N enc N π PT5 (Pctice Poblem) Show tht the tooidl inductnce is L N h b ln π Figue Coutesy MT Open Cousewe The Finite Solenoid We conside tightly wound finite line solenoid with N windings ove distnce of l We will detemine the mgnetic field long the xis The numbe of windings pe length is n N l We use ou esult fom befoe fo the mgnetic field long the xis fom cicle o cuent Ou fomul fom befoe, ( ) π π ( + ) 3/ kˆ, gets modified to dπ π ( ') + d( ') kˆ 3/ with d ( )( nd ') l / nd ' π l / π ( ') + 3/ Michel J Rui, Cetive Commons AttibutionNonCommecilSheAlike 3 Unpoted License
8 l / n d ' l/ ( ') + 3/ We cn do this integl using ou integl fiend fom eilie dx x ( x + ) x + 3/ Let x ', whee is constnt Then dx d ' d ' ' ( ' ) + ( ' ) + 3/ l / n d ' l/ ( ') + n ' ( ' ) + 3/ l / l / n ( l / ) ( l / ) l l ( ) + ( ) + n ( l / ) ( l / ) ( l / ) + ( + l / ) + + PT6 (Pctice Poblem) Obtin the infinite solenoid esult fom this fomul PT7 (Pctice Poblem) Find () Wht is the fist coection tem in the expnsion fo smll dius? You leding tem should be the infinite solenoid esult Michel J Rui, Cetive Commons AttibutionNonCommecilSheAlike 3 Unpoted License
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