1.8.22 · D1Biot-Savart law — magnetic field from current element
Wire ka ek tiny piece jisme electric current flow kar raha hai, woh thodi si magnetic field bahar phenkta hai, aur woh field wire ke around circles mein wrap karti hai — wire ki taraf ya usse door nahi jaati. Yeh poora topic ek careful recipe hai — Biot–Savart law — jo measure karti hai ki woh choti si field kitni badi hai aur kis taraf curl karti hai, taaki hum saare tiny pieces ko jodkar ek total nikal sakein.
Parent formula padhne se pehle
d B = 4 π μ 0 r 2 I d l × r ^ , har ek mark ka poora knowledge hona chahiye. Yeh page unhe ek ek karke, bilkul scratch se build karta hai. Neeche kuch bhi yeh assume nahi karta ki tumne pehle kisi letter ke upar arrow ya "cross" dekha hai.
Definition Vector — ek quantity jisme size AUR direction dono hainEk vector ek arrow hai. Uski ek length hoti hai (kitna) aur woh kisi direction mein point karta hai (kis taraf). Hum ise letter ke upar ek chota arrow likhkar dikhate hain: B l r B r length batata hai — ek single number, koi direction nahi.
Socho ek dart table par rakha hai. Ek complete description ke liye do facts chahiye: woh kitna lamba hai, aur uski tip kis taraf point karti hai. Woh pair hi vector hai.
Intuition Topic ko vectors ki zaroorat kyun haiKisi point par magnetic field sirf "kitni strong hai" woh nahi batata — wahan ek compass needle bhi ek direction ki taraf turn karti hai . Strength + direction = vector. Isliye B zaroor arrow carry karta hai. d l
I Current I ampere (A) hai.
Zyada I I
Definition Current elementd l d infinitely small amount ." Toh d l tiny straight snippet hai, jo ek arrow ki tarah draw kiya gaya hai. Iska length snippet ki length hai; iska direction us taraf point karta hai jis taraf current us mein se flow karti hai.
Intuition Wire ko pieces mein kyun kaatein?Ek bent ya lambi wire complicated hoti hai. Lekin ek tiny piece almost ek seedha chota dart hota hai — simple. Hum ek tiny piece ka field nikalte hain, phir saare pieces ko jod dete hain . Yahi poori strategy hai. I d l
Ab humein yeh batana hai ki hum field kahan measure kar rahe hain. Us jagah ko P
r r current element d l field point P
Common mistake #1 trap, seedha shuru meinr nahi hota. Yeh jis bhi tiny piece d l r har piece ke liye badalta hai . Yeh galat samjho toh har integral galat ho jaata hai.
Is ek arrow se teen cheezein milti hain:
r r ^
r r length , yaani piece se P r ^ unit vector : wahi arrow jaise r 1 direction rakhta hai aur size hata deta hai.
Formula: r ^ = r r
Intuition Direction aur size ko alag kyun karein?Law ko P direction chahiye (woh r ^ P distance alag chahiye (woh r r 2
squared kyun hoti hai
Socho spray paint ek nozzle se aa rahi hai. Do guna door, wahi paint 2 × 2 = 4 4 1/ ( distance ) 2 r 2 1
Yahi 1/ r 2 Coulomb's Law mein electric charge ke liye bhi aata hai — ek useful familiar cousin.
θ d l r ^ θ P
sin θ sin θ 0 1 P kitna sideways hai.
P aage : θ = 0 ∘ sin θ = 0 koi field nahi .P side mein : θ = 9 0 ∘ sin θ = 1 maximum field .
Intuition Magnetism ko "sideways" ki parwah kyun hai?Ek moving charge apna magnetic influence apni motion ke perpendicular direction mein phenkta hai — kabhi aage nahi, kabhi peeche nahi. Toh ek piece magnetically seedhe aage blind hai aur apni sides mein loudest hai. sin θ
Hum sin cos sin seedhe aage zero hai aur side mein one — bilkul wahi behaviour jo humne abhi describe kiya. Yahi match hai isliye yeh specific tool yahan aata hai.
Yeh sabse strange symbol hai, isliye hum ise poori tarah build karte hain.
A × B cross product ek naya arrow produce karta hai, do rules ke saath:
Length = ∣ A ∣ ∣ B ∣ sin θ sin θ Direction = perpendicular . Do perpendicular ways mein se kaunsa? Right-Hand Rule decide karta hai.
Definition Right-Hand Rule (is cross product ke liye)Apne right hand ki ungliyan pehle arrow (d l doosre (r ^ thumb ab d l × r ^ d B
Kyunki result d l uske around wrap kar sakti hai. Woh wrapping hi magnetism ki poori personality hai: fields loops banati hain, spray-out stars nahi. Zyada practice ke liye Right-Hand Rule dekho.
Intuition Cross product kyun, ordinary multiplication kyun nahi?Do numbers ka ordinary multiplication ek number deta hai jisme koi direction nahi. Lekin humein d B sideways aur circling chahiye. Cross product woh ek operation hai jo (a) automatically sin θ
μ 0 μ 0 = 4 π × 1 0 − 7 T⋅m/A kitni strongly magnetic field mein convert hoti hai. Yeh amperes se teslas ka "exchange rate" hai.
4 π 4 π 4 π 4 π Coulomb's Law mein milte ho.
∫ ∫ Sum ke liye. Iska matlab hai: kisi cheez ko infinitely many tiny bits mein kaato, aur har bit ka contribution jodo. ∫ − ∞ ∞
Intuition Ek piece par kyun nahi ruk sakteBare law sirf ek snippet ka d B unhe sab sum karna hoga : woh hai integral. In summed loop-fields ko add karna hi woh tarika hai jisse Magnetic Field of a Solenoid bhi build hota hai.
Common mistake Arrows ko add karna vs. numbers ko add karnaTum chote d B plain numbers ki tarah add kar sakte ho jab saare d B P vectors ki tarah add karo — pehle components mein split karo, warna cheezein galat cancel ya over-count ho jaengi.
Har mark ka ab ek meaning hai, ek picture hai, aur ek reason hai. Tum parent note ke liye ready ho: the topic note .
Vectors arrows with size and direction
Current I charge per second
Current element I dl tiny wire arrow
Separation r from element to point P
Unit vector r-hat direction only
1 over r squared spreading in space
sin theta sideways factor
Cross product perpendicular and circling
Right-hand rule picks the direction
Integral sum of all pieces
Kisi letter ke upar arrow (jaise B B Arrow = vector (size AUR direction); plain letter = sirf uski length (ek single number).
Current element d l Wire ka ek tiny straight snippet, ek arrow ki tarah draw kiya gaya jiska length snippet ki length hai aur jiska direction current ka flow hai.
Arrow r Yeh current element
d l se start hota hai aur field point
P par khatam hota hai — aur har element ke liye badalta hai.
r ^ r ko length 1 tak shrink karta hai, sirf uska direction rakhta hai:
r ^ = r / r .
Field 1/ r 2 Influence sphere ki area pe spread hoti hai, jo distance-squared ki tarah badhti hai, isliye 1/ r 2
sin θ Jab θ = 0 ∘ P
Cross product A × B Ek length
∣ A ∣∣ B ∣ sin θ , aur ek direction jo dono inputs ke perpendicular hai (right-hand rule se choose ki jaati hai).
Cross product kyun, ordinary multiplication kyun nahi? Yeh automatically sideways (sin θ
Integral sign ∫ Wire ko infinitely many tiny pieces mein kaato aur har piece ka
d B jodo.
Chote d B Sirf jab saare
d B arrows same direction mein point karein; warna unhe vectors ki tarah (components se) add karo.
