1.8.32
Ek changing electric field current ki tarah kaam karta hai — woh magnetic field produce karta hai bilkul waisi hi tarah jaise flowing charge karta hai. Maxwell ne yeh "current" add ki thi ek contradiction fix karne ke liye jo original Ampère's law mein thi, aur aise karte hue unhone electromagnetic waves predict ki.
Original Ampère's law kehti hai: ek closed loop ke around circulate karne wala magnetic field us kisi bhi surface se guzarne wale current ke barabar hota hai jo us loop se bounded ho.
∮ B ⋅ d l = μ 0 I e n c
Intuition "Koi bhi surface" kyun?Ek closed loop bas ek wire ring hoti hai. Aap us par infinitely many soap-film surfaces stretch kar sakte ho jaise ek bubble. Ampère's law ko un sab ke liye same answer dena chahiye — warna law ka koi matlab nahi.
Charging capacitor ka paradox. Socho ek wire ek parallel-plate capacitor ko charge kar rahi hai.
Surface 1 (flat disc jo wire ko cut karta hai): conduction current I ∮ B ⋅ d l = μ 0 I Surface 2 (bulging surface jo plates ke beech se guzarti hai): gap ke across koi charge flow nahi karta → I e n c = 0 ∮ B ⋅ d l = 0
Same loop, do alag answers. Contradiction. Gap mein kuch missing hai.
Common mistake "Charge gap ko jump karta hai, toh current phir bhi flow karti hai" — yeh kyun sahi lagta hai, aur fixYeh sahi lagta hai kyunki wire current bahar se continuous hai. Lekin vacuum gap ke across koi charge cross nahi hota — electrons ek plate par jam jaate hain, doosri se nikalte hain. Real fix moving charge nahi hai; woh yeh hai ki plates ke beech badhta hua electric field current ka role play karta hai. (Steel-man corrected.)
Definition Displacement currentQuantity I d = ε 0 d t d Φ E Φ E = ∫ E ⋅ d A nahi hai — yeh ek changing electric flux ka magnetic-field-produce-karne-wala effect hai.
Intuition Deep WHY — symmetry → wavesFaraday: changing B E E B ek doosre ko feed karte hain — ek self-sustaining ripple jo c = 1/ μ 0 ε 0 I d koi light nahi .
Worked example 1. Charging capacitor ke andar B-fieldCircular plates radius R = 5 I = 2 r = 3 B
Step 1 — symmetry: B r ∮ B ⋅ d l = B ( 2 π r ) Kyun? Wire jaisi hi symmetry.
Step 2 — radius r I d I d , e n c = I π R 2 π r 2 = I R 2 r 2 Kyun? Chhote loop ke through flux, total flux ka r 2 / R 2 Step 3 — law apply karo: B ( 2 π r ) = μ 0 I R 2 r 2 B = 2 π R 2 μ 0 I r = 2 π ( 0.05 ) 2 ( 4 π × 1 0 − 7 ) ( 2 ) ( 0.03 ) = 4.8 × 1 0 − 6 T Yeh form kyun? Uniform current density wale wire ke andar B
Worked example 2. Voltage ramp se numericalI d C = 2 μ F d V / d t = 1 0 6 I d
Step 1 — Q = C V ⇒ I d = d t d Q = C d t d V Kyun? Humne prove kiya I d = I = d Q / d t Step 2 — I d = ( 2 × 1 0 − 6 ) ( 1 0 6 ) = 2 Kyun important hai: Tumne field directly kabhi measure nahi kiya, phir bhi displacement current mil gaya — yeh hamesha cap ko feed karne wale conduction current ke barabar hota hai.
Worked example 3. Directly flux seEk region ke through electric flux d Φ E / d t = 4 × 1 0 6 V⋅m/s I d
I d = ε 0 d t d Φ E = ( 8.85 × 1 0 − 12 ) ( 4 × 1 0 6 ) = 3.5 × 1 0 − 5 Kyun? Direct definition — bilkul bhi charges ki zaroorat nahi (yeh pure vacuum waves mein kaam karta hai).
Recall Feynman: 12-saal ke bacche ko samjhaoSocho do metal plates hain jinke beech gap hai. Tum ek plate mein electricity push karte ho. Gap ke across koi spark cross nahi hoti, phir bhi gap ke paas ek compass kaanpti hai — magnetic field appear hoti hai! Kyun? Kyunki gap mein invisible electric field grow kar raha hai, aur Maxwell ne realize kiya ki growing electric field bilkul real electric current ki tarah behave karta hai. Toh magnetic field "gap ko notice nahi karta." Yahi trick hai jis wajah se light exist karti hai: ek electric wiggle ek magnetic wiggle banati hai jo ek electric wiggle banati hai... hamesha ke liye, space mein race karti hui.
"Changing E secret current hai."
I d = ε 0 Φ ˙ E Epsilon-zero times flux-dot ."
Faraday: moving B → E . Maxwell: moving E → B . Mirror twins.
Original Ampère's law capacitor gap par kyun fail hoti hai?
Kya displacement current mein moving charge hoti hai?
Gap mein I d I c
μ 0 ε 0
What flaw in Ampère's law did Maxwell fix? Charging capacitor ke liye, same loop par do surfaces alag-alag I e n c
Define displacement current. I d = ε 0 d Φ E / d t
Why must I d = I c Charge conservation: I = d Q / d t Φ E = Q / ε 0 ε 0 d Φ E / d t = d Q / d t = I
State the Ampère–Maxwell law. ∮ B ⋅ d l = μ 0 I c + μ 0 ε 0 d Φ E / d t .
What does displacement current predict physically? Self-sustaining EM waves jo
c = 1/ μ 0 ε 0 par travel karti hain.
Does a spark/charge cross the capacitor gap? Nahi — sirf electric field change hota hai; woh change current ka role play karta hai.
B inside a charging circular capacitor at radius r < R B = μ 0 I r / ( 2 π R 2 )
I d I d = C d V / d t
Charging capacitor paradox
Derived from charge conservation
