Displacement current — Maxwell's addition to Ampere's law: Visual walkthrough

1.8.32 · D2 · Physics › Electromagnetism › Displacement current — Maxwell's addition to Ampere's law

Hum sab se concrete cheez se shuru karte hain: ek wire, do plates, aur ek compass.

Step 1 — Ek loop aur ek wire: "circulation" ka matlab kya hai

Figure — Displacement current — Maxwell's addition to Ampere's law

KYA kiya: ek ring ke around "field-along-the-walk" ko sum karke jo number milta hai, use naam diya. KYU: Ampère's law iss exact number ke baare mein ek statement hai — isliye ise use karne se pehle hume iska feel hona chahiye. PICTURE: s01 mein, loop cyan ring hai. Amber arrow ek step $d l$ hai; white arrow wahan ka $B$ hai. Sirf overlap (unka dot product) sum mein jaata hai.

Step 2 — "Loop se guzarna" mein ek trap chupi hai: kaunsi surface?

KYA: dikhaya ki ek rim infinitely many surfaces carry kar sakti hai. KYU: Yahi freedom poora game hai. Agar do legal surfaces disagree karein, toh law toot jaata hai. PICTURE: s02 mein, ek cyan rim, do white surfaces — ek flat disc aur ek bulging cap — usi rim ko share karte hue.

$S u r f a ce choice must not matter ::: K y u nk i e k l oo p s i r f e k r imhai; " e n c l ose d c u r r e n t " ha r u ss u r f a ce k e l i y e i d e n t i c a l h o na c hahi y e j i se w o hb o u n d k a r t ihai, w a r na A m p \overset{e}{ˋ} r e^{'} s l a w se l f - co n t r a d i c t or y hai .$

Step 3 — Capacitor trap ko expose karta hai

Ab ek loop wire ke around draw karo, aur usse bound karne wali do surfaces chuno:

Flat surface — wire se seedha guzarta hai. Conduction current $I$ ise pierce karta hai.
Bulging surface — side se ghoom kar plates ke beech, empty gap se guzarta hai.

KYA: Step 2 ki freedom ko ek real circuit par apply kiya. KYU: contradiction ko act mein pakadne ke liye. PICTURE: s03 — amber current arrow flat surface ko spear karta hai (left). Bulging surface (right) gap mein slip karta hai jahan kuch bhi flow nahi karta.

Step 4 — Gap mein kya zinda hai? Ek badhta hua electric field

Pehle humhe field chahiye. Gauss's Law ke zariye, charge $Q$ aur area $A$ wali plates ke beech uniform field hai:

KYA: gap field compute kiya aur notice kiya ki woh badhta hai. KYU: ek aisi physical quantity dhundhne ke liye jo missing current ki jagah le sake. PICTURE: s04 — parallel field lines (cyan) jo $Q$ badhne ke saath thicken ho rahi hain; amber $E$ arrow time ke saath bada hota hai.

Step 5 — Flux: field ko ek number mein bottle karna

Step 4 ka field plug in karo:

KYA: poore field pattern ko ek number mein collapse kiya jo directly $Q$ se linked hai. KYU: missing "current" flux ki rate nikalni hai — isliye pehle flux chahiye. PICTURE: s05 — bulging surface saari field lines pakad rahi hai; ek counter $Φ_{E} = Q / ε_{0}$ read karta hai.

Step 6 — Time-rate lo: flux exactly current ki tarah change karta hai

KYA: static flux ko ek rate mein convert kiya, aur recognize kiya ki woh rate wire current se set hota hai. KYU: kyunki current matlab charge-per-second hai, aur yahan $d Q / d t = I$ gap ko wire se link karta hai. PICTURE: s06 — wire current $I$ (amber) plate ko feed kar raha hai; $Φ_{E}$ vs time ka ek rising graph jiska slope $I / ε_{0}$ hai.

Step 7 — $I_{d}$ define karo taaki dono surfaces agree karein

KYA: missing piece ko naam diya aur size kiya taaki dono surfaces reconcile ho sakein. KYU: law ki consistency yahi definition force karti hai — isme kuch arbitrary nahi hai. PICTURE: s07 — dono surfaces ab same value $μ_{0} I$ ke saath labelled hain; ek bridge jisme " $I_{d} = I$ " likha hai gap ko span karta hai.

Step 8 — Edge cases: har scenario, checked

PICTURE: s08 — teen mini-panels: (a) steady $Q$ , flat flux graph, $I_{d} = 0$ ; (b) falling $Q$ , $I_{d}$ reversed; (c) oscillating field, koi charge nahi, $I_{d}$ wiggling.

Step 9 — Completed law

Faraday's Law of Induction (changing $B$ se $E$ banta hai) ke saath milkaar, yeh naya term (changing $E$ se $B$ banta hai) loop ko ek self-feeding ripple mein close karta hai — isliye $I_{d}$ speed $c=1/\sqrt{\mu_0\varepsilon_0}$ par light predict karta hai, aur isliye yeh Maxwell's Equations complete karta hai.

Ek picture mein summary

Poori kahani ek frame mein: do surfaces (flat + bulge) wala ek loop, left par wire current, right par badhta hua field, aur bridge $I_{d} = ε_{0} d Φ_{E} / d t = I$ jo dono surfaces ko agree karaata hai.

Recall Feynman retelling — walkthrough simple words mein

Wire ke around ek ring draw karo. Ampère kehta hai: ring ke around looping magnetism ek constant times "ussse guzarne wali current" ke barabar hai. Lekin kisse guzarna? Tum ring ko kisi bhi surface se cap kar sakte ho — ek drum-skin ya ek bada soap bubble. Bubble ko itna phulao ki woh ek capacitor ke gap mein ghus jaaye. Ab wire ki current drum-skin ko pierce karti hai lekin bubble se miss ho jaati hai — bubble empty gap mein baitha hai jahan koi charge flow nahi karta. Do surfaces, do answers: paradox! Maxwell ne gap ke andar dekha aur paya ki woh physics se khali nahi tha: wahan ka electric field badh raha tha. Usne us growth ko "flux rate" ki tarah measure kiya, nature ke constant $ε_{0}$ se multiply kiya, aur — miraculously — ek aisa number paya jo exactly wire ki current ke barabar tha. Usne ise displacement current kaha. Use add karo, aur bubble ab drum-skin ke jaisa hi "count" karta hai. Paradox heal. Aur kyunki changing electric field ab magnetism banata hai (jabki changing magnetic field pehle se electricity banata tha), dono khali space mein ek doosre ko hamesha ke liye chase kar sakte hain — woh chase hi light hai.

Active Recall

Why does taking

d / d t

turn flux into a current-like thing?

Kyunki "current" matlab charge (ya flux) per second hota hai; rate

ε_{0} d Φ_{E} / d t

ki units amperes hain aur yeh

B

bilkul real current ki tarah produce karta hai.

On the bulging gap surface, what are

I_{c}

and

I_{d}

I_{c} = 0

(koi charge cross nahi karta),

I_{d} = I

(badhte field se), total

μ_{0} I

— wire surface ke jaisa hi.

When the capacitor is fully charged, what is

I_{d}

Zero, kyunki

d Φ_{E} / d t = 0

; aur

I_{c} = 0

bhi hai, toh dono surfaces

0

read karti hain — phir bhi consistent.