Foundations — Capacitance — parallel plate derivation, cylindrical, spherical

1.8.11 · D1 · Physics › Electromagnetism › Capacitance — parallel plate derivation, cylindrical, spheri

Is machine ko chalte dekhne se pehle, tumhe har wo gear chahiye jo wo use karta hai. Ye page har symbol aur idea ko bilkul zero se build karta hai, usi order mein jisme wo stack hote hain. Yahan kuch bhi assume nahi kiya gaya — agar parent note ne kuch use kiya tha, to hum use neeche forge karenge.

0. Charge — woh cheez jo hum pump karte hain

Capacitor mein hum hamesha ek conductor par $+Q$ aur doosre par $-Q$ daalte hain — same size, opposite sign. Equal aur opposite kyun? Kyunki capacitor charge karne ke liye tum electrons ko ek plate se doosri plate par move karte ho: jo ek plate khoye, doosri wahi paaye. Kuch create nahi hota.

1. Electric field — invisible push

Topic ko ye kyun chahiye. Charge akela invisible hai. Field woh zariya hai jisse wo charge gap ke paar doosri plate se "baat" karta hai. Har derivation charge → field → voltage ke order mein jaati hai, isliye field woh middle man hai jise hum skip nahi kar sakte.

2. Vector arrow aur dot product $\vec{E}\cdot d\vec{l}$

Parent note $\vec{E}\cdot d\vec{l}$ aur $\vec{E}\cdot d\vec{A}$ likhta hai. Woh dots multiplication nahi hain — ye dot product hain, aur tumhe pata hona chahiye ki ye kya poochhta hai.

Do cases jo tumhe turant pehchanne chahiye:

• Same direction ($\varphi =0°$, $\cos 0=1$): $\vec{E}\cdot d\vec{l}=Edl$ — poora effect. Ye tab hota hai jab tum field ke seedhe saath chalo, jaise hum capacitor plates ke beech karte hain.
• Perpendicular ($\varphi =90°$, $\cos 90=0$): $\vec{E}\cdot d\vec{l}=0$ — koi effect nahi. Field ke across chalna kuch cost nahi karta.

Topic ko ye kyun chahiye. Voltage $\vec{E}\cdot d\vec{l}$ ko ek path ke along add karke banta hai. Dot product woh tool hai jo answer karta hai "field kitna actually meri walk ke along point kar raha hai?" — koi aur operation exactly yahi nahi poochhta.

3. Tiny pieces: $d\vec{l}$, $d\vec{A}$, aur integral sign $\int$

$d\vec{l}$ aur $d\vec{A}$ par chhota arrow sirf yaad dilata hai ki har tiny piece ki bhi ek direction hoti hai — step direction, ya woh direction jis taraf patch face karta hai — taaki ye dot product mein enter ho sake.

4. Potential difference — voltage $V$

Topic ko ye kyun chahiye. Voltage woh cheez hai jo battery set karti hai aur meter read karta hai. Capacitance cause ($Q$) ko is measurable effect ($V$) se jodata hai. Poori machinery ke liye dekho Electric Potential — yahan hume bas itna chahiye ki $V$ ek path ke along sum kiya hua field hai.

5. Charge densities: $\sigma$ aur $\lambda$

Jab charge kisi surface ya wire par faila ho, to hum describe karte hain ki ye kitna crowded hai, ek density se.

Topic ko ye kyun chahiye. Gauss's law enclosed charge count karta hai. Densities humein "poori plate ka charge" ko "mere chhote Gaussian box ke andar charge" mein cleanly convert karne deti hain — $A$ ya $L$ phir cancel ho jaata hai, aur exactly yahi wajah hai ki $C$ pure geometry nikalta hai.

6. Constant ${\epsilon }_0$

Kabhi kabhi tum $k=\frac{1}{4\pi {\epsilon }_0}=8.99\times 10^9$ dekhoge — Coulomb constant. Ye wahi ${\epsilon }_0$ hai ek alag hat mein, spheres ke liye convenient.

7. Gauss's law — engine

Ab pieces combine hoke woh tool banate hain jo field dhundhta hai.

Ye tool kyun, koi aur kyun nahi? Hum Coulomb's law ko charge ke har speck par add kar sakte the — brutal integrals. Lekin jab charge symmetric ho (flat plane, long cylinder, sphere), Gauss's law humein constant $E$ ko integral se bahar kheenchne aur ek line mein solve karne deta hai. Ye shortcut sirf tab kaam karta hai jab symmetry humein ek aisi surface deti hai jahan $E$ constant ho. Poori treatment: Gauss's Law.

8. Teen geometries, teen field powers kyun

Har shape Gauss ko ek alag area formula ki surface deta hai, aur wahi decide karta hai ki field kaise fall off karta hai:

Aur field ki shape phir decide karti hai woh integral jo $V$ banata hai:

• constant field → $V=Ed$ (sirf multiply karo),
• $1/r$ field → $\int dr/r=\ln$ (ek logarithm), dekho Coaxial Cable Transmission Lines,
• $1/r^2$ field → $\int dr/r^2=-1/r$ (woh $1/a-1/b$ combo).

Woh ek chain — area → field power → integral type — teeno derivations ka skeleton hai.

Prerequisite map

Jo bhi upstream hai woh do arrows $Q$ aur $V$ ko feed karta hai jo $C$ define karte hain. Upstream boxes master karo aur parent note ki teeno derivations ek hi walk ban jaayengi, teen baar.

Related building blocks

• Dielectrics and Polarization — factor $\kappa$ ${\epsilon }_0$ ke saath kya karta hai.
• Energy Stored in a Capacitor — $C$ aur $V$ ko stored joules mein badle.
• Capacitors in Series and Parallel — kayi $C$'s ko combine karna.
• Parent par wapas: the Hinglish version.

Equipment checklist

Khud test karo — right side cover karo. Agar koi bhi answer fuzzy lage, derivations chhoone se pehle uska section upar phir padho.