Dielectrics — polarization, dielectric constant, effect on capacitance: Foundations

1.8.14 · D1 · Physics › Electromagnetism › Dielectrics — polarization, dielectric constant, effect on c

Parent note ki ek bhi line follow karne se pehle, tumhe symbols ka ek toolbox chahiye. Neeche har symbol, term, aur notation hai jo yeh topic use karta hai — har ek ka seedha matlab, uski picture, aur yeh bhi ki topic uske bina kyon nahi chal sakta. Inhe is tarah order kiya gaya hai ki har ek sirf upar walon par hi depend kare.

0. Shuruaati picture: do charged plates

Figure — Dielectrics — polarization, dielectric constant, effect on capacitance

Do flat metal plates ek doosre ke saamne hain, beech mein thodi si gap hai. Ek plate extra positive charge se bhari hai, doosri negative charge se. Gap mein, andeekhi "force lines" seedhi + plate se − plate tak jaati hain. Woh gap aur woh lines woh stage hai jahan poora topic hota hai. Poori build ke liye dekho Electric Field of Parallel Plates aur Capacitance and Capacitors.

1. Charge $Q$ aur charge-per-area $σ$

Picture: agar $Q$ coulombs $A$ area ki ek plate par evenly paint kiye gaye hain, toh $σ$ tumhe batata hai ki paint kitni densely spread hai — har square metre par coulombs.

Topic ko yeh kyun chahiye: plates ke beech ka field sirf is baat par depend karta hai ki charge kitna crowded hai, total par nahi. Toh $σ$ , $Q$ nahi, woh hai jo seedha field banata hai. Baad mein hum do flavours milenge — free $σ_{f}$ (battery se deliver hota hai) aur bound $σ_{b}$ (insulator ke andar create hota hai).

2. Electric field $E$ aur uski magnitude $E$

Picture: gap mein seedhi lines hi $E$ hain. Denser lines = zyada strong field. Bade parallel plates ke beech field har jagah same hota hai (uniform), isliye figure mein saare arrows equal length ke hain.

Topic ko yeh kyun chahiye: dielectric ka poora kaam $E$ ko shrink karna hai. Jis cheez ko tum shrink hote nahi dekha, uski baat kaise karoge.

3. Free space ki permittivity $ε_{0}$

Picture: socho ise "empty space ki stiffness" ki tarah. Bada $ε_{0}$ matlab charge sirf ek weak field produce karta. Yeh "charge crowding $σ$ " aur "field strength $E$ " ke beech conversion rate hai.

Topic ko yeh kyun chahiye: yeh $E_{0} = σ_{f} / ε_{0}$ aur capacitance formula mein aata hai. Units ka ek sanity check karte hain: $σ_{f} = 8.85 \times 1 0^{- 6} C/m^{2}$ ki ek plate charge deta hai

$E_{0} = \frac{8.85 \times 1 0 ^{- 6}}{8.85 \times 1 0 ^{- 12}} = 1 0^{6} V/m,$ ek clean, yaad rakhne mein aasaan round number.

4. Voltage $V$ aur gap $d$

Picture: voltage gap ke paar ki walk par field-strength ka sum hai. Same field zyada wide gap par = zyada voltage; same field patelee gap par = kam voltage. Yeh ek line isliye hai ki "field $κ$ se drop hota hai" turant "voltage $κ$ se drop hota hai" ban jaata hai (jab $d$ frozen ho).

5. Capacitance $C$

Picture: bada bucket thodi si level rise ke liye bahut saara paani hold karta hai; bade- $C$ capacitor mein voltage thodi si rise ke liye bahut saara charge hota hai. $C$ bucket ki width hai.

Topic ko yeh kyun chahiye: poore topic ka headline result hai $C = κ C_{0}$ . $C$ kya hai yeh jaanna zaroori hai usse badh jaate dekh ke impress hone se pehle.

6. Electric dipole $p$ — molecule ka tiny arrow

Picture: ek see-saw jisme ek end par + baccha hai aur doosre par − baccha. Field mein + end ek taraf push hota hai, − end doosri taraf, toh see-saw field ke saath tip karta hai — yeh ek torque hai (dekho Electric Dipoles and Torque).

Topic ko yeh kyun chahiye: ek dielectric in see-saws se bhara ek room hai. Unka collective tipping poora mechanism hai. Aur dekho Electric Susceptibility ke liye ki woh kitni aasaani se tip karte hain.

7. Polarization $P$ — volume per dipoles ki count

Picture — units C/m² kyun hain: jab slab ke andar har see-saw ek hi direction mein tip karta hai, toh ek ka + end apne neighbour ke − end ke saath baithta hai aur woh pairs mein cancel ho jaate hain. Sirf do outer faces par uncancelled charge bachta hai. Toh yeh saari internal tipping ek har face par charge ki sheet ke roop mein dikhti hai — ek charge per area.

Formula

$σ_{b} = P$ derive karna (cancellation, exactly) $A$ face-area aur $t$ thickness ki ek slab lo, uniformly polarized magnitude $P$ se (dipoles sab thickness ke paar point kar rahe hain). Slab ka total dipole moment do tarike se compute karo aur equal karo. Tarika 1 — per volume: dipole moment per m³ $P$ hai, aur volume $A t$ hai, toh $total moment = P \cdot (A t) .$ Tarika 2 — do exposed sheets ke roop mein: interior cancellation ke baad, poora slab ek face par $+$ sheet aur doosre par $-$ sheet ki tarah dikhta hai, dono $Q_{b} = σ_{b} A$ charge carry karte hain, thickness $t$ se separated. Do opposite sheets $t$ doori par hain hi ek giant dipole jiska moment $total moment = Q_{b} \cdot t = (σ_{b} A) t .$ Dono ko equal karo (same physical slab, same moment): $P (A t) = σ_{b} A t ⟹ σ_{b} = P .$ $A$ aur $t$ cancel ho jaate hain — isliye face charge $P$ ke equal hai, chahe slab kitni bhi badi ya moti ho.

Topic ko yeh kyun chahiye: $P$ "molecules tipping" se "ek extra charged sheet jo field se ladti hai" tak ka bridge hai. Yeh poori derivation ka pivot hai.

8. Bound vs free charge: $σ_{b}$ aur $σ_{f}$

Picture: har plate par, metal ka free charge; ek baal ki doori par, dielectric ka bound charge opposite sign ka. + plate ka free charge dielectric ke − bound face ke paas baithta hai. Kyunki woh opposite hain, bound sheet ka field free sheet ke field ke khilaf point karta hai.

Topic ko yeh kyun chahiye: field cancellation exactly $E = (σ_{f} - σ_{b}) / ε_{0}$ hai. Free aur bound charge ko alag piles mein rakhna baad mein ek shortcut field (often $D$ likha jaata hai, "displacement field") ko bhi motivate karta hai jo sirf free charge ko respond karne ke liye bana hai — lekin woh apne aap mein ek alag topic hai, poori tarah Gauss's Law and the D-field mein develop kiya gaya hai. Is page par tumhe sirf itna jaanna hai ki do piles alag se count hote hain.

9. Susceptibility $χ_{e}$ aur dielectric constant $κ$

Picture: $κ = 1$ (empty space, kuch tip nahi hota) → koi shrink nahi. Paani, $κ \approx 80$ → field apni bare value ki $1/80$ tak choke ho jaata hai. Dono pure numbers hain — koi units nahi.

Topic ko yeh kyun chahiye: $κ$ woh single number hai jo har consequence carry karta hai — $E ↓$ by $κ$ , $V ↓$ by $κ$ , $C ↑$ by $κ$ .

10. Stored energy $U$

Picture: work jo tumne like charges ko unke repulsion ke khilaf ek saath push karte kiya, ab release hone ke liye ready baithi hai. Kaun sa formula use karna hai yeh depend karta hai kya fixed hai — $Q$ (battery disconnected) ya $V$ (battery connected). Dekho Energy Stored in a Capacitor.

Topic ko yeh kyun chahiye: do-regime subtlety (energy girna vs badhna) in dono forms mein se sahi ek choose karne se hi decide hoti hai.

Prerequisite map

Neeche diagram mermaid se draw ki gayi hai; agar yeh tumhare app mein render nahi hoti, toh neeche plaintext version padho — dono ek hi baat kehte hain.

Map ka plaintext version (upar se neeche padho):

Charge $Q$ → spread out → surface density $σ$ .
$σ$ permittivity $ε_{0}$ ke saath → field $E$ .
Field $E$ gap $d$ par → voltage $V = E d$ .
$Q$ aur $V$ saath mein → capacitance $C = Q / V$ .
Dipole $p$ → volume per packed → polarization $P$ → bound charge $σ_{b}$ .
Free charge $σ_{f}$ (plates par $σ$ se) aur bound charge $σ_{b}$ → field cancellation.
Cancellation + susceptibility $χ_{e}$ → dielectric constant $κ$ .
$κ$ jo $C$ par act karta hai → final $C = κ C_{0}$ → stored energy $U$ feed karta hai.

Equipment checklist

Daayein side cover karo aur zor se jawab do; check karne ke liye reveal karo.

σ

ka kya matlab hai aur uski units?

Charge per unit area,

σ = Q / A

, C/m² mein.

E

par arrow tumhe kya batata hai?

Woh direction jisme ek positive test charge push hota hai; magnitude

E

V/m mein strength hai.

ε_{0}

plain words mein kya hai?

Free space ki permittivity, ≈

8.85 \times 1 0^{- 12}

F/m — "empty space ki stiffness" jo

σ

E

se link karti hai.

Vacuum mein parallel plates ka field batao.

E_{0} = σ_{f} / ε_{0}

Uniform field ke liye

V

E

d

kaise related hain?

V = E d

Molecular separation

ℓ

kyun likha jaata hai

d

nahi?

Kyunki

d

pehle se plate gap ka matlab rakhta hai; ek letter ko do bahut alag lengths ke liye use karna errors invite karta hai.

Capacitance define karo aur vacuum plate formula do.

C = Q / V

;

C_{0} = ε_{0} A / d

Dipole moment

p

kya hai?

p = q ℓ

, ek arrow

- q

+ q

tak jo separation × charge measure karta hai.

Polarization

P

kya hai aur uski units?

Volume per dipole moment; units C·m/m³ = C/m².

σ_{b} = P

kyun hota hai dikhao.

Total moment

= P (A t)

(per-volume × volume) equals

= σ_{b} A \cdot t

(do face sheets

t

doori par);

A, t

cancel ho ke

σ_{b} = P

milta hai.

General bound-charge formula aur uska zero case?

σ_{b} = P \cdot \overset{n}{^}

; side faces par jahan

\overset{n}{^} ⊥ P

woh zero hota hai.

σ_{f}

aur

σ_{b}

mein kya fark hai?

σ_{f}

battery se free plate charge hai;

σ_{b}

tipped molecules se bound face charge hai, paas mein opposite sign ka.

χ_{e}

kya hai aur

κ

usse kaise related hai?

Susceptibility = polarize hone ki eagerness;

κ = 1 + χ_{e} \geq 1

Stored energy ki do forms aur kab use karein?

U = Q^{2} /2 C

(Q fixed) aur

U = \frac{1}{2} C V^{2}

(V fixed).

Connections

1.8.14 Dielectrics — polarization, dielectric constant, effect on capacitance (Hinglish)
Capacitance and Capacitors
Electric Field of Parallel Plates
Gauss's Law and the D-field
Electric Dipoles and Torque
Energy Stored in a Capacitor
Electric Susceptibility