1.8.14
Intuition Ek-sentence mein pictureEk dielectric ek insulator hai jo, jab external field mein rakha jata hai, chhote aligned charge separations (dipoles ) develop karta hai — jinki collective field external field ko oppose karti hai — isliye andar net field shrink ho jaata hai , same charge ke liye voltage girta hai , aur isliye capacitance κ .
Ek capacitor charge Q V C = Q / V zyada charge, kam voltage par (aur nahi chahte ki plates ke beech spark ho). Plates ke beech insulator daalna dono kaam karta hai. Kyun samajhne ke liye hume samajhna hoga ki electric field neutral matter ko kya karta hai.
Jab ek dielectric field E Polarization vector P dipole moment per unit volume :
P = Volume ∑ p i [ m 3 C⋅m = m 2 C ]
Do mechanisms hain (dono "stretching" jaisi feel dete hain):
Electronic / non-polar (jaise O 2 Orientational / polar (jaise H 2 O torque hoke partial alignment mein aa jaate hain.
P uncharged surfaces exposed reh jaati hain . Isliye polarization ek surface charge ke roop mein manifest hoti hai. Yeh exactly ek charge per area hai — same units. Yeh sab kuch samajhne ki key bridge hai.
Socho ek uniformly polarized slab hai, field right ki taraf point kar raha hai. Har molecule = (−)(+) pair jo d cancellation . Sirf dono faces par uncancelled charges rehte hain.
Yeh step kyun? Humne ek hi total dipole moment likhne ke do tarike equate kiye — "per-volume times volume" ka "exposed charge times its lever arm" ke barabar hona zaroori hai. Geometry baki sab kar deta hai.
Plates free charge σ f bound charge σ b nearby plate ke saath opposite sign mein (+ plate dipoles ke − ends ko attract karta hai).
Definition Dielectric constantκ = 1 + χ e = E E 0 ≥ 1 χ e κ κ = 1 ≈ 1.0006 ≈ 80
Intuition Do regimes — kya fixed rehta hai?
Battery disconnected (Q fixed): dielectric insert karo → V κ E κ U = 2 C Q 2 girta hai (slab andar khicha jaata hai, kaam karta hua).Battery connected (V fixed): Q = C V κ E unchanged rehta hai (battery V U = 2 1 C V 2 badhta hai (battery extra charge pump karta hai).
Worked example 1 — Basic capacitance boostC 0 = 4 μ F κ = 5 C C = κ C 0 = 5 × 4 = 20 μ F Yeh step kyun? A , d only factor κ
Worked example 2 — Field & bound chargePlates ke saath σ f = 8.85 × 1 0 − 6 C/m 2 κ = 4 E 0 = σ f / ε 0 = 8.85 × 1 0 − 6 /8.85 × 1 0 − 12 = 1 0 6 E = E 0 / κ = 2.5 × 1 0 5 Kyun? Field κ σ b = σ f ( 1 − 1/ κ ) = 8.85 × 1 0 − 6 ( 1 − 0.25 ) = 6.64 × 1 0 − 6 Kyun? E = ( σ f − σ b ) / ε 0 = σ f / ( κ ε 0 )
Worked example 3 — Battery connected, energyC 0 = 2 μ F V = 100 κ = 3 C = 6 μ F Q = C V = 600 μ C 200 400 μ C U = 2 1 C V 2 = 2 1 ( 6 × 1 0 − 6 ) ( 100 ) 2 = 30 10 Kyun? V C ↑ Q , U ↑
Worked example 4 — Forecast-then-verifyForecast: disconnected capacitor, dielectric insert karo — kya stored energy badhega ya ghategaa? Padhne se pehle predict karo.
Verify: Q U = Q 2 /2 C C U girta hai . Energy ka difference woh kaam hai jo slab ko andar kheenchne mein hota hai. ✅
Common mistake "Dielectric andar field badhata hai."Kyun sahi lagta hai: tumne "stuff add kiya," aur stuff usually field add karta hai. Fix: induced bound charge free charge ko oppose karta hai, isliye net field κ decrease hota hai. Yahan zyada matter ka matlab zyada opposition hai.
Common mistake Battery connected hone parE = E 0 / κ Kyun sahi lagta hai: slab-shrinks-the-field rule memory mein burn hai. Fix: battery ke saath, V E = V / d forced constant hota hai; instead Q kya fixed hai?
Common mistake Bound charge ko free charge ki tarah Gauss's law mein treat karna.Kyun sahi lagta hai: charge toh charge hota hai. Fix: E total charge use karta hai; clean trick hai D = ε 0 κ E
Common mistake Yeh sochna kiκ Fix: κ = 1 + χ e χ e ≥ 0 κ ≥ 1
Recall Feynman: 12-saal ke bacche ko explain karoSocho capacitor plates do bheed hain, ek sab "khush" (+), ek sab "grumpy" (−), ek kamre mein chhote see-saws (molecules) ko kheeench rahe hain. Har see-saw is taraf jhukta hai ki uska khush wala end grumpy bheed ki taraf ho. Ab see-saws ki apni chhoti khichain bheed ki khichain ko partly cancel kar deti hain, toh kamre ke paar tug kamzor lagta hai. Kamzor tug = tum zyada khush aur grumpy logon ko pack kar sakte ho pehle dangerous bane. Woh "zyada packing" higher capacitance hai, aur jo number batata hai tug kitna kamzor hua woh κ
Mnemonic Effects yaad rakho"Dielectrics Cut the field, so they Crank the C."
E ↓ κ C ↑ κ "κ = 1 + χ → "Kappa hai one plus the eagerness to polarize."
Polarization P Dielectric ka electric dipole moment per unit volume; units C/m².
Dielectric insert karne se andar field kyun reduce hota hai? Aligned dipoles bound surface charge produce karte hain jo free charge ko oppose karta hai, use partially cancel karta hua.
Bound surface charge aur polarization ke beech relation? σ b = P ⋅ n ^ (magnitude
σ b = P ).
Fields ke terms mein dielectric constant define karo. κ = E 0 / E ≥ 1
Susceptibility ke terms mein κ κ = 1 + χ e
Dielectric ke saath parallel-plate capacitance? C = κ ε 0 A / d = κ C 0
Battery DISCONNECTED, dielectric insert karo: kya badlta hai? Q E V κ C U = Q 2 /2 C
Battery CONNECTED, dielectric insert karo: kya badlta hai? V E Q U = 2 1 C V 2 κ
κ σ b = σ f ( 1 − 1/ κ )
Kya κ < 1 Nahi, κ = 1 + χ e ≥ 1
torques/stretches molecules
opposes free charge field
Dielectric constant kappa
