Foundations — Energy stored in capacitor U = ½CV²

1.8.13 · D1 · Physics › Electromagnetism › Energy stored in capacitor U = ½CV²

Yeh page parent note ke har symbol ko, bilkul zero se, ek aise order mein build karta hai jahan har ek sirf pehle wale par lean karta hai. Agar parent page par koi word ya squiggle ne confuse kiya, toh woh yahan define hai.

0 · Charge $q$ aur $Q$ — woh stuff jo move ho raha hai

• Lowercase $q$ = plate par abhi is waqt charge, filling ke beech mein (running total).
• Uppercase $Q$ = final charge jab hum filling band kar dete hain.

Topic ko yeh kyun chahiye: poora sawaal hai "charge $Q$ build karne mein kitna kaam lagta hai?", toh hum sirf end ki baat nahin kar sakte, har intermediate moment $q$ par charge ki baat karni padegi.

Figure mein beads dekho: pile ki height hi $q$ hai. Jab hum beads add karna band karte hain, height $Q$ hai.

1 · Voltage $V$ (aur instantaneous $v$) — "pahaad ki unchai"

• Lowercase $v$ = hill ki unchai abhi, charging ke dauran.
• Uppercase $V$ = final hill ki unchai.

Phir do symbols kyun? Kyunki derivation ki poori trick yeh hai ki jaise-jaise fill karte ho pahaad barhta jaata hai. Jab plate empty hai toh hill flat hai ($v=0$, charge add karna free hai); jab full hai toh hill apni tallest height par hai ($v=V$).

2 · Capacitance $C$ — bucket kitni "spacious" hai

Iske baare mein sochne ke do tarike:

• Ek chaudi bucket (bada $C$) thodi si level rise mein bahut zyada charge hold karti hai — voltage low rehta hai.
• Ek narrow test-tube (chhota $C$) apna level jaldi bhar leta hai — thoda charge, bada voltage.

Topic ko yeh kyun chahiye: $C$ woh fixed constant hai jo charge aur voltage ko link karta hai. Kyunki $v=q/C$ har moment par hota hai, capacitance woh number hai jo "abhi kitna charge hai" ko "hill abhi kitni tall hai" mein convert karta hai.

3 · Proportionality $Q=CV$ — ek straight line

Kyunki $C$ constant hai, charge $q$ ko voltage $v$ ke against plot karne par origin se ek straight line milti hai jiska slope $C$ hai.

Is relation ko puri tarah dekhne ke liye Capacitance and Q = CV dekho.

4 · Kisi cheez ka "$d$" ($dq$, $dW$) — ek tiny sliver

Topic ko yeh kyun chahiye: hill ki height lagatar badal rahi hai, isliye hum poore kaam ke liye ek fixed voltage use nahin kar sakte. Hum kaam ko itne tiny slivers mein kaatate hain ki har sliver ke dauran voltage $v$ effectively constant ho. Tab hum safely keh sakte hain $dW=vdq$.

5 · Integral $\int$ — "saare slivers jodo"

Puri line ko simple shabdon mein padho:

"Add up karo (kaam per sliver = current hill height $\times$ charge ka sliver), empty se full tak."

Kyunki hill ek straight ramp ki tarah barhti hai, yeh sum §3 ke figure mein triangle ke area ke barabar hai — isliye jawab $U=\frac{Q^2}{2C}$ mein denominator mein $2$ hai (triangle = rectangle ka aadha).

6 · ${\epsilon }_0$, $A$, $d$, $E$ — parallel-plate cast

Yeh tab aate hain jab parent note energy ko energy density mein convert karta hai. Inhe ek baar milo:

Ek parallel plate capacitor ke liye yeh combine hokar $C=\frac{{\epsilon }_{0}A}{d}$ deta hai — bade plates ya chhote gaps zyada roomy bucket banate hain. Poori kahani: Parallel plate capacitor C = ε₀A/d.

Topic ko yeh kyun chahiye: yeh hume stored energy ko $u=\frac{1}{2}{\epsilon }_0{E}^2$ mein rewrite karne dete hain aur reveal karte hain ki energy field mein rehti hai, metal mein nahin. Dekho Energy density of electric field.

7 · $u$ vs $U$ — total energy vs energy per volume

Topic ko dono kyun chahiye: $U$ sawaal ka jawaab deta hai "kul kitni energy?"; $u$ ka jawaab hai "woh energy space mein kaise spread hai?". Same letter, alag case, alag matlab — inhe alag rakho.

Prerequisite map

Har arrow kehta hai "right box samajhne se pehle tumhe left box chahiye". Inhe upar se neeche follow karo aur tum, poori tarah equipped hokar, parent topic par pahunch jaoge: the topic note.

Related building blocks: Dielectrics and capacitance, Work done by a battery and Joule heating, Capacitors in series and parallel.

Equipment checklist

Khud ko test karo — dayi taraf cover karo aur zor se jawab do.