RC circuits — charging, discharging, time constant τ = RC: Foundations

1.8.19 · D1 · Physics › Electromagnetism › RC circuits — charging, discharging, time constant τ = RC

Yeh page kuch bhi assume nahin karta. Har woh letter jis par parent note depend karta hai, yahan pehle ek picture se build kiya gaya hai. Derivations se pehle ise padho, aur unmen se koi bhi tumhe surprise nahin karega.

1. Charge $Q$ — woh "cheez" jo move karti hai

Picture. Ek bucket imagine karo. Paani ka level woh charge hai jo capacitor par stored hai. Ek khali bucket ka $Q = 0$ hota hai; ek bhari hui bucket ka koi maximum $Q_{m a x}$ hota hai.

Topic ko iska zaroorat kyun hai. RC circuit ki poori kahani yeh hai ki "paani ka level $Q$ time ke saath kaise change karta hai?" $Q$ is show ka star hai — woh quantity jisko humaari final equation solve karti hai.

2. Current $I$ — charge in motion, aur yeh ek rate kyun hai

Ab ek important move. Agar $Q$ paani ka level hai aur $I$ kitni tezi se paani pour ho raha hai, tab $I$ literally woh rate hai jis par $Q$ change karta hai. Hum ise likhte hain

$I = \frac{d Q}{d t}$

Yeh tool kyun aur sirf " $Q$ divided by $t$ " kyun nahin? Kyunki flow steady nahin hota — yeh har pal change karta hai. Simple " $Q / t$ " sirf tab kaam karta hai jab rate constant ho (jaise fixed speed par chal rahi car). Jis pal rate vary kare, humen instantaneous rate $d Q / d t$ ki zaroorat padti hai, jo ki derivative hai. Yeh exactly wahi tool hai jo tum Exponential Decay and Differential Equations mein milte ho.

Figure — RC circuits — charging, discharging, time constant τ = RC

Figure dekho: kisi bhi moment par $Q$ -versus-time curve ki steepness us moment ka current hai. Jahan curve steep hai, current bada hai; jahan yeh flat hota hai, current chhota hai.

3. Voltage $V$ — woh "push"

Do special voltages baar baar aate hain:

$E$ (script E) — battery ka emf, woh steady push jo yeh supply karta hai.
$V_{C}$ — capacitor ke across voltage, jo hum dekhenge steady nahin hota: yeh badhta jaata hai jaise charge accumulate hota hai.

Topic ko iska zaroorat kyun hai. Push (voltage) hi woh cheez hai jo flow (current) cause karti hai. Topic ka har rule ek bookkeeping statement hai ki yeh pushes kaise add up hote hain.

4. Resistance $R$ — flow par throttle

Ise ek story ki tarah padho: resistor cross karte waqt jo voltage use up hota hai woh us mein se jaane wale current aur uske resistance ke barabar hota hai. Resistance double karo aur usi flow ke liye double push chahiye — ya usi push ke liye aadha flow milega.

Topic ko iska zaroorat kyun hai. Resistor hi woh cheez hai jo charging ko time lagaata hai. Iske bina, current infinite hota aur capacitor instant bhar jaata. $R$ woh referee hai jo race ko slow karta hai. Yeh Ohm's Law and Resistance mein poori tarah se build kiya gaya hai.

5. Capacitance $C$ — bucket kitni badi hai

Topic ko iska zaroorat kyun hai. $V_{C} = Q / C$ woh link hai jo capacitor ke charge ko ek voltage mein turn karta hai jise hum loop rule mein plug kar saken. Yahi wajah hai ki capacitor jaise jaise bharta hai "zyada fight back" karta hai. Capacitors and Capacitance mein poori tarah se build kiya gaya hai.

6. Kirchhoff's Voltage Law — accounting rule

Picture. Ek pahadi par hiking loop ki socho. Battery par tum upar chadhte ho (ek rise, $+ E$ ), phir resistor ke across neeche chalte ho (ek drop, $- I R$ ) aur capacitor ke across phir neeche (ek drop, $- Q / C$ ). Kyunki tum starting point par wapas aate ho, saare ups ko saare downs exactly cancel karne chahiye:

$E - I R - \frac{Q}{C} = 0$

Topic ko iska zaroorat kyun hai. Yahi woh single equation hai jahan se poori derivation shuru hoti hai. Upar ke har symbol yahan ek line mein milte hain. Kirchhoff's Voltage Law mein poori tarah se build kiya gaya hai.

7. Exponential $e^{- t / τ}$ — "self-slowing" ki shape

Parent note ke answers sabhi ek hi coat pahante hain: $e^{- t / τ}$ . Yeh raha woh coat, zero se.

Yeh tool kyun aur straight line kyun nahin? §2 se yaad karo ki current $Q$ -curve ka slope hai, aur §5 se ki fuller capacitor zyada pushback karta hai, driving voltage aur isliye current ko shrink karta hai. Toh change ka rate iss par proportional hai ki hum kitne se finished hain. Woh quantity jiska rate khud ke proportional ho wahi exactly $e^{- t / τ}$ describe karta hai — koi polynomial, koi straight line, yeh nahin karta. Yehi shape Newton's Law of Cooling aur radioactive decay ko govern karti hai.

Sab cases jo tumhe pehchanne chahiye (curve kabhi surprise nahin karta):

$t = 0$ par: $e^{0} = 1$ (full value, kuch decay nahin hua).
$t = τ$ par: $e^{- 1} \approx 0.368$ (37% tak down).
$t = 2 τ$ par: $e^{- 2} \approx 0.135$ .
$t = 5 τ$ par: $e^{- 5} \approx 0.0067$ (essentially zero — "done").
$t \to \infty$ par: $e^{- t / τ} \to 0$ lekin kabhi exactly $0$ nahin.
Rising version $1 - e^{- t / τ}$ mirror image hai: start mein $0$ , $t = τ$ par $0.632$ , $1$ ki taraf climb karta hua.

8. Logarithm $ln$ — woh tool jo exponential ko undo karta hai

Topic ko iska zaroorat kyun hai. §5 ke worked examples mein hum level jaante hain aur time chahte hain — jaise "kab $V_{C}$ 5 V tak girta hai?" Unknown $t$ $e^{- t / τ}$ ke exponent mein trapped hai. Isse neeche kheenchne ka ek hi tarika hai us operation ko apply karo jo $e$ ko undo karta hai: logarithm. Toh $ln$ woh key hai jo $t$ ko unlock karta hai.

$e^{- t / τ} = k ⟹ - \frac{t}{τ} = ln k ⟹ t = - τ ln k$

9. Foundations topic ko kaise feed karti hain

Upar se neeche padho: charge aur voltage raw ideas hain; current charge ki rate hai; Ohm's law aur capacitor relation unhe voltages mein turn karte hain; Kirchhoff un voltages ko ek differential equation mein sum karta hai; exponential ise solve karta hai; logarithm times wapas padhta hai.

Equipment checklist

Self-test: kya tum reveal se pehle har ek ka jawab de sakte ho? Agar nahin, toh matching section phir se padho.

Q

kya stand karta hai aur uski unit kya hai?

Electrical charge, coulombs (

C

) mein measured — bucket mein "paani ka level."

I = \frac{d Q}{d t}

ka seedha matlab kya hai?

Current woh rate hai jis par charge change hota hai — charge-versus-time curve ki steepness.

Ohm's law batao aur ek sentence ki tarah padho.

V_{R} = I R

: ek resistor cross karte waqt use hone wala voltage current times resistance ke barabar hai.

V_{C} = Q / C

tumhe kya batata hai?

Capacitor ka voltage uska stored charge uski capacitance se divided hai — zyada charge ya chhota bucket matlab bada back-push.

Ek loop ke liye Kirchhoff's Voltage Law batao.

Ek closed loop ke around ek baar jaane par, saare voltage rises aur drops zero tak add up hote hain.

e^{- t / τ}

ko define karne wali special property kya hai?

Iska fall ka rate uss se proportional hai jo bacha hai — self-slowing decay jo 1 se start hoti hai, 0 ki taraf approach karti hai.

t = τ

par

e^{- t / τ}

ki value kya hai?

e^{- 1} \approx 0.368

(approximately 37% bacha hai).

ln

kya karta hai aur yahan iska kyun zaroorat hai?

Yeh exponential ko undo karta hai, unknown time

t

ko free karta hai jo exponent mein baitha hai.

τ

kya hai aur uski unit roughly kya hai?

Ek characteristic time (seconds) jo exponential ko stretch ya squeeze karta hai; yahan

τ = R C

Connections

Capacitors and Capacitance — jahan $V_{C} = Q / C$ aata hai.
Ohm's Law and Resistance — jahan $V_{R} = I R$ aata hai.
Kirchhoff's Voltage Law — loop accounting rule.
Exponential Decay and Differential Equations — $e^{- t / τ}$ machinery aur derivative $d Q / d t$ .
Newton's Law of Cooling — wahi self-slowing exponential ek thermal disguise mein.
Energy Stored in a Capacitor · LR Circuits — jahan yeh symbols dobara aate hain.
Parent par wapas: RC circuits (main note).