RL circuit — growth and decay of current: Foundations

1.8.29 · D1 · Physics › Electromagnetism › RL circuit — growth and decay of current

Parent note RL circuit — growth and decay of current padhne se pehle, us story ka har letter tumhare liye kuch maana rakhna chahiye. Yeh page har ek ko usi order mein zero se build karta hai jis order mein woh ek doosre par depend karte hain — koi bhi symbol use nahi hoga jab tak define na ho.

1. Current $I$ — charge kitni tezi se flow karti hai

Ek pipe mein pani flow hone ka imagine karo. Current yeh nahi hai ki kitna pani pipe mein baitha hai — balki yeh hai ki ek marked line se kitne litre har second guzarte hain. Ek wire mein "pani" electrons hote hain.

Figure 1 neeche exactly yahi dikhata hai: violet charges ek wire ke saath drift karte hain, aur magenta "counting line" woh jagah hai jahan hum count karte hain ki har second kitne guzarte hain. Orange flow arrows follow karo — us line ko per second cross karne wala number hi $I$ hai.

Figure — RL circuit — growth and decay of current

Fig 1 — Current as charge crossing a counting line each second: magenta line tally point hai, orange arrows charges ki flow direction dikhate hain.

2. Rate of change $\frac{d I}{d t}$ — flow khud kitni tezi se change ho rahi hai

Ab ek subtle jump. Hum sirf current ki care nahi karte — hum care karte hain current kitni tezi se change ho rahi hai.

Ek car ke speedometer ke baare mein socho. Position woh hai jahan tum ho; speed yeh hai ki position kitni tezi se change hoti hai. Yahan:

$I$ = current (the "position").
$\frac{d I}{d t}$ = woh current kitni tezi se chadh ya gir rahi hai (the "speed").

Figure 2 magenta mein current curve dikhata hai aur, ek instant par, violet dashed tangent line jiska steepness hi $d I / d t$ hai. Annotations notice karo: jahan curve steep hai wahan current fast change hoti hai; jahan yeh flat ho jaati hai, $d I / d t$ zero ki taraf shrink ho jaati hai.

Fig 2 — Current curve ki slope $d I / d t$ hai: steep curve = current fast change ho rahi hai (orange note), flat curve = current muskil se change ho rahi hai (navy note).

3. Voltage $V$ — electrical push

Koi bhi law jo $V$ use kare usse pehle, hume batana hoga ki $V$ hai kya.

Pipe wali picture mein, $V$ woh height difference hai jo pani ko neeche flow karne par majboor karti hai: jitna bada drop, utna zyada pani push hota hai. Koi height difference nahi → koi flow nahi.

4. Resistance $R$ aur Ohm's law — friction

Ab jab $V$ (Section 3) aur $I$ (Section 1) dono define ho gaye hain, Ohm's law unhe $R$ se jodta hai:

$V = I R$

5. EMF $ε$ — battery ki push

Yeh sirf ek special voltage hai: source voltage, woh cheez jo poore circuit ko drive karti hai. Pipe wali picture mein $ε$ woh pump hai jo pani ko waapas upar uthata hai.

6. Inductance $L$ — "electrical inertia"

Yeh topic ka star hai. Poori story ke liye Inductance and self-induction dekho; yahan working picture hai.

Jab ek coil mein current change hoti hai, coil ek aisi voltage create karti hai jo change se fight karti hai. Us fight-back voltage ka size exactly hai

$V_{L} = L \frac{d I}{d t} .$

Figure 3 coil (violet) ko dikhata hai jisme orange arrow rising current enter hoti dikhata hai, aur magenta arrow coil ki khud ki fight-back voltage $V_{L}$ ko opposite direction mein mark karta hai — woh push jo woh us rise ko oppose karne ke liye create karti hai.

Fig 3 — Coil ek rising current (orange arrow) ka jawab opposing voltage $V_{L} = L d I / d t$ (magenta arrow) se deti hai: jitna tezi se change, utna bada fight-back.

7. Faraday & Lenz — fight-back kahan se aati hai

Do prerequisite laws explain karte hain ki kyun coil $V_{L}$ produce karti hai aur, crucially, woh kis direction mein push karta hai.

Faraday's law of electromagnetic induction — coil mein magnetic flux ka change ek voltage create karta hai. Changing current se changing flux banta hai, toh changing current se voltage banti hai. Yahi $L d I / d t$ ka origin hai.
Lenz's law — created voltage hamesha us change ko oppose karti hai jisne use cause kiya. Toh jab current grow hoti hai, coil ki voltage battery ki push ke khilaf point karti hai.

8. Kirchhoff's voltage law — bookkeeping rule

Section 7 mein jo signs hum ne settle kiye unhe use karte hue:

battery potential ko $ε$ se uthata hai (plus),
resistor ise $I R$ se girata hai (minus),
inductor $L d I / d t$ se oppose karta hai (minus).

Unhe zero mein add karo:

$ε - I R - L \frac{d I}{d t} = 0.$

Yahi woh single equation hai jo poora parent note solve karta hai. Isme har symbol ab tumhara hai.

9. Exponential $e^{- t / τ}$ — "easing in" ki shape

Us equation ka solution $e^{- t / τ}$ contain karta hai, jahan $τ$ ek timing number hai jो neeche define hoga. Tumhe yeh shape pehchanni chahiye.

Figure 4 $τ$ ke units mein time ke khilaf dono shapes plot karta hai: violet decay curve $e^{- t / τ}$ 1 se girta hua, aur magenta growth curve $1 - e^{- t / τ}$ 1 ki taraf utha hua. Orange dashed line $t = τ$ mark karti hai, jahan dotted guides $63.2%$ aur $36.8%$ read off karte hain.

Fig 4 — Growth (magenta) aur decay (violet) mirror curves hain jo exactly $t = τ$ par (orange line) $63.2%$ / $36.8%$ marks cross karti hain.

$t = τ$ par: $e^{- 1} \approx 0.368$ . Toh decay $36.8%$ tak gir gayi hai; growth $1 - 0.368 = 63.2%$ tak chadh gayi hai.

10. Time constant $τ = L / R$ — ek number sab rule karta hai

11. Coil mein energy — jahan story khatam hoti hai

$\frac{1}{2}$ aur square isliye appear hote hain kyunki energy tab accumulate hoti hai jab current build hoti hai — ek picture jise tum us linked note mein unpack karoge.

Prerequisite map

Equipment checklist

Recall Kya main ready hoon? (answers hide karo)

$I$ kya measure karta hai, aur kis unit mein? ::: Charge flowing per second; amperes (A). $\frac{d I}{d t}$ ka plain words mein kya matlab hai? ::: Current-vs-time graph ki slope — current abhi kitni tezi se change ho rahi hai. $V$ kya hai, aur uski unit? ::: Voltage — do points ke beech potential difference ya electrical push; volts (V). Ohm's law state karo aur har symbol ka naam lo. ::: $V = I R$ ; $V$ voltage (volts), $I$ current (A), $R$ resistance (ohms). $ε$ kya hai aur $ε / R$ special kyun hai? ::: Battery ki EMF (push, volts mein); $ε / R$ woh final steady current hai jab aur kuch fight back nahi karta. Ek inductor kis quantity par react karta hai, aur woh kaunsi voltage banata hai? ::: Yeh current $d I / d t$ ke change par react karta hai; yeh $V_{L} = L d I / d t$ banata hai us change ko oppose karte hue. Faraday aur Lenz kahan enter karte hain? ::: Faraday: changing current ek voltage banati hai. Lenz: woh voltage change ko oppose karti hai (sign fix karta hai). Inductor term KVL mein minus sign ke saath kyun enter karta hai? ::: Jab current grow hoti hai, Lenz's law coil ko hamare khilaf push karti hai, toh potential $L d I / d t$ se girta hai — ek minus. RL circuit ke liye KVL loop equation likho. ::: $ε - I R - L \frac{d I}{d t} = 0$ . $e^{- t / τ}$ ki shape kaisi hai aur RL current ise kyun follow karta hai? ::: 1 se 0 ki taraf ek smooth decay; kyunki change ki rate remaining gap ke proportional hai. $τ$ kya hai, uska formula, aur uski unit? ::: Time constant $τ = L / R$ , seconds mein. $e^{- 1}$ ki value aur woh do percentages jo yeh deta hai? ::: $\approx 0.368$ ; decay $36.8%$ tak girta hai, growth ek $τ$ par $63.2%$ tak chadh jaata hai.

Connections

Inductance and self-induction — $L$ aur induced voltage $L d I / d t$ define karta hai.
Faraday's law of electromagnetic induction — changing current voltage kyun banata hai.
Lenz's law — woh voltage change ko kyun oppose karti hai.
Kirchhoff's voltage law — loop-sum rule jo hamari equation deta hai.
RC circuit — charging and discharging — is topic ka exponential dual.
Energy stored in a magnetic field — $\frac{1}{2} L I^{2}$ endpoint.
Parent topic — jahan yeh foundations use hoti hain.