RL circuit — growth and decay of current

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WHAT is happening physically

WHY does current not jump instantly? An inductor produces a back-EMF $-L\,\dfrac{dI}{dt}$. If current tried to change infinitely fast, $dI/dt\to\infty$, demanding infinite voltage — impossible. So current changes smoothly.

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HOW to derive the growth equation (from scratch)

Apply Kirchhoff's voltage law around the loop. Going around: the battery raises potential by $\varepsilon$; the resistor drops $IR$; the inductor opposes with $L\,dI/dt$.

$\varepsilon - IR - L\frac{dI}{dt} = 0$

Rearrange to isolate the rate:

$\frac{dI}{dt} = \frac{\varepsilon - IR}{L}$

Separate variables. Let me write $I_0 = \varepsilon/R$ (the final steady current):

$\frac{dI}{\varepsilon - IR} = \frac{dt}{L}$

Why this step? We want all $I$ on one side, $t$ on the other, so we can integrate each independently.

Integrate from $I=0$ at $t=0$ to $I$ at $t$:

$\int_0^I \frac{dI}{\varepsilon - IR} = \int_0^t \frac{dt}{L}$

The left integral: let $u = \varepsilon - IR$, $du = -R\,dI$, so $\int \frac{dI}{\varepsilon-IR} = -\frac{1}{R}\ln(\varepsilon - IR)$.

$-\frac{1}{R}\Big[\ln(\varepsilon - IR) - \ln\varepsilon\Big] = \frac{t}{L}$

$\ln\!\left(\frac{\varepsilon - IR}{\varepsilon}\right) = -\frac{Rt}{L}$

Exponentiate:

$\frac{\varepsilon - IR}{\varepsilon} = e^{-Rt/L}\quad\Rightarrow\quad IR = \varepsilon\left(1 - e^{-Rt/L}\right)$

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HOW to derive the decay equation

Now remove the battery and short the loop ($\varepsilon = 0$):

$0 - IR - L\frac{dI}{dt} = 0 \;\Rightarrow\; L\frac{dI}{dt} = -IR$

$\frac{dI}{I} = -\frac{R}{L}\,dt$

Why this step? Again separate $I$ and $t$ to integrate. Integrate from initial current $I_0$ to $I$:

$\ln\frac{I}{I_0} = -\frac{Rt}{L}$

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The time constant — the 80/20 core

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Worked examples

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Common mistakes

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Active recall

Recall Test yourself (hide answers)

• What is $\tau$ for an RL circuit, with units check? ::: $\tau = L/R$; H/Ω = s.
• Behavior of inductor at $t=0$ and $t\to\infty$ during growth? ::: $t=0$: open circuit ($I=0$); $t\to\infty$: wire ($I=\varepsilon/R$).
• Fraction of max current at $t=\tau$ (growth)? ::: $1-e^{-1}\approx 63.2\%$.

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Recall Explain to a 12-year-old (Feynman)

Imagine pushing a heavy shopping cart. When you start pushing (flip the switch), it doesn't zoom to full speed instantly — its weight makes it speed up gradually. The inductor is that heaviness for electric current: it resists changing how fast the electrons flow. When you stop pushing (remove the battery), the cart keeps rolling and slowly stops — that's the current decaying. The number $\tau = L/R$ tells you how "heavy and sticky" the cart is: bigger means slower to start and slower to stop.

Connections

• Inductance and self-induction — source of the $L\,dI/dt$ term.
• Faraday's law of electromagnetic induction — origin of back-EMF.
• Lenz's law — why the induced EMF opposes change.
• RC circuit — charging and discharging — dual structure, $\tau=RC$, capacitor opposite to inductor.
• Energy stored in a magnetic field — $U=\tfrac12 LI^2$.
• Kirchhoff's voltage law — the loop equation we started from.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, RL circuit ka pura funda ek hi baat par tika hai: inductor ko current ka achanak change pasand nahi. Jaise hi switch on karte ho, current seedha apni maximum value ($\varepsilon/R$) tak nahi pahunchta — inductor ek back-EMF $L\,dI/dt$ banata hai jo battery ka virodh karta hai, isliye current dheere-dheere exponential tarike se badhta hai. Ye smoothness Faraday aur Lenz ke kaaran aati hai.

Growth ka formula hai $I = \frac{\varepsilon}{R}(1 - e^{-t/\tau})$ aur decay ka $I = I_0 e^{-t/\tau}$, jahan time constant $\tau = L/R$. Yaad rakho: ek $\tau$ ke baad growth mein current 63.2% tak chadhta hai, aur decay mein 37% tak gir jaata hai. Ye dono complementary hain. Lagbhag $5\tau$ ke baad transient khatam, current settle ho jaata hai.

Sabse common galti: log sochte hain $t=0$ par inductor short circuit hai. Galat! $t=0$ par inductor open circuit ki tarah behave karta hai (current zero), aur bahut der baad simple wire ban jaata hai. Ye capacitor ke bilkul ulta hai. Aur units se hamesha check karo — $L/R$ ke units seconds aate hain, $LR$ ke nahi.

Intuition pakad lo: inductor current ke liye "weight" ya inertia jaisa hai. Bhaari coil (bada $L$) → current slow start aur slow stop. Zyada $R$ → jaldi settle. Bas itna samajh gaye to numericals aram se ban jaayenge.