Derivation and Formula for Current Growth and Decay in LR Circuits
Growth and decay of current in LR circuits describe how the current changes when a circuit containing both an inductor and a resistor is switched on or off. This topic is essential in JEE Main, as it explains key time-dependent processes in electrical circuits. LR circuit transients appear in relays, car ignition systems, and many control circuits. Understanding their formulas, time constant, and graphical behavior builds a strong foundation for advanced electricity problems.
When a battery is connected to a series LR circuit, the current does not immediately reach its maximum value due to the inductor's opposition to change. Instead, the current increases gradually, a process called growth of current. When the battery is removed, the current decreases exponentially, known as decay of current. These phenomena are directly governed by Kirchhoff’s law and the properties of inductance.
In an LR circuit, an inductor (symbol L) creates a back emf proportional to the rate of change of current. The resistor (R) limits steady-state current. At the instant voltage is applied, the inductor resists sudden changes, enforcing a smooth exponential approach to the maximum value. For comparison, RC circuits involve capacitors, but here, inductors are the focus.
Let's start with a simple circuit: a battery of emf E is connected in series with R and L. Using Kirchhoff’s law, the voltage equation is:
E = L (dI/dt) + IR
Initially (\( t = 0 \)), current (\( I \)) is zero. As time passes, current increases according to the following steps.
Growth and Decay of Current in LR Circuits: Stepwise Derivation
- Start with E = L (dI/dt) + IR.
- Rearrange: (E - IR) = L (dI/dt).
- Separation of variables: dI/(E/R - I) = (R/L) dt.
- Integrate from \( t = 0, I = 0 \) to \( t = t, I = I \):
- Obtain: I = (E/R)[1 - e^{-t/τ}], where \( τ = L/R \).
- For decay, initial current \( I_0 = E/R \): I = I_0 e^{-t/τ}.
Graphical Representation of Current Growth and Decay in LR Circuits
The nature of current change in LR circuits is best visualized using current vs. time graphs. The growth phase curve starts at zero and asymptotically approaches maximum, while the decay phase falls steadily to zero.
Both curves show exponential behavior. The time constant determines how steep or gradual these curves appear for different values of L and R.
Time Constant and Its Significance in Growth and Decay of Current in LR Circuits
The time constant (τ = L/R) is the crucial parameter for analyzing LR circuit transients. It tells us how quickly current changes:
- After one time constant (\( t = τ \)), current grows to 63% of maximum.
- After \( t = τ \) during decay, current falls to 37% of its initial value.
- Larger L or smaller R means a slower current change.
- Standard units: L in henry (H), R in ohm (Ω), \( τ \) in second (s).
- Same τ applies to both growth and decay phases.
Below is a table summarizing important quantities in LR circuits:
| Quantity | Formula | Unit |
|---|---|---|
| Time constant (τ) | L/R | s |
| Growth of current | I = I0(1–e–t/τ) | A |
| Decay of current | I = I0e–t/τ | A |
| Steady-state current (I0) | E/R | A |
Worked Example: Growth and Decay of Current in LR Circuits
A 12 V battery, 6 Ω resistor, and 3 H inductor are connected in series. Calculate:
- The time constant of the circuit
- Current after 1 s when switch is closed
- Current after 1 s if switch is opened at maximum current
- \( τ = L/R = 3/6 = 0.5\,s \)
- Maximum current, \( I_0 = 12/6 = 2\,A \)
- After 1 s (growth): \( I = 2 \times (1 - e^{-1/0.5}) = 2 \times (1 - e^{-2}) \approx 2 \times (1 - 0.135) = 1.73\,A \)
- After 1 s (decay): \( I = 2 \times e^{-1/0.5} = 2 \times e^{-2} \approx 2 \times 0.135 = 0.27\,A \)
So, current after 1 s during growth is 1.73 A, and after switch off, decays to 0.27 A in 1 s.
Common Pitfalls and Tips for Growth and Decay of Current in LR Circuits
- Never assume current changes instantly in an LR circuit; the inductor opposes sudden change.
- For growth, \( I \) starts at zero; for decay, it starts at maximum value (\( E/R \)).
- Always use \textbf{SI} units: henry for inductance, ohm for resistance, second for time.
- Logarithm errors often occur in integration steps in derivations.
- Keep an eye on sign conventions when applying exponential formulas.
- Compare with capacitor versus inductor behavior for conceptual clarity.
Besides academics, growth and decay of current in LR circuits appear in real-world devices: relays, timers, and circuits with magnetic coils. The time constant directly affects the speed of devices like electromagnetic door locks or automotive relays. Practising questions on electromagnetic induction and alternating currents will further strengthen your understanding.
For more on steady-state conditions, self-inductance, and the stepwise reasoning behind formulas, review inductor properties and self-inductance derivations. Vedantu content aligns with latest JEE trends and covers all solved examples for confidence-building.
Summary: growth and decay of current in LR circuits are ruled by exponential change with time constant τ = L/R. The current approaches its steady-state or falls to zero predictably, enabling reliable circuit control and timing in physics and engineering.
- Review ohm’s law and resistance for resistor effects in LR circuits.
- Sharpen your circuit-solving with kirchhoff’s circuit law strategy.
- Explore wheatstone bridge for advanced question variety.
- Test yourself with revision notes on electromagnetic induction.
- Check your LR circuit knowledge through this mock test on electromagnetic induction.
- See circuit analogies via lr and rc circuits comparisons.
- Build depth using current electricity basics and types of resistance connections.
- Browse all jee physics syllabus essentials for exam scope.
FAQs on Growth and Decay of Current in LR Circuits Explained
1. What is the growth and decay of current in RL circuit?
Growth and decay of current in an RL circuit refer to how the current increases or decreases exponentially when the circuit is switched on or off.
Key points:
- During growth, current builds up from zero to its maximum value.
- During decay, current drops from its maximum to zero after removing the supply.
- These changes follow exponential formulas involving LR circuit time constant (τ = L/R).
2. How do you derive the formula for current in an LR circuit?
The formula for current in an LR circuit is derived using Kirchhoff’s Voltage Law and inductor properties.
Steps:
- Apply KVL to the circuit: V = L (dI/dt) + IR
- Rearrange and integrate: dI/(I_0 - I) = (R/L) dt
- For current growth: I = I_0(1 - e^{-t/τ})
- For current decay: I = I_0 e^{-t/τ}
3. What happens to current when you disconnect the battery in an LR circuit?
When the battery is disconnected from an LR circuit, the current does not drop instantly but decays exponentially.
Process:
- Current gradually decreases from its initial value to zero.
- This decay is governed by I = I_0 e^{-t/τ}.
- Induced emf in the inductor opposes the sudden change due to Lenz's Law.
4. Why does current take time to reach maximum in LR circuits?
Current in LR circuits takes time to reach its maximum value because the inductor resists changes in current.
Reasons:
- Inductance (L) creates a back emf opposing the rise in current.
- This causes the current to increase gradually following I = I_0(1 - e^{-t/τ}).
- The rate of growth depends on the time constant (τ = L/R).
5. How is the time constant calculated for an LR circuit?
The time constant (τ) of an LR circuit is calculated as:
τ = L/R
- L is the inductance (in henry, H)
- R is the resistance (in ohm, Ω)
- Time constant represents the time taken for current to reach 63% of its maximum or fall to 37% during decay.
6. Can the current in an LR circuit rise or fall instantaneously?
No, the current in an LR circuit cannot change instantly due to the presence of inductance.
Key points:
- Inductors always oppose sudden changes in current.
- Current changes occur gradually following the exponential law.
- This behaviour is essential in protecting circuits from current surges.
7. Is the time constant the same for current growth and decay in an LR circuit?
Yes, the time constant (τ = L/R) remains the same for both current growth and decay phases in an LR circuit.
- τ determines how fast current rises or falls.
- It depends on the circuit's inductance and resistance.
- Bigger L or smaller R means a slower process; smaller τ means faster change.
8. What is the physical meaning of “steady-state” current in this context?
In LR circuits, steady-state current is the final, constant current value after all transients have died out.
Details:
- For current growth: steady-state = maximum current (I_0 = V/R)
- For decay: steady-state = zero current
- At steady state, inductor acts as a simple conductor with no induced emf.
9. How do you identify growth vs. decay phases in LR circuit diagrams?
You can identify growth and decay phases in LR circuit diagrams by the position of the switch and circuit setup.
Indicators:
- Growth phase: Switch connects battery across LR circuit; current starts from zero.
- Decay phase: Switch disconnects battery and connects LR loop alone; current drops from maximum value.
10. What are common mistakes when applying exponential formulas in LR circuit numericals?
Common mistakes when solving LR circuit numericals include:
- Using incorrect time constant (τ = L/R must be calculated correctly).
- Mixing up growth and decay formulas.
- Forgetting units for inductance (H) and resistance (Ω).
- Not checking initial and final current values.
- Forgetting the effect of switch positions on circuit behaviour.
11. What is the formula for the growth and decay of current in an LR circuit?
The growth and decay of current in an LR circuit are described by exponential formulas:
- Growth: I = I_0(1 - e^{-t/τ})
- Decay: I = I_0 e^{-t/τ}
- Where I_0 = V/R and τ = L/R is the time constant.
12. Where are LR circuits used in real life?
LR circuits are used in various real-life applications including:
- Electrical relays and timers
- Filtering signals in audio systems
- Inductive kick in ignition systems
- Surge protection devices
- Switching circuits and electronic chokes
