JEE/NEET Physics · Alternating Current series · Part 2 of 5 · All parts →
- Resistor: current in step with voltage — V = IR at every instant
- Inductor: current LAGS voltage by 90° — X_L = ωL grows with frequency
- Capacitor: current LEADS voltage by 90° — X_C = 1/ωC falls with frequency
- Reactances (Ω) oppose AC like resistance — but shift phase and dissipate nothing
- At DC: L is a short, C is an open — at high AC: the reverse
Feed the same AC to a resistor, a coil, and a capacitor, and each responds with a different personality: the resistor answers in step, the inductor dawdles behind, the capacitor jumps ahead. These phase dances build every filter and tuner you own. Part 2 of the Alternating Current series.
- The resistor: no drama
- The inductor: the laggard
- The capacitor: the rusher
- Reactance vs frequency
- Solved examples
- Common mistakes
- This physics in your daily life
- Practice set
- Recap
The Resistor: No Drama
v = V₀sinωt across R → i = (V₀/R)sinωt: same phase, same shape. Ohm’s law instant by instant. Average power: full V_rmsI_rms — resistors are AC’s honest workers.
The Inductor: The Laggard
Inductors oppose current CHANGE (the EMI card) — and AC changes forever. Result: current peaks a quarter-cycle LATE, and the opposition X_L = ωL grows with frequency: fast swings, furious opposition.
The Capacitor: The Rusher
Capacitor current depends on how fast voltage is CHANGING (i = C dv/dt): at the voltage’s steepest zero-crossings the current PEAKS. Current leads by 90°, and opposition X_C = 1/ωC falls with frequency: fast swings meet an easy path.
| Element | Phase (i vs v) | At DC | At high f |
|---|---|---|---|
| R | in step (0°) | R | R |
| L | lags 90° | short circuit | huge X_L |
| C | leads 90° | open circuit | tiny X_C |
Solved Examples
X_L = 2π×50×0.1 ≈ 31.4 Ω; X_C = 1/(2π×50×10⁻⁵) ≈ 318 Ω.
✔
Answer: 31.4 Ω; 318 Ω
X_L doubles; X_C halves — inductive paths tighten, capacitive paths open: the basis of every filter.
✔
Answer: X_L ×2; X_C ×½
ωL = 1/ωC → ω = 1/√(LC) = 1/√(10⁻⁶) = 1000 rad/s → f ≈ 159 Hz.
This is the resonance frequency — the next two parts’ hero.
✔
Answer: ≈159 Hz (resonance)
- Reactance = resistance. Both are in ohms, but reactance stores and returns energy (zero average dissipation) while resistance burns it.
- Phase direction swap. ELI the ICE man: EMF Leads I (inductor); I leads Capacitor EMF — chant it once, never swap again.
- DC behaviour forgotten. L at steady DC = plain wire (short); C at steady DC = gap (open): the ω = 0 limits of the formulas.
- Peak vs RMS mixing in Ohm-like calculations. Be consistent: V₀ = I₀X or V_rms = I_rmsX — never cross.
This Physics in Your Daily Life
- Crossover networks in speakers — inductors route bass to the woofer, capacitors route treble to the tweeter: your music filtered by phase personalities.
- Power supply smoothing — capacitors short the ripple (high f) to ground while blocking DC: clean power by X_C contrast.
- Radio tuners — L and C combining so only one frequency passes easily: every station you’ve ever heard selected by reactances.
- Dimmer switches and motor starters — inductive reactance limiting current without burning watts: control by lag.
- ‘Block DC, pass AC’ coupling capacitors in amplifiers — stage-to-stage audio handshakes through C’s frequency favouritism.
The inductor is a creature of habit: it fights any change in current, so when voltage commands ‘rise!’, current grudgingly follows late — a quarter cycle of hesitation forever. The capacitor is impulsive: it responds most when voltage is CHANGING fastest, sprinting through the transitions and resting when voltage rests.
0.1 H at 50 Hz: 31 Ω of opposition; at 5000 Hz: 3140 Ω — a coil progressively slamming doors. 10 μF: 318 Ω at 50 Hz, 3.2 Ω at 5000 Hz — the capacitor opening every gate. One element tightens with speed, the other loosens: filters in two formulas.
Draw the three waveforms on one time axis: R’s current hugging the voltage, L’s current sliding a quarter-period right, C’s current a quarter-period left. The two arrows flanking the honest one: the phase family portrait.
Practice set (answers hidden — try first)
(NEET-level) X_L at 100 Hz for 0.2 H:
(JEE Main-level) C = 20 μF at 50 Hz: X_C ≈
(NEET-level) In a pure inductor, current
(Concept) At DC steady state, a capacitor is
(JEE Main-level) Doubling f: X_L
- R: in phase, real power
- X_L = ωL: i lags 90°
- X_C = 1/ωC: i leads 90°
- L short at DC; C open at DC
- reactances store, don’t burn
- 🔁 three phase personalities
- 🔁 reactance formulas
- 🔁 frequency dependence
- 🧠 Chant (ELI the ICE man): ‘E leads I in L; I leads E in C’.
- 🧠 Speed rule: ‘L tightens, C loosens with frequency’.
- 🏠 Daily: speaker crossovers filter by reactance.
- 🏠 Daily: ‘block DC, pass AC’ is C’s motto.
Quick revision
- Resistor: current in step with voltage — V = IR at every instant
- Inductor: current LAGS voltage by 90° — X_L = ωL grows with frequency
- Capacitor: current LEADS voltage by 90° — X_C = 1/ωC falls with frequency
- Reactances (Ω) oppose AC like resistance — but shift phase and dissipate nothing
- At DC: L is a short, C is an open — at high AC: the reverse
- The inductor: the laggard
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