Purely Inductive AC Circuit 🌀⚡
A coil (inductance L) is hooked to an AC source that delivers \(v = v_m \sin \omega t\). Kirchhoff’s loop rule says the applied voltage must balance the self-induced emf, so
\[ v – L\frac{di}{dt}=0 \] :contentReference[oaicite:0]{index=0}
Finding the current
Rewrite the loop rule as \[ \frac{di}{dt}= \frac{v_m}{L}\sin \omega t. \quad (1) \] Integrate (1):
\[ i = -\frac{v_m}{\omega L}\cos \omega t + C. \] The current swings symmetrically about zero, so the constant C is zero. Use the identity \(\cos\omega t = \sin\bigl(\omega t + \frac{\pi}{2}\bigr)\) to get
\[ \boxed{\;i = i_m \sin\!\Bigl(\omega t – \frac{\pi}{2}\Bigr)\;} \qquad\text{with}\qquad i_m = \frac{v_m}{\omega L}. \] :contentReference[oaicite:1]{index=1}
Key takeaway ⏰: the current lags the voltage by exactly \(\frac{\pi}{2}\) rad (one-quarter cycle). :contentReference[oaicite:2]{index=2}
Inductive Reactance
Define the opposition offered by a coil as its inductive reactance
\[ \boxed{\;X_L = \omega L\;} \] :contentReference[oaicite:3]{index=3}
- Same unit as resistance: ohm (Ω).
- Bigger L or higher frequency → larger \(X_L\), so smaller current.
Instantaneous and Average Power 🔌
Instantaneous power is \[ p = vi = v_m i_m \sin\omega t \, \sin\!\Bigl(\omega t – \tfrac{\pi}{2}\Bigr) = -\tfrac{1}{2}v_m i_m \sin 2\omega t. \]
The average of \(\sin 2\omega t\) over a full cycle is zero, so
\[ \boxed{\;P_{\text{avg}} = 0\;} \]
Thus, an ideal inductor stores energy in its magnetic field during part of the cycle and returns it later; it never consumes energy. :contentReference[oaicite:4]{index=4}
Worked Example 🎯
Given: \(L = 25.0\,\text{mH}\), \(V_{\text{rms}} = 220\,\text{V}\), \(f = 50\,\text{Hz}\).
- \(X_L = 2\pi f L = 7.85\,\Omega\).
- \(I_{\text{rms}} = \dfrac{V_{\text{rms}}}{X_L} = 28\,\text{A}\).
Even a modest inductance lets only \(\approx 28\,\text{A}\) flow at mains frequency—proof that reactance behaves like frequency-dependent “resistance.” :contentReference[oaicite:5]{index=5}
Visualising with Phasors 🧭
Picture two arrows spinning at angular speed \(\omega\): the voltage arrow leads, and the current arrow trails by \(90^{\circ}\). Their vertical shadows trace the sine curves of \(v\) and \(i\). This rotating-vector trick makes phase relationships crystal-clear. :contentReference[oaicite:6]{index=6}
Quick Recap 📌
- Current lags voltage by \(\tfrac{\pi}{2}\).
- Inductive reactance \(X_L = \omega L\) limits AC current.
- \(I_{\text{rms}} = V_{\text{rms}}/X_L\).
- Average power in an ideal inductor is zero.
High-Yield Ideas for NEET 🎓
- Phase difference: coil current lags supply voltage by \(90^{\circ}\).
- Inductive reactance formula \(X_L = \omega L\) and its frequency dependence.
- Expression for peak current \(i_m = v_m/(\omega L)\).
- Zero average power in a purely inductive circuit.
