6.7 Inductance 🌀

An electric current can create or feel a change in magnetic flux either because of a neighboring coil or because of its own changing current. In both cases the flux through a coil stays directly proportional to the current flowing in (or near) the coil – a relationship that lies at the heart of inductance. :contentReference[oaicite:0]{index=0}

Core Idea 🌟

  • For a tightly wound N-turn coil the “flux linkage” is \(N\Phi_B\propto I\) . The constant of proportionality is called inductance. :contentReference[oaicite:1]{index=1}
  • Inductance depends only on the coil’s shape and the magnetic material inside – just as capacitance depends on plate geometry and dielectric. :contentReference[oaicite:2]{index=2}
  • Unit: henry (H) ✏️. Dimensions: \([M L^2 T^{-2} A^{-2}]\). Named after Joseph Henry. :contentReference[oaicite:3]{index=3}

6.7.1 Mutual Inductance (\(M\)) 🤝

If a current in one coil (S2) sets up a flux in a neighboring coil (S1), the flux linkage is

\[ N_1\Phi_1 = M I_2. \tag{6.7} \]

For two long coaxial solenoids of common length \(l\)

\[ M = \mu_0\,n_1 n_2 \,\pi r_1^{2}\,l, \tag{6.9} \]

and remarkably \(M_{12}=M_{21}\) – current direction doesn’t matter! :contentReference[oaicite:4]{index=4}

  • Filling the space with a material of relative permeability \(\mu_r\) multiplies M by \(\mu_r\).
  • Mutual inductance also changes with coil spacing and orientation.
Quick Example 📐 (Example 6.8)

Two concentric rings with \(r_1\ll r_2\):

\[ M \;=\; \mu_0\,\frac{\pi r_1^{2}}{2r_2}. \]

Induced emf between coils ⚡

If the current in the primary coil varies with time,

\[ \varepsilon_1 = -\,M\,\dfrac{\mathrm d I_2}{\mathrm dt}. \]

:contentReference[oaicite:8]{index=8}

6.7.2 Self-Inductance (\(L\)) 🔄

A single coil resists changes in its own current; the flux linkage follows

\[ N\Phi_B = L I. \tag{6.13} \]

Changing the current produces the “back emf”

\[ \boxed{\;\varepsilon = -\,L\,\dfrac{\mathrm d I}{\mathrm dt}\;} \tag{6.14} \]

  • The back emf always fights the change (Lenz’s rule in action) 🛡️. :contentReference[oaicite:9]{index=9}

Solenoid as an Inductor

For a long solenoid of cross-sectional area \(A\), length \(l\), and turn density \(n\):

\[ L = \mu_0\,n^2\,A\,l. \tag{6.15} \]

With a magnetic core of permeability \(\mu_r\):

\[ L = \mu_0\mu_r\,n^2\,A\,l. \tag{6.16} \]

:contentReference[oaicite:10]{index=10}

Energy Stored in an Inductor 💡

Building up current takes work, stored as magnetic energy:

\[ U = \tfrac12\,L I^{2}. \tag{6.17} \]

In a solenoid this equals

\[ U = \frac{B^{2}}{2\mu_0}\,A l, \]

so the energy density in any magnetic field is

\[ u = \frac{B^{2}}{2\mu_0}. \tag{6.18} \]

:contentReference[oaicite:11]{index=11}

The inductor’s tendency to oppose current change is analogous to mass resisting acceleration – inductance acts like electrical inertia. :contentReference[oaicite:12]{index=12}

High-Yield Picks for NEET 🔥

  1. Mutual inductance equality: \(M_{12}=M_{21}\) for any pair of coils.
  2. Self-induced emf formula \( \varepsilon = -L\,\dfrac{\mathrm d I}{\mathrm dt}\).
  3. Solenoid inductance \(L=\mu_0 n^{2}Al\) (or \(\mu_0\mu_r n^{2}Al\) with core).
  4. Energy in an inductor \(U=\tfrac12 LI^{2}\) and energy density \(u = B^{2}/(2\mu_0)\).
  5. Mutual emf between coils \( \varepsilon = -M\,\dfrac{\mathrm d I}{\mathrm dt}\) and its dependence on geometry and medium.

Happy studying! 🧲✨