Self and Mutual Inductance — Revision Notes

⚡ 30-Second Revision

Self-Inductance ($L$): — Property of a coil to oppose current change in itself. Unit: Henry (H).
Induced EMF (Self): — $\mathcal{E} = -L \frac{dI}{dt}$
Self-Inductance of Solenoid: — $L = \mu_0 \frac{N^2 A}{l}$ (for air core)
Energy Stored in Inductor: — $U = \frac{1}{2}LI^2$
Mutual Inductance ($M$): — Property of two coils where current change in one induces EMF in other. Unit: Henry (H).
Induced EMF (Mutual): — $\mathcal{E}_2 = -M \frac{dI_1}{dt}$
Coefficient of Coupling ($k$): — $M = k \sqrt{L_1 L_2}$ (where $0 \le k \le 1$)
Lenz's Law: — Induced EMF opposes the change in current/flux causing it.

2-Minute Revision

Self and mutual inductance are crucial concepts in electromagnetic induction. **Self-inductance ($L$)** is a coil's inherent property to resist changes in its own current. When current in a coil changes, it creates a changing magnetic flux through itself, inducing a 'back EMF' ($\mathcal{E} = -L \frac{dI}{dt}$) that opposes this change, as per Lenz's Law.

The self-inductance of a solenoid depends on its geometry ($L = \mu_0 \frac{N^2 A}{l}$) and core material. Inductors store energy in their magnetic fields, given by $U = \frac{1}{2}LI^2$. **Mutual inductance ($M$)** describes the magnetic coupling between two separate coils.

A changing current in one coil (primary) induces an EMF in the other coil (secondary) ($\mathcal{E}_2 = -M \frac{dI_1}{dt}$). $M$ depends on the geometry, orientation, and separation of both coils, as well as the core material.

The coefficient of coupling ($k$) quantifies this interaction: $M = k \sqrt{L_1 L_2}$. Both $L$ and $M$ are measured in Henrys (H). Remember that inductance opposes *change* in current, not current itself.

5-Minute Revision

Let's quickly review Self and Mutual Inductance, focusing on key formulas and concepts for NEET. **Self-inductance ($L$)** is the property of a single coil to oppose any change in the current flowing through it.

This opposition arises from an induced EMF, often called 'back EMF', which is given by $\mathcal{E} = -L \frac{dI}{dt}$. The negative sign is crucial, signifying Lenz's Law – the induced EMF always acts to oppose the change in current.

For a long solenoid, its self-inductance is $L = \mu_0 \frac{N^2 A}{l}$, where $N$ is the number of turns, $A$ is the cross-sectional area, and $l$ is the length. Note that $L$ is independent of the current itself, but depends on the coil's geometry and the permeability of its core.

An inductor stores energy in its magnetic field, calculated as $U = \frac{1}{2}LI^2$. This energy is released when the current is switched off. For example, if a $0.5,\text{H}$ inductor carries $4,\text{A}$, it stores $U = \frac{1}{2}(0.

5)(4^2) = 4, ext{J}$.

**Mutual inductance ($M$)** describes the phenomenon where a changing current in one coil (the primary) induces an EMF in a nearby, separate coil (the secondary). The induced EMF in the secondary coil is $\mathcal{E}_2 = -M \frac{dI_1}{dt}$, where $dI_1/dt$ is the rate of change of current in the primary.

Mutual inductance depends on the geometry of both coils, their relative orientation, separation, and the permeability of the medium between them. It's a reciprocal property, meaning $M_{12} = M_{21}$.

The degree of magnetic coupling between two coils is quantified by the coefficient of coupling $k$, where $M = k \sqrt{L_1 L_2}$. $k$ ranges from 0 (no coupling) to 1 (perfect coupling). For instance, if $L_1 = 2,\text{H}$, $L_2 = 8,\text{H}$, and $k=0.

75$, then$M = 0.75 \sqrt{2 \times 8} = 0.75 \sqrt{16} = 0.75 \times 4 = 3, ext{H}$. Both self and mutual inductance are measured in Henrys (H). Remember to apply Lenz's Law correctly for the direction of induced EMF and current.

Prelims Revision Notes

Self and Mutual Inductance: NEET Quick Recall

1. Magnetic Flux ($\Phi_B$):

Definition: Number of magnetic field lines passing through an area.
Formula: $\Phi_B = BA \cos\theta$.
Unit: Weber (Wb).

2. Self-Inductance ($L$):

Definition: Property of a coil to oppose change in its own current.
Relation to flux: $\Phi_B = LI$ (Total flux linkage).
Induced EMF: $\mathcal{E} = -L \frac{dI}{dt}$. (Negative sign due to Lenz's Law).
Unit: Henry (H). $1,\text{H} = 1,\text{Wb/A} = 1,\text{V s/A}$.
**Factors affecting $L$ (for a solenoid):**

* Number of turns ($N$): $L \propto N^2$. * Cross-sectional area ($A$): $L \propto A$. * Length ($l$): $L \propto 1/l$. * Permeability of core material ($\mu$): $L \propto \mu$.

Formula for long solenoid (air core): — $L = \mu_0 \frac{N^2 A}{l}$.
Energy stored in an inductor: — $U = \frac{1}{2}LI^2$. (Stored in magnetic field).

3. Mutual Inductance ($M$):

Definition: Property of two coils where a changing current in one induces EMF in the other.
Relation to flux: $\Phi_{B2} = M I_1$ (Flux in coil 2 due to current in coil 1).
Induced EMF: $\mathcal{E}_2 = -M \frac{dI_1}{dt}$. (Similarly, $\mathcal{E}_1 = -M \frac{dI_2}{dt}$).
Unit: Henry (H).
**Factors affecting $M$:**

* Geometry of both coils. * Relative orientation and separation between coils. * Permeability of the medium/core.

Formula for two coaxial solenoids: — $M = \mu_0 \frac{N_1 N_2 A_2}{l_1}$ (if inner coil 2 is shorter and its area $A_2$ is used).
Coefficient of Coupling ($k$): — $M = k \sqrt{L_1 L_2}$.

* $0 \le k \le 1$. * $k=1$ for perfect coupling (e.g., ideal transformer).

4. Lenz's Law:

The direction of induced EMF/current always opposes the cause producing it.
Crucial for determining the sign of induced EMF and direction of current.

5. Key Distinctions:

Self-inductance: Single coil, opposes *its own* current change.
Mutual inductance: Two coils, one induces EMF in the *other*.
Inductors oppose *change* in current, not current itself (unlike resistors).

Vyyuha Quick Recall

To remember the factors affecting self-inductance of a solenoid: 'N.A.L.I.M.A.'

N — Number of turns ($N^2$)
A — Area of cross-section ($A$)
L — Length of the solenoid ($1/l$)
I — *Independent* of Current (This is the trick! Current *causes* flux, but $L$ is the ratio, not dependent on $I$)
M — Material of the core ($\mu$)
A — (Just to complete the name, no specific factor)