Electromagnetic Induction — Revision Notes

⚡ 30-Second Revision

Magnetic Flux: — $\Phi_B = BA\cos\theta$ (Unit: Weber, Wb)
Faraday's Law: — $\epsilon = -N\frac{d\Phi_B}{dt}$
Lenz's Law: — Induced EMF opposes the cause of flux change.
Motional EMF (linear): — $\epsilon = BLv$ (if $\vec{B}$, $\vec{L}$, $\vec{v}$ are mutually perpendicular)
Motional EMF (rotating rod): — $\epsilon = \frac{1}{2}B\omega L^2$ (if $\vec{B}$ is perpendicular to plane of rotation)
Self-Inductance: — $\Phi_B = LI$, $\epsilon = -L\frac{dI}{dt}$ (Unit: Henry, H)
Self-Inductance of solenoid: — $L = \frac{\mu_0 N^2 A}{l}$
Energy stored in inductor: — $U = \frac{1}{2}LI^2$
Mutual Inductance: — $\Phi_{B2} = MI_1$, $\epsilon_2 = -M\frac{dI_1}{dt}$ (Unit: Henry, H)
Eddy Currents: — Circulating currents in bulk conductors due to changing flux; cause heating, minimized by lamination.

2-Minute Revision

Electromagnetic Induction (EMI) is the generation of EMF and current by a changing magnetic flux. Magnetic flux ($\Phi_B = BA\cos\theta$) is the key quantity. Faraday's Law states that induced EMF is proportional to the rate of change of flux: $\epsilon = -N\frac{d\Phi_B}{dt}$.

The negative sign is explained by Lenz's Law, which says the induced current opposes the change that caused it, upholding energy conservation. This means if flux increases, the induced field opposes it; if flux decreases, it aids it.

Motional EMF ($BLv$) occurs when a conductor moves in a magnetic field. For a rotating rod, it's $\frac{1}{2}B\omega L^2$. **Self-inductance ($L$)** is a coil's property to induce EMF in itself due to its own changing current ($\epsilon = -L\frac{dI}{dt}$).

Energy stored in an inductor is $U = \frac{1}{2}LI^2$. **Mutual inductance ($M$)** describes EMF induced in one coil due to changing current in a nearby coil ($\epsilon_2 = -M\frac{dI_1}{dt}$). Eddy currents are induced loops in bulk conductors, causing heating (energy loss), but used in applications like induction furnaces.

They are minimized by laminating cores.

5-Minute Revision

Electromagnetic Induction (EMI) is the core principle behind electricity generation. It hinges on magnetic flux ($\Phi_B$), which is the measure of magnetic field lines passing through an area. The fundamental law is Faraday's Law of Induction, stating that an EMF is induced whenever magnetic flux changes, and its magnitude is $\epsilon = -N\frac{d\Phi_B}{dt}$.

The rate of change is crucial. For instance, if a coil's area changes from $A_1$ to $A_2$ in time $\Delta t$ in a field $B$, $\epsilon = -NB\frac{(A_2-A_1)}{\Delta t}$.

Lenz's Law provides the direction: the induced current creates a magnetic field that opposes the *change* in flux. This is vital for energy conservation. If you push a North pole towards a coil, the induced current makes the coil's face a North pole to repel it. If you pull it away, it becomes a South pole to attract it. Always think 'opposition to change'.

Motional EMF is a special case. A conductor of length $L$ moving with velocity $v$ perpendicular to a magnetic field $B$ induces $\epsilon = BLv$. For a rod rotating with angular velocity $\omega$ in a perpendicular field, $\epsilon = \frac{1}{2}B\omega L^2$. Remember the conditions for these formulas.

**Self-inductance ($L$)** is a coil's inherent property to resist changes in its own current. When current $I$ changes, it induces a 'back EMF' $\epsilon = -L\frac{dI}{dt}$. A solenoid's inductance is $L = \frac{\mu_0 N^2 A}{l}$. Inductors store energy in their magnetic field: $U = \frac{1}{2}LI^2$. This energy is released when the current decreases.

**Mutual inductance ($M$)** occurs between two coils. A changing current in one coil ($I_1$) induces an EMF in the other ($\epsilon_2 = -M\frac{dI_1}{dt}$). Transformers operate on this principle. The value of $M$ depends on the geometry and relative positioning of the coils.

Eddy currents are induced circulating currents in bulk conductors due to changing flux. They cause heating (energy loss) and are minimized by laminating transformer cores. However, they are useful in induction furnaces and magnetic braking. Always review the units: Weber (Wb) for flux, Henry (H) for inductance, Volt (V) for EMF.

Prelims Revision Notes

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Magnetic Flux ($\Phi_B$): — Scalar quantity. $\Phi_B = \vec{B} \cdot \vec{A} = BA\cos\theta$. Unit: Weber (Wb). $1\,\text{Wb} = 1\,\text{T}\cdot\text{m}^2$. Change in flux can be due to change in $B$, $A$, or $\theta$.
2
Faraday's Laws:

* First Law: EMF is induced when magnetic flux changes. * Second Law: $\epsilon = -N\frac{d\Phi_B}{dt}$. The magnitude is $|\epsilon| = N|\frac{d\Phi_B}{dt}|$.

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Lenz's Law: — Direction of induced current opposes the change in magnetic flux that produced it. This ensures conservation of energy. (e.g., if flux increases, induced field opposes; if flux decreases, induced field aids).
2
Motional EMF:

* Straight conductor of length $L$ moving with velocity $v$ perpendicular to uniform $B$: $\epsilon = BLv$. Polarity by right-hand rule (for force on positive charge). * Rod of length $L$ rotating with angular velocity $\omega$ in uniform $B$ (perpendicular to plane of rotation): $\epsilon = \frac{1}{2}B\omega L^2$.

1
Eddy Currents: — Induced circulating currents in bulk conductors. Cause heating (Joule heating, $I^2R$ loss). Applications: induction furnace, magnetic braking, electromagnetic damping. Minimized by laminating cores (thin insulated sheets).
2
Self-Inductance (L): — Property of a coil to oppose change in its own current. $\Phi_B = LI$. Induced EMF: $\epsilon = -L\frac{dI}{dt}$. Unit: Henry (H). $1\,\text{H} = 1\,\text{Wb/A}$.

* For a long solenoid: $L = \frac{\mu_0 N^2 A}{l}$. (If core material present, replace $\mu_0$ with $\mu = \mu_r \mu_0$). * Energy stored in an inductor: $U = \frac{1}{2}LI^2$.

1
Mutual Inductance (M): — Property of two coils where a changing current in one induces EMF in the other. $\Phi_{B2} = MI_1$. Induced EMF in secondary: $\epsilon_2 = -M\frac{dI_1}{dt}$. Unit: Henry (H).

* For two coaxial solenoids: $M = \frac{\mu_0 N_1 N_2 A_1}{l_1}$ (where $A_1$ is area of inner solenoid). * Coefficient of coupling $k = \frac{M}{\sqrt{L_1 L_2}}$. For perfectly coupled coils, $k=1$, so $M = \sqrt{L_1 L_2}$.

1
Important Points:

* Induced EMF depends on *rate of change* of flux, not just flux. * Lenz's Law is a consequence of energy conservation. * Inductors oppose *changes* in current, not current itself (in DC steady state, $X_L=0$).

Vyyuha Quick Recall

For Lazy Men, Some Money Earns:

Faraday's Law (magnitude of EMF)
Lenz's Law (direction of EMF)
Motional EMF (moving conductors)
Self-inductance (single coil, its own current)
Mutual inductance (two coils, coupled)
Eddy currents (bulk conductors, energy loss)