Electromagnetic Induction & Lenz's Law: Master Notes

Chapter Overview

Electromagnetic Induction (EMI) couples mechanical motion with electrical currents. In IIT-JEE (Advanced) and Olympiads, the most challenging problems combine motional EMF with Newton's 2nd Law (Terminal Velocity on Rails), induced non-conservative electric fields ($\oint {\overrightarrow{E}}_{\text{ind}}\cdot d\overrightarrow{\ell }=-\dot{\mathrm{\Phi }}$), RL transient dynamics, and mutual inductance matrices.

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1. Faraday's Flux Rule & Motional EMF

$$\mathcal{E}=-\frac{d{\mathrm{\Phi }}_B}{dt},{\mathrm{\Phi }}_B={\iint }_{\mathcal{S}}\overrightarrow{B}\cdot d\overrightarrow{A}$$

1.1 Motional EMF Derivations

• Straight Conductor Translating in Uniform $\overrightarrow{B}$:$$\boxed{{\displaystyle \mathcal{E}=(\overrightarrow{v}\times \overrightarrow{B})\cdot \overrightarrow{L}=BvL\sin \theta }}$$
• Rod of Length $L$ Rotating about One End with Angular Speed $\omega$ in $\overrightarrow{B}\perp \text{plane}$:$$\mathcal{E}={\int }_0^L(\omega rB)dr=\boxed{{\displaystyle \frac{1}{2}B\omega L^2}}$$
• Arbitrary Rotating Rigid Body: The potential difference between the rotation center and any rim point is $\mathrm{\Delta }V=\frac{1}{2}B\omega R^2$, completely independent of the shape of the perimeter!

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2. Induced Non-Conservative Electric Fields (${\overrightarrow{E}}_{\text{ind}}$)

When magnetic flux changes with time ($\frac{dB}{dt}\ne 0$) inside a cylindrical region of radius $R$:

$${\oint }_{\mathcal{C}}{\overrightarrow{E}}_{\text{ind}}\cdot d\overrightarrow{\ell }=-\frac{d{\mathrm{\Phi }}_B}{dt}=-\frac{dB}{dt}\cdot \text{Area}$$

Multi-Mode DiagramElectromagnetic Induction: Faraday Law & Rod on Rails

Option 1: Publication-Grade Scientific Vector SVG

Induced electric field loops and magnetic braking of conducting rod sliding on rails with terminal velocity v_t = mgR / (B^2 L^2).

Motional EMF: $\mathcal{E} = B L v$

Terminal Velocity: $v_t = \frac{m g R \sin\theta}{B^2 L^2}$

Option 2: Interactive 60 FPS Canvas Physics Engine

Simulate sliding rod magnetic braking and current induction.

Field B: 4 T

Induced EMF ($\mathcal{E}$):7.20 V

Option 3: Interactive 3D Spatial Viewport

3D rail circuit in transverse magnetic field.

+X

+Y

+Z

👆 Click and drag anywhere to rotate in 3D space

Azimuth Angle (ψ):35°

Elevation Angle (θ):-22°

Faraday's Law:$\mathcal{E} = -\frac{d\Phi}{dt}$

Option 5: Reactive Parametric Physics with Live Calculus

Terminal velocity calculator.

Field B: 3 T

: 45

v_t ($v_t$):$5.6 m/s$

Active Mode:Option 1 (Print & Derivation Vector SVG)Switch modes using the tabs above to explore different visualizations.

Region	Induced Electric Field $E_{\text{ind}}(r)$	Field Line Shape
Inside ($r\le R$)	$\boxed{E_{\text{in}} = \frac{r}{2}\left	\frac{dB}{dt}\right
Outside ($r>R$)	$\boxed{E_{\text{out}} = \frac{R^2}{2r}\left	\frac{dB}{dt}\right

::: pitfall ❌ Electric Potential Does NOT Exist for Induced Fields! Because $\mathrm{\nabla }\times {\overrightarrow{E}}_{\text{ind}}\ne \overrightarrow{0}$, work done depends on the path. Never define a scalar potential $V$ for induced electric fields! :::

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3. Rod Sliding on Rails (Magnetic Braking Dynamics)

A conducting rod of mass $m$, length $L$, and resistance $R$ slides on frictionless rails under external force $F_0$ or gravity $mg$ in perpendicular field $B$:

<!-- Left Resistor R --> <line x1="60" y1="40" x2="60" y2="60" stroke="var(--vp-c-text-1)" stroke-width="2.5"/> <rect x="50" y="60" width="20" height="40" fill="var(--vp-c-bg)" stroke="var(--vp-c-brand-1)" stroke-width="2"/> <text x="56" y="85" fill="var(--vp-c-brand-1)" font-size="12" font-weight="700">R</text> <line x1="60" y1="100" x2="60" y2="120" stroke="var(--vp-c-text-1)" stroke-width="2.5"/> <!-- Sliding Rod --> <rect x="250" y="25" width="16" height="110" fill="#3b82f6" stroke="var(--vp-c-text-1)" stroke-width="1.5" rx="3"/> <text x="275" y="60" fill="#3b82f6" font-size="12" font-weight="700">Rod (m, L)</text> <!-- Velocity Vector --> <line x1="266" y1="80" x2="330" y2="80" stroke="#10b981" stroke-width="3" marker-end="url(#arrow-green)"/> <text x="335" y="85" fill="#10b981" font-size="13" font-weight="700">v</text> <!-- Magnetic field cross markers --> <text x="140" y="70" fill="var(--vp-c-text-3)" font-size="14">&otimes; &otimes; &otimes;</text> <text x="140" y="100" fill="var(--vp-c-text-3)" font-size="14">&otimes; &otimes; &otimes; B</text> 

Figure Conducting rod of length L sliding with velocity v across parallel rails with resistance R in magnetic field B.

1. Induced Current: $I=\frac{\mathcal{E}}{R}=\frac{BvL}{R}$
2. Magnetic Retarding Force (Lenz's Law):$$F_{\text{mag}}=ILB=\left(\frac{BvL}{R}\right)LB=\frac{B^2{L}^2}{R}v$$
3. Equation of Motion:$$m\frac{dv}{dt}=F_0-\frac{B^2{L}^2}{R}v$$
4. Terminal Steady-State Speed ($a=0$):$$\boxed{{\displaystyle v_t=\frac{F_{0}R}{B^2{L}^2}\text{or}{v}_t=\frac{mgR\sin \theta }{B^2{L}^2}}}$$

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4. Inductance & RL Transients

• Self-Inductance of Solenoid: $L={\mu }_0{n}^{2}A\ell =\frac{{\mu }_0{N}^{2}A}{\ell }$
• Energy Stored in Magnetic Field: $U_B=\frac{1}{2}L{I}^2=\int \frac{B^2}{2{\mu }_0}dV$
• RL Current Growth: $i(t)=I_0(1-e^{-t/\tau })$ where $\tau =\frac{L}{R}$
• RL Current Decay: $i(t)=I_0{e}^{-t/\tau }$

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5. Authentic Previous Years Questions (PYQs)

PYQ 1: JEE Advanced 2022 (Paper 2) — Rod on Rails with Capacitor

Question:
A conducting rod of mass $m$ and length $L$ slides on frictionless horizontal rails connected across a capacitor of capacitance $C$ in a uniform vertical magnetic field $B$. A constant horizontal force $F$ is applied to the rod. Find the acceleration $a$ of the rod.

Step-by-Step Solution:

1. When the rod reaches speed $v$, the induced EMF across the capacitor is $\mathcal{E}=BvL$.
2. Charge on capacitor: $q=C\mathcal{E}=CBLv$.
3. Current flowing into capacitor:$$i=\frac{dq}{dt}=CBL\frac{dv}{dt}=CBLa$$
4. Magnetic retarding force on the rod:$$F_{\text{mag}}=iLB=(CBLa)LB=(C{B}^2{L}^2)a$$
5. Newton's Second Law for the rod:$$F-F_{\text{mag}}=ma⟹F-(C{B}^2{L}^2)a=ma$$$$F=(m+C{B}^2{L}^2)a⟹\boxed{{\displaystyle a=\frac{F}{m+C{B}^2{L}^2}}}$$(Notice that the capacitor acts as an additional "virtual mass" $m_{\text{virtual}}=C{B}^2{L}^2$!)

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PYQ 2: JEE Advanced 2020 — Induced Electric Field Torque on Ring

Question:
A non-conducting thin ring of mass $M$, radius $R$, carrying uniform charge $Q$ is free to rotate about its central axis. A uniform magnetic field $B(t)$ perpendicular to the ring changes at a constant rate $\frac{dB}{dt}=\alpha$. Find the angular acceleration ${\alpha }_{\text{rot}}$ of the ring.

Step-by-Step Solution:

1. Induced electric field at radius $R$:$$E_{\text{ind}}=\frac{R}{2}\left|\frac{dB}{dt}\right|=\frac{R\alpha }{2}$$
2. Force on ring charge $Q$:$$F=Q{E}_{\text{ind}}=Q\left(\frac{R\alpha }{2}\right)$$
3. Torque on ring:$$\tau =F\cdot R=\left(Q\frac{R\alpha }{2}\right)R=\frac{1}{2}Q\alpha R^2$$
4. Applying $\tau =I{\alpha }_{\text{rot}}$ where $I_{\text{ring}}=M{R}^2$:$$\frac{1}{2}Q\alpha R^2=(M{R}^2){\alpha }_{\text{rot}}⟹\boxed{{\displaystyle {\alpha }_{\text{rot}}=\frac{Q\alpha }{2M}}}$$

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6. High-Yield Formula Sheet

Entity	Formula	Notes
Rotating Rod EMF	$\mathcal{E}=\frac{1}{2}B\omega L^2$	Pinned at one end
Induced E-Field (Inside)	$E=\frac{r}{2}\dot{B}$	Circular non-conservative
Terminal Rail Speed	$v_t=\frac{FR}{B^2{L}^2}$	Constant force $F$
Virtual Mass of Capacitor	$m_{\text{eff}}=m+C{B}^2{L}^2$	Rod on rails with capacitor
Magnetic Energy Density	$u_B=\frac{B^2}{2{\mu }_0}$	Stored in B-field