Electromagnetic Induction (JEE

Cheatsheet Content

### Magnetic Flux ($\Phi_B$) - **Definition:** The number of magnetic field lines passing through a given area. - **Formula:** $\Phi_B = \vec{B} \cdot \vec{A} = BA\cos\theta$ - $B$: Magnetic field strength (Tesla, T) - $A$: Area vector, magnitude is area, direction is normal to the surface ($m^2$) - $\theta$: Angle between $\vec{B}$ and $\vec{A}$ - **For varying field/area:** $\Phi_B = \int \vec{B} \cdot d\vec{A}$ - **Unit:** Weber (Wb) or Tesla-meter$^2$ ($T m^2$) - **Nature:** Scalar quantity. ### Faraday's Law of Electromagnetic Induction - **Statement:** The magnitude of the induced electromotive force (EMF) in a circuit is directly proportional to the time rate of change of magnetic flux through the circuit. - **Formula:** $\mathcal{E} = -\frac{d\Phi_B}{dt}$ - $\mathcal{E}$: Induced EMF (Volts, V) - $\frac{d\Phi_B}{dt}$: Rate of change of magnetic flux - **For N turns coil:** $\mathcal{E} = -N\frac{d\Phi_B}{dt}$ - **Integral form (generalized):** $\oint \vec{E} \cdot d\vec{l} = -\frac{d}{dt} \int \vec{B} \cdot d\vec{A}$ - This shows that a changing magnetic field induces a non-conservative electric field. ### Lenz's Law - **Statement:** The direction of the induced current (or EMF) is such that it opposes the cause producing it. - **Implication:** The negative sign in Faraday's law represents Lenz's Law. - **Conservation of Energy:** Lenz's law is a consequence of the conservation of energy. If the induced current aided the change, it would lead to a perpetual increase in energy. - **Application:** - If $\Phi_B$ increases, induced current creates $\vec{B}_{ind}$ opposite to original $\vec{B}$. - If $\Phi_B$ decreases, induced current creates $\vec{B}_{ind}$ in the same direction as original $\vec{B}$. ### Motional EMF - **Definition:** EMF induced across a conductor moving in a magnetic field. - **Formula for a straight conductor:** $\mathcal{E} = (\vec{v} \times \vec{B}) \cdot \vec{L}$ - If $\vec{v}$, $\vec{B}$, and $\vec{L}$ are mutually perpendicular: $\mathcal{E} = BLv$ - $\vec{v}$: Velocity of the conductor - $\vec{B}$: Magnetic field - $\vec{L}$: Length vector of the conductor - **Mechanism:** Magnetic force on charges ($F_m = q(\vec{v} \times \vec{B})$) separates them, creating an electric field and thus an EMF. - **Rod rotating in B-field:** For a rod of length $L$ rotating with angular velocity $\omega$ about one end in a uniform magnetic field $B$ perpendicular to the plane of rotation: $$\mathcal{E} = \frac{1}{2} B \omega L^2$$ ### Induced Current, Power & Force - **Induced Current:** $I = \frac{\mathcal{E}}{R}$ (where R is the resistance of the circuit) - **Power Dissipated:** $P = I\mathcal{E} = I^2R = \frac{\mathcal{E}^2}{R}$ - **Force on moving conductor:** When a current $I$ flows in a conductor of length $L$ moving in a magnetic field $B$, a magnetic force $F = I L B$ acts on it (if perpendicular). This force opposes the motion (Lenz's Law). - **Mechanical Power:** To maintain constant velocity, an external force must be applied, doing work at a rate $P_{mech} = F_{ext} v$. This mechanical power is converted into electrical power. ### Self-Induction - **Definition:** The phenomenon where an EMF is induced in a coil due to a change in the current flowing through the *same* coil. - **Self-Inductance (L):** The ratio of magnetic flux linked with a coil to the current flowing through it. - $\Phi_B \propto I \implies \Phi_B = LI$ - Unit: Henry (H) - **Induced EMF:** $\mathcal{E}_L = -L\frac{dI}{dt}$ - **Energy Stored in Inductor:** $U_L = \frac{1}{2}LI^2$ - **Inductors in Series:** $L_{eq} = L_1 + L_2 + ...$ (if no mutual inductance) - **Inductors in Parallel:** $\frac{1}{L_{eq}} = \frac{1}{L_1} + \frac{1}{L_2} + ...$ (if no mutual inductance) - **RL Circuit (DC source):** - **Current growth:** $I(t) = I_0(1 - e^{-t/\tau})$ where $I_0 = \frac{\mathcal{E}}{R}$ and $\tau = \frac{L}{R}$ (time constant) - **Current decay:** $I(t) = I_0 e^{-t/\tau}$ ### Mutual Induction - **Definition:** The phenomenon where an EMF is induced in one coil due to a change in current flowing through an *adjacent* coil. - **Mutual Inductance (M):** The ratio of magnetic flux linked with one coil to the current flowing in the other coil. - $\Phi_{21} \propto I_1 \implies \Phi_{21} = M_{21}I_1$ - $\Phi_{12} \propto I_2 \implies \Phi_{12} = M_{12}I_2$ - It can be shown that $M_{12} = M_{21} = M$. - Unit: Henry (H) - **Induced EMF:** - $\mathcal{E}_2 = -M\frac{dI_1}{dt}$ (EMF in coil 2 due to change in current in coil 1) - $\mathcal{E}_1 = -M\frac{dI_2}{dt}$ (EMF in coil 1 due to change in current in coil 2) - **Coefficient of Coupling (k):** $M = k\sqrt{L_1L_2}$ where $0 \le k \le 1$. - $k=1$ for perfectly coupled coils. ### AC Generator (Dynamo) - **Principle:** Converts mechanical energy into electrical energy based on Faraday's law. - **Working:** A coil rotates in a uniform magnetic field, causing the magnetic flux through it to change, inducing an EMF. - **Formula for induced EMF:** - If a coil of $N$ turns and area $A$ rotates with angular velocity $\omega$ in a uniform magnetic field $B$: - $\Phi_B = NBA\cos(\omega t)$ - $\mathcal{E} = -\frac{d\Phi_B}{dt} = NBA\omega\sin(\omega t)$ - $\mathcal{E} = \mathcal{E}_{max}\sin(\omega t)$ where $\mathcal{E}_{max} = NBA\omega$ ### Eddy Currents - **Definition:** Circulating currents induced in bulk conductors (like metal sheets or cores) when they are subjected to changing magnetic flux. - **Direction:** Follows Lenz's law, opposing the change in magnetic flux. - **Effects:** - **Heating:** Due to $I^2R$ losses, leading to energy dissipation. - **Damping:** Opposes motion (e.g., electromagnetic brakes). - **Applications:** Electromagnetic brakes, induction furnaces, speedometers. - **Minimization:** Laminating the core (using thin insulated sheets stacked together) increases resistance to eddy currents, reducing their magnitude. ### Key Formulas Summary - **Magnetic Flux:** $\Phi_B = BA\cos\theta$ - **Faraday's Law:** $\mathcal{E} = -N\frac{d\Phi_B}{dt}$ - **Motional EMF:** $\mathcal{E} = BLv$ (for perpendicular case) - **Motional EMF (rotating rod):** $\mathcal{E} = \frac{1}{2} B \omega L^2$ - **Self-Inductance:** $\Phi_B = LI$, $\mathcal{E}_L = -L\frac{dI}{dt}$ - **Energy in Inductor:** $U_L = \frac{1}{2}LI^2$ - **Mutual Inductance:** $\Phi_{21} = MI_1$, $\mathcal{E}_2 = -M\frac{dI_1}{dt}$ - **AC Generator EMF:** $\mathcal{E} = NBA\omega\sin(\omega t)$