Electromagnetic Induction
Magnetic Flux:
PYQ-2023-AC-Q6, PYQ-2023-Magnetic-Effects-Of-Current-Q17
$$\phi=\int \vec{B} \cdot d \vec{s}$$
Faraday’s Laws Of Electromagnetic Induction:
PYQ-2023-EMI-Q1, PYQ-2023-Gravitation-Q13
$$E=-\frac{d \phi}{d t}$$
Lenz’s Law:
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Conservation of energy principle.
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According to this law, emf will be induced in such a way that it will oppose the cause which has produced it.
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Motional emf.
Induced Emf Due To Rotation:
PYQ-2023-EMI-Q2, PYQ-2023-EMI-Q3, PYQ-2023-EMI-Q4
Emf induced in a conducting rod of length I rotating with angular speed $\omega$ about its one end, in a uniform perpendicular magnetic field $B$ is $1 / 2 B \omega v^{2}$.
EMF Induced in a rotating disc :
Emf between the centre and the edge of disc of radius $r$ rotating in a magnetic field $B=\frac{B \omega r^{2}}{2}$
Fixed loop in a varying magnetic field:
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If magnetic field changes with the rate $\frac{\mathrm{dB}}{\mathrm{dt}}$, electric field is generated whose average tangential value along a circle is given by $E=\frac{r}{2} \frac{d B}{d t}$
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This electric field is non conservative in nature. The lines of force associated with this electric field are closed curves.
Self induction:
PYQ-2023-AC-Q12, PYQ-2023-EMI-Q5
$$\varepsilon=-\frac{\Delta(\mathrm{N} \phi)}{\Delta \mathrm{t}}=-\frac{\Delta(\mathrm{LI})}{\Delta \mathrm{t}}=-\frac{\mathrm{L} \Delta \mathrm{I}}{\Delta \mathrm{t}} $$
The instantaneous emf is given as $$\varepsilon=-\frac{\mathrm{d}(\mathrm{N} \phi)}{\mathrm{dt}}=-\frac{\mathrm{d}(\mathrm{LI})}{\mathrm{dt}}=-\frac{\mathrm{LdI}}{\mathrm{dt}}$$
Self inductance of solenoid $$L=\mu_{0} n^{2} \pi r^{2} \ell.$$
(i) Inductor:
Electrical equivalence of loop
$$V_{A}-L \frac{dl}{dt}=V_{B} $$
Energy stored in an inductor $$ U=\frac{1}{2} \mathrm{LI}^{2}$$
Growth Of Current in Series R-L Circuit:
- If a circuit consists of a cell, an inductor $L$ and a resistor $R$ and a switch $S$ ,connected in series and the switch is closed at $t=0$, the current in the circuit $I$ will increase as
$$I=\frac{\varepsilon}{R}\left(1-e^{\frac{-R t}{L}}\right)$$
- The quantity L/R is called time constant of the circuit and is denoted by $\tau$. The variation of current with time is as shown.
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Final current in the circuit $=\frac{\varepsilon}{R}$, which is independent of $L$.
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After one time constant, current in the circuit = 63% of the final current.
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More time constant in the circuit implies slower rate of change of current.
Decay of current in the circuit containing resistor and inductor:
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Let the initial current in a circuit containing inductor and resistor be $\mathrm{I}_{0}$.
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Current at a time $t$ is given as $I=I_{0} e^{\frac{-R \mathrm{t}}{\mathrm{L}}}$
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Current after one time constant : $\mathrm{I}=\mathrm{I}_{0} \mathrm{e}^{-1}=0.37$ % of initial current.
Mutual Inductance:
Mutual inductance is induction of EMF in a coil (secondary) due to change in current in another coil (primary). If current in primary coil is I, total flux in secondary is proportional to I, i.e. $\mathrm{N} \phi$ (in secondary) $\propto \mathrm{I}$.
$$\mathrm{N} \phi =\mathrm{M} \mathrm{I}$$
The emf generated around the secondary due to the current flowing around the primary is directly proportional to the rate at which that current changes.
Equivalent self inductance :
(i) Series combination :
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$L=L_1+L_2 \quad$ ( neglecting mutual inductance)
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$L=L_{1}+L_{2}+2 M$ (if coils are mutually coupled and they have winding in same direction)
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$L=L_{1}+L_{2}-2 M \quad$ (if coils are mutually coupled and they have winding in opposite direction)
(ii) Parallel Combination :
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$\frac{1}{L}=\frac{1}{L_1}+\frac{1}{L_2} \quad$ ( neglecting mutual inductance)
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For two coils which are mutually coupled it has been found that $M \leq \sqrt{L_{1} L_{2}}$ or $M=k \sqrt{L_{1} L_{2}}$ where $k$ is called coupling constant and its value is less than or equal to 1.
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$\frac{E_{s}}{E_{p}}=\frac{N_{s}}{N_{p}}=\frac{I_{p}}{I_{s}}$, where denotations have their usual meanings. $N_S>N_{P}$
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$ E_{S}>E_{P}$, for step up transformer.
LC Oscillations:
$$\omega^{2}=\frac{1}{\mathrm{LC}}$$