Magnetic Flux:

$$\phi=\int \vec{B} \cdot d \vec{s}$$

Faraday’s Laws Of Electromagnetic Induction:

$$E=-\frac{d \phi}{d t}$$

Lenz’s Law:

• Conservation of energy principle.

• According to this law, emf will be induced in such a way that it will oppose the cause which has produced it.

• Motional emf.

Induced Emf Due To Rotation:

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:

• 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 $E=\frac{r}{2} \frac{d B}{d t}$

• This electric field is non conservative in nature. The lines of force associated with this electric field are closed curves.

Self induction:

$$\varepsilon=-\frac{\Delta(\mathrm{N} \phi)}{\Delta \mathrm{t}}=-\frac{\Delta(\mathrm{LI})}{\Delta \mathrm{t}}=-\frac{\mathrm{L} \Delta \mathrm{I}}{\Delta \mathrm{t}} $$

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.

• Final current in the circuit $=\frac{\varepsilon}{R}$, which is independent of $L$.

• After one time constant, current in the circuit = 63% of the final current.

• More time constant in the circuit implies slower rate of change of current.

Decay of current in the circuit containing resistor and inductor:

• Let the initial current in a circuit containing inductor and resistor be $\mathrm{I}_{0}$.

• Current at a time $t$ is given as $I=I_{0} e^{\frac{-R \mathrm{t}}{\mathrm{L}}}$

• 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 :

• $L=L_1+L_2 \quad$ ( neglecting mutual inductance)

• $L=L_{1}+L_{2}+2 M$ (if coils are mutually coupled and they have winding in same direction)

• $L=L_{1}+L_{2}-2 M \quad$ (if coils are mutually coupled and they have winding in opposite direction)

(ii) Parallel Combination :

• $\frac{1}{L}=\frac{1}{L_1}+\frac{1}{L_2} \quad$ ( neglecting mutual inductance)

• 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.

• $\frac{E_{s}}{E_{p}}=\frac{N_{s}}{N_{p}}=\frac{I_{p}}{I_{s}}$, where denotations have their usual meanings. $N_S>N_{P}$

• $ E_{S}>E_{P}$, for step up transformer.

LC Oscillations:

$$\omega^{2}=\frac{1}{\mathrm{LC}}$$