physics-class-12-unit-06-chapter-04-faradays-law-of-induction-induced-emf-l-4-4-dajijnypcoi

### <Success style="font-size:25px"> Faradays Law Of Induction Induced Emf L-4 </Success>
### <primary style="font-size:24px"> Example </primary>
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- Uniform magnetic field $\vec B$

- $\vec B$ is increasing with time, $\frac {dB}{dt}$ = 0.04 T/s

- Radius of the conducting loop r = 5cm

- Let the resistances of the loop R = 5$\Omega$

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<footer style = "font-size: 18px">Faraday's and Lenz's Law $\rightarrow$ Faraday's and Lenz's Law $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Induced EMF $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Faradays Law Of Induction Induced Emf L-4 </Success>
### <primary style="font-size:24px"> Induced EMF </primary>
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- $\varepsilon =-\frac{d {\Phi}_B}{d t} $

- ${\Phi}_B  = B A = B \pi r^2 $

- $\varepsilon =-\pi r^2 \frac{d B}{d t} $

- $ =-\pi \times 25 \times 10^{-4} \times 0.04 $

- = 0.314 mV

- Induced current $I=\frac{\varepsilon}{R}=\frac{0.314 \times 10^{-3}}{5} \simeq 63 \mu A$

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<footer style = "font-size: 18px">Faraday's and Lenz's Law $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Induced EMF </span> $\rightarrow$ Example $\rightarrow$ Total Flux through Solenoid </footer>

### <Success style="font-size:25px"> Faradays Law Of Induction Induced Emf L-4 </Success>
### <primary style="font-size:24px"> Example </primary>
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- A very long solenoid $S_1$ with $N_1$ turns/unit length.

- Current through $S_1$ = $I_1$

- Magnetic fields within $S_1$ = $\mu_0 N_1 I_1$

- Uniform within the solenoid

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<footer style = "font-size: 18px">Example $\rightarrow$ Induced EMF $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Total Flux through Solenoid $\rightarrow$ Induced EMF </footer>

### <Success style="font-size:25px"> Faradays Law Of Induction Induced Emf L-4 </Success>
### <primary style="font-size:24px"> Induced EMF </primary>
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- $ \varepsilon =-\frac{d \phi_ B}{d t} =-\mu_ 0 N_ 1 N_ {2 T} \pi R_ 2^2 \frac{d I_1}{d t}$

- $N_ 1$ = 100 turns/cm, $ I_1 $ = 1 A

- $ N_ {2}$ = 100, $R_2$ = 1 cm

- Current $I_1$ changes from 1A to zero in 10ms

- $\frac{d I_1}{d t} = -\frac{1}{10^{-2}}$ = - 100 A/s

- **Induced EMF**

- $\varepsilon = + 4 \pi \times 10^{-7} \times 10^{4} \times 100 \times \pi \times 10^{-4} \times 100 \simeq + 39.5 mv$

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<footer style = "font-size: 18px">Example $\rightarrow$ Total Flux through Solenoid $\rightarrow$ <span style = "color:blue"> Induced EMF </span> $\rightarrow$ Example $\rightarrow$ Induced EMF </footer>

### <Success style="font-size:25px"> Faradays Law Of Induction Induced Emf L-4 </Success>
### <primary style="font-size:24px"> Example </primary>
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- N : Number of turns per unit length

- Area of solenoid = $\pi r^2$

- very long solenoid B = $\mu_0 N I$

- Flux through the conducting loop = $B \pi r^2$ = $\mu_0 N I \pi r^2$

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<footer style = "font-size: 18px">Total Flux through Solenoid $\rightarrow$ Induced EMF $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Induced EMF $\rightarrow$ EMF </footer>

### <Success style="font-size:25px"> Faradays Law Of Induction Induced Emf L-4 </Success>
### <primary style="font-size:24px"> EMF </primary>
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- For $\frac{d I}{d t}>0$

- $\varepsilon<0 $

- $ \varepsilon=\oint_C \vec{E} \cdot \vec{dl}=-\frac{d}{d t} \int_A \vec{B} \cdot d \vec{A} $

- $ \oint \vec{\varepsilon} \cdot \overrightarrow{d l} = 2 \pi R \varepsilon=-\mu_0 N A \frac{d I}{d t} $

- $ \varepsilon=-\frac{\mu_0 N A}{2 \pi R} \frac{d I}{d t}$

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<footer style = "font-size: 18px">Example $\rightarrow$ Induced EMF $\rightarrow$ <span style = "color:blue"> EMF </span> $\rightarrow$ Example $\rightarrow$ Induced Electric Field </footer>

### <Success style="font-size:25px"> Faradays Law Of Induction Induced Emf L-4 </Success>
### <primary style="font-size:24px"> Example </primary>
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- $\oint \vec{E} \cdot \vec{dl} \neq 0$

- Electrostatic field $\oint \vec{E} \cdot \vec{dl}=0$

- Number of turns $N=1000 / m$ per unit length

- $\frac{d I}{d t}=100 A / s$

- Area of the solenoid $=\pi \times 25 \times 10^{-4} {m}^2$ (radius of solenoid $ =5 \mathrm{~cm}$)

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<footer style = "font-size: 18px">Induced EMF $\rightarrow$ EMF $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Induced Electric Field $\rightarrow$ Potential Difference </footer>

### <Success style="font-size:25px"> Faradays Law Of Induction Induced Emf L-4 </Success>
### <primary style="font-size:24px"> Induced Electric Field </primary>
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- Induced electric field at R = 10cm ?

- $ \varepsilon =-\frac{\mu_0 N A}{2 \pi R} \frac{d I}{dt} $

- $ =\frac{-4 \pi \times 10^{-7} \times 1000 \times \pi \times 25 \times 10^{-4} \times 100}{2 \pi \times 0.1} $

- $ \simeq-1.57 \times 10^{-3} \mathrm{~V} / m z4$

- For $\frac{d I}{d t}>0$, $\varepsilon< 0$

- $\varepsilon=\oint_ C \vec{\varepsilon} \cdot \vec{dl}$ =-$\frac{d}{d t} \int_A \vec{B} \cdot d \vec{A}$

- $\oint \vec{\varepsilon} \cdot \overrightarrow{d l}=2 \pi R \varepsilon=-\mu_0 N A \frac{d I}{d t}$

- $\varepsilon=-\frac{\mu_0 N A}{2 \pi R} \frac{d \underline{I}}{d t}$

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<footer style = "font-size: 18px">EMF $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Induced Electric Field </span> $\rightarrow$ Potential Difference $\rightarrow$ Work Done </footer>