physics-class-12-unit-06-chapter-03-self-inductance-and-energy-in-magnetic-field-electromagnet-l-3-4-gojrnp5sopy

<h3 id="success-stylefont-size25px-self-inductance-and-energy-in-magnetic-field-electromagnet-l-3-success"><Success style="font-size:25px"> Self Inductance And Energy In Magnetic Field Electromagnet L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-mutual-inductance-primary"><primary style="font-size:24px"> Mutual Inductance </primary></h3>
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<p>$ \phi_2 = M_{21} I_1$</p>
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<p><strong>Mutual inductances</strong></p>
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<p>$M_{21} = M_{12}$</p>
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<footer style = "font-size: 18px"> $\rightarrow$ Self Inductance and Energy in Magnetic Field Electromagnet $\rightarrow$ <span style = "color:blue"> Mutual Inductance </span> $\rightarrow$ Self Inductance $\rightarrow$ Toroid </footer>

### <Success style="font-size:25px"> Self Inductance And Energy In Magnetic Field Electromagnet L-3 </Success>
### <primary style="font-size:24px"> Self Inductance </primary>
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- $\phi $ = L I

- L = self inductances

- Back EMF

- Unit for Inductance **HENRY**

- |H| = $\frac {T m^2} {A}$

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<footer style = "font-size: 18px">Self Inductance and Energy in Magnetic Field Electromagnet $\rightarrow$ Mutual Inductance $\rightarrow$ <span style = "color:blue"> Self Inductance </span> $\rightarrow$ Toroid $\rightarrow$ Magnetic Flux </footer>

### <Success style="font-size:25px"> Self Inductance And Energy In Magnetic Field Electromagnet L-3 </Success>
### <primary style="font-size:24px"> Toroid </primary>
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- Mean radius = r

- Area of cross section =  A

- Assume : Magnetic field is uniform.

- $ \oint \vec B . \vec {dl} = \mu_0 I_{enc}$

- $ 2 \pi r B = \mu _0 N_t I$

- B = $\frac {\mu_0 N_t}{2 \pi r} . I$

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<footer style = "font-size: 18px">Mutual Inductance $\rightarrow$ Self Inductance $\rightarrow$ <span style = "color:blue"> Toroid </span> $\rightarrow$ Magnetic Flux $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Self Inductance And Energy In Magnetic Field Electromagnet L-3 </Success>
### <primary style="font-size:24px"> Example </primary>
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- $L=\frac{\mu_0 N_t^2 A}{2 \pi r}$

- $ N_t=200$

- $ A=5 \mathrm{cm}^2=5 \times 10^{-4} \mathrm{m}^2 $

- r = 10 cm = 0.1m

- $L = \frac{4 \pi \times 10^{-7} \times 4 \times 10^4 \times 5 \times 10^{-4}}{2 \pi \times 0.1} $
- $ =40 \times 10^{-6} $ H = 40 $\mu$ H

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<footer style = "font-size: 18px">Toroid $\rightarrow$ Magnetic Flux $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Example $\rightarrow$ Energy in Magnetic Fields </footer>

### <Success style="font-size:25px"> Self Inductance And Energy In Magnetic Field Electromagnet L-3 </Success>
### <primary style="font-size:24px"> Energy in Magnetic Fields </primary>
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- Coil with self inductances: L

- For current changing with time, induced emf

- E = -L $\frac {dI}{dt}$

- E = Work done in moving a unit charge through the circuit

- Work done by the external agent = -E

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<footer style = "font-size: 18px">Example $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Energy in Magnetic Fields </span> $\rightarrow$ Energy in Magnetic Fields $\rightarrow$ Solenoid </footer>

<h3 id="success-stylefont-size25px-self-inductance-and-energy-in-magnetic-field-electromagnet-l-3-success-1"><Success style="font-size:25px"> Self Inductance And Energy In Magnetic Field Electromagnet L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-energy-in-magnetic-fields-primary"><primary style="font-size:24px"> Energy in Magnetic Fields </primary></h3>
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<p>I : represent total charge energy the circuit per unit time.</p>
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<p>Work done per unit time.</p>
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<p>${dw}{dt} = -E I = - L I \frac{dI}{dt}$</p>
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<p>Total work done in increasing the current from 0 to I</p>
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<p>W = $L \int_{0}^{I} . IdI$</p>
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<p>W = $\frac{1}{2} L I^2$</p>
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<footer style = "font-size: 18px">Example $\rightarrow$ Energy in Magnetic Fields $\rightarrow$ <span style = "color:blue"> Energy in Magnetic Fields </span> $\rightarrow$ Solenoid $\rightarrow$ Energy Stores </footer>

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### <primary style="font-size:24px"> Solenoid </primary>
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* Closely bound

* Very long

- Magnetic field is uniform within the solenoid. B = $\mu_0 N I$

- Self Inductances. L =$\mu_0 N^2 \pi r^2 l$

- N : Number of turns per unit length.

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<footer style = "font-size: 18px">Energy in Magnetic Fields $\rightarrow$ Energy in Magnetic Fields $\rightarrow$ <span style = "color:blue"> Solenoid </span> $\rightarrow$ Energy Stores $\rightarrow$ Energy Density </footer>

<h3 id="success-stylefont-size25px-self-inductance-and-energy-in-magnetic-field-electromagnet-l-3-success-2"><Success style="font-size:25px"> Self Inductance And Energy In Magnetic Field Electromagnet L-3 </Success></h3>
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<p><strong>Energy stores</strong> = $\frac{1}{2} L I^2 $</p>
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<p>$ =\frac{1}{2} \mu_0 N^2 \pi r^2 l \cdot I^2 $</p>
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<p>$ =\frac{1}{2} \mu_0 N^2 I^2 \pi r^2 l$</p>
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<p>$ =\frac{1}{2 \mu_0}\left(\mu_0 N I\right)^2\left(\pi r^2 l\right) $</p>
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<p>$ =\frac{1}{2 \mu_0} B^{2} \times volume$</p>
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<footer style = "font-size: 18px">Energy in Magnetic Fields $\rightarrow$ Solenoid $\rightarrow$ <span style = "color:blue"> Energy Stores </span> $\rightarrow$ Energy Density $\rightarrow$ Example </footer>

<h3 id="success-stylefont-size25px-self-inductance-and-energy-in-magnetic-field-electromagnet-l-3-success-3"><Success style="font-size:25px"> Self Inductance And Energy In Magnetic Field Electromagnet L-3 </Success></h3>
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<p><strong>Magnetic fields</strong>, $B=I T$</p>
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<p><strong>Energy density</strong> $=\frac{1}{2 \mu_0} B^2$</p>
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<p>$ =\frac{1}{2 \times 4\pi  \times 10^{-7}} \times 1 $</p>
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<p>$ =\frac{1}{8 \pi} \times 10^7 \mathrm{J} / \mathrm{m}^3$</p>
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<footer style = "font-size: 18px">Energy Stores $\rightarrow$ Energy Density $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Solenoid $\rightarrow$ Example </footer>

<h3 id="success-stylefont-size25px-self-inductance-and-energy-in-magnetic-field-electromagnet-l-3-success-4"><Success style="font-size:25px"> Self Inductance And Energy In Magnetic Field Electromagnet L-3 </Success></h3>
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<p>N  = 1000 turns/m</p>
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<p>I  = 1 A</p>
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<p>B  =$\mu_0 N I $</p>
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<p>=$4 \pi \times 10^{-7} \times 10^3 $ =$4 \pi \times 10^{-4} T$</p>
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<p>$U_B =\frac{1}{2 \mu_0} B^2 = \frac{1}{2 \mu} \mu_0^2 N^2 I^2 $</p>
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<p>$ =\frac{\mu_0 N^2 I^2}{2} = \frac{4 \pi \times 10^{-7} \times 10^6 \times 1}{2} $</p>
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<p>$=2 \pi \times 10^{-1} \mathrm{~J} / \mathrm{m}^3$</p>
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<footer style = "font-size: 18px">Energy Density $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Solenoid </span> $\rightarrow$ Example $\rightarrow$ Energy Density of Magnetic Field </footer>

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- **Non uniform magnetic field**

- $ \oint \vec{B} \cdot \overrightarrow{d L}=\mu_0 I_{enc} $

- $ 2 \pi r \cdot B=\mu_0 I $

- $ B =\frac{\mu_0 I}{2 \pi r}$

-  a < r < b

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<footer style = "font-size: 18px">Example $\rightarrow$ Solenoid $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Energy Density of Magnetic Field $\rightarrow$ Total Magnetic Energy </footer>

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- $U_B =\frac{1}{2 \mu_0} B^2 $

- $ =\frac{1}{2 \mu_0}\left(\frac{\mu_0 I}{2 \pi r}\right)^2 $

- $ =\frac{\mu_0 I^2}{8 \pi^2 r^2}$

- **Energy in a length l**

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<footer style = "font-size: 18px">Solenoid $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Energy Density of Magnetic Field </span> $\rightarrow$ Total Magnetic Energy $\rightarrow$ Total Magnetic Energy </footer>

### <Success style="font-size:25px"> Self Inductance And Energy In Magnetic Field Electromagnet L-3 </Success>
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- Elementary volume = $2 \pi .r .d r .l$

- Total magnetic energy =$\int_a^b U_B 2 \pi. r. d r. l $

- $ =\frac{\mu_0 I^2}{8 \pi^2} 2 \pi \int_a^b \frac{r .d r}{r^2} l $

- $ =\frac{\mu_0 I^2}{4 \pi} l \int_a^b \frac{d r}{r}$

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<footer style = "font-size: 18px">Example $\rightarrow$ Energy Density of Magnetic Field $\rightarrow$ <span style = "color:blue"> Total Magnetic Energy </span> $\rightarrow$ Total Magnetic Energy $\rightarrow$ Coaxial Cable </footer>

### <Success style="font-size:25px"> Self Inductance And Energy In Magnetic Field Electromagnet L-3 </Success>
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- Total magnetic energy =$\frac{\mu_0 l}{4 \pi} \ln \left(\frac{b}{a}\right) I^2 = \frac{1}{2} L I^2$

- **Self Inductance**

- L = $\frac{\mu_0 l}{2 \pi} ln \frac{b}{a}$

- Self inductance per unit length =$\frac{\mu_0}{2 \pi} \ln \left(\frac{b}{a}\right)$

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<footer style = "font-size: 18px">Energy Density of Magnetic Field $\rightarrow$ Total Magnetic Energy $\rightarrow$ <span style = "color:blue"> Total Magnetic Energy </span> $\rightarrow$ Coaxial Cable $\rightarrow$ Inductance </footer>

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### <primary style="font-size:24px"> Inductance </primary>
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- $\Phi_B  =\int \vec{B} \cdot  \vec{dA} $

- $ =\frac{\mu_0 I}{2 \pi} l .\ln (\frac{b}{a}) $ = L I 
 
- L =$\frac{\mu_0 L }{2 \pi} \ln(\frac{b}{a})$

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<footer style = "font-size: 18px">Total Magnetic Energy $\rightarrow$ Coaxial Cable $\rightarrow$ <span style = "color:blue"> Inductance </span> $\rightarrow$ Thank You $\rightarrow$  </footer>