physics-class-12-unit-04-chapter-02-generalization-of-amperes-law-and-its-applications-l-2-7-bhcxkuxznrs

### <Success style="font-size:25px"> Generalization Of Amperes Law And Its Applications L-2 </Success>
### <primary style="font-size:24px"> Magnetic Field due to Wire of Infinite Length </primary>
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- $\vec{B}=-\frac{\mu_0 I}{2 \pi x} \hat{k}$

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

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<footer style = "font-size: 18px"> $\rightarrow$ Generalization of Amperes Law and its Application $\rightarrow$ <span style = "color:blue"> Magnetic Field due to Wire of Infinite Length </span> $\rightarrow$ Magnetic Field Enclosed Loop $\rightarrow$ Ampere's Law </footer>

### <Success style="font-size:25px"> Generalization Of Amperes Law And Its Applications L-2 </Success>
### <primary style="font-size:24px"> Magnetic Field Enclosed Loop </primary>
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- $\oint \vec{B} . d\vec{A} = 0$

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<footer style = "font-size: 18px">Generalization of Amperes Law and its Application $\rightarrow$ Magnetic Field due to Wire of Infinite Length $\rightarrow$ <span style = "color:blue"> Magnetic Field Enclosed Loop </span> $\rightarrow$ Ampere's Law $\rightarrow$ Magnetic Field Subtending an Angle </footer>

### <Success style="font-size:25px"> Generalization Of Amperes Law And Its Applications L-2 </Success>
### <primary style="font-size:24px"> Ampere's Law </primary>
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- $\oint \vec{B} . \vec{dl} = {\mu}_{0}I$

- $|\vec{dl}| =r d \phi$

- $\vec{B} . \vec{d l}  =B d l=B r d \phi$

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

- $\oint \vec{B}.\vec{dl} = \frac{\mu_0 I}{2 \pi} \oint d \phi = \mu_0 I$

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<footer style = "font-size: 18px">Magnetic Field due to Wire of Infinite Length $\rightarrow$ Magnetic Field Enclosed Loop $\rightarrow$ <span style = "color:blue"> Ampere's Law </span> $\rightarrow$ Magnetic Field Subtending an Angle $\rightarrow$ Magnetic Field in the Closed Path </footer>

### <Success style="font-size:25px"> Generalization Of Amperes Law And Its Applications L-2 </Success>
### <primary style="font-size:24px"> Magnetic Field Subtending an Angle </primary>
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- $\vec{B} . \vec{dl} = \tau d l \cos \theta$

- $ d l \cos \theta = r d \phi$

- $\vec{B} . \vec{dl} = B r d \phi$

- $ =\frac{\mu_0 I}{2 \pi r} . r d \phi = \frac{\mu_0 I}{2 \pi} d \phi$

- $\oint \vec{B} . \vec{dl} = \frac{\mu_0 I}{2 \pi} \oint d \phi=\mu_0 I$

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<footer style = "font-size: 18px">Magnetic Field Enclosed Loop $\rightarrow$ Ampere's Law $\rightarrow$ <span style = "color:blue"> Magnetic Field Subtending an Angle </span> $\rightarrow$ Magnetic Field in the Closed Path $\rightarrow$ Magnetic Field in the Closed Path </footer>

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### <primary style="font-size:24px"> Magnetic Field in the Closed Path </primary>
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- $\oint_{C_{2}} \vec{B} . \vec{dl} = {-\mu_0 I}$

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

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<footer style = "font-size: 18px">Ampere's Law $\rightarrow$ Magnetic Field Subtending an Angle $\rightarrow$ <span style = "color:blue"> Magnetic Field in the Closed Path </span> $\rightarrow$ Magnetic Field in the Closed Path $\rightarrow$ Magnetic Field Subtending Angle </footer>

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### <primary style="font-size:24px"> Magnetic Field in the Closed Path </primary>
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- $\oint \vec{B} . \vec{dl} = \oint (\vec{B_1} + \vec{B_2}) . \vec{dl}$

- $= \oint \vec{B_1} . \vec{dl} + \oint \vec{B_2} . \vec{dl})$

- $= {\mu I_1} + {\mu I_2}$

- $= {\mu_0} (I_1 + I_2)$

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<footer style = "font-size: 18px">Magnetic Field Subtending an Angle $\rightarrow$ Magnetic Field in the Closed Path $\rightarrow$ <span style = "color:blue"> Magnetic Field in the Closed Path </span> $\rightarrow$ Magnetic Field Subtending Angle $\rightarrow$ Ampere's Law </footer>

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- $= \oint \vec{B} . \vec{dl} = \oint B r d \phi$

- $= \oint \frac{\mu_0 I}{2 \pi r} . r d \phi$

- $ = \frac{\mu_0 I}{2 \pi} \int_{\phi_1}^{\phi_2} d \phi + \int_{\phi_2}^{\phi_1} dl$

- $= \frac{\mu_0 I}{2 \pi} (\phi_2 - \phi_1 + \phi_1 - \phi_2) = 0$

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<footer style = "font-size: 18px">Magnetic Field in the Closed Path $\rightarrow$ Magnetic Field in the Closed Path $\rightarrow$ <span style = "color:blue"> Magnetic Field Subtending Angle </span> $\rightarrow$ Ampere's Law $\rightarrow$ Problem </footer>

### <Success style="font-size:25px"> Generalization Of Amperes Law And Its Applications L-2 </Success>
### <primary style="font-size:24px"> Ampere's Law </primary>
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- $ \oint \vec{B} . \vec{dl} = {\mu_0} I_{enc}$

- $ \oint \vec{B} . \vec{dl} = {\mu}(I_1 - I_2 - I_3)$

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<footer style = "font-size: 18px">Magnetic Field in the Closed Path $\rightarrow$ Magnetic Field Subtending Angle $\rightarrow$ <span style = "color:blue"> Ampere's Law </span> $\rightarrow$ Problem $\rightarrow$ Example </footer>

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### <primary style="font-size:24px"> Problem </primary>
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- Find the values of $\oint \vec{B} . \vec{dl}$ for paths, $c_1$ and $c_2$.
 
- Draw paths for which $\oint \vec{B} . \vec{dl}$ is maximum a positive and maximum a negative.
 
- Draw another path having a same value of $\oint \vec{B} . \vec{dl}$ as over $c_1$.

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<footer style = "font-size: 18px">Magnetic Field Subtending Angle $\rightarrow$ Ampere's Law $\rightarrow$ <span style = "color:blue"> Problem </span> $\rightarrow$ Example $\rightarrow$ Magnetic Field due to Infinite Wire </footer>

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### <primary style="font-size:24px"> Example </primary>
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- Infinitely long and straight current carrying conductor.

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

- $\vec{B} \|| \vec{dl}$

- $\vec{B} . \vec{dl} = B dl$

- $dl = r d \phi$

- $\vec{B} . \vec{dl} = B r d \phi$

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<footer style = "font-size: 18px">Ampere's Law $\rightarrow$ Problem $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Magnetic Field due to Infinite Wire $\rightarrow$ Example </footer>

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### <primary style="font-size:24px"> Example </primary>
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- Current distributed uniformally over the cross section of a cylindrical wire of circular cross section and infinitely long.

- $\vec{B}(r)$

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<footer style = "font-size: 18px">Example $\rightarrow$ Magnetic Field due to Infinite Wire $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Magnetic Field due to Enclosed Current $\rightarrow$ Magnetic Field due to Enclosed Current </footer>

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### <primary style="font-size:24px"> Magnetic Field due to Enclosed Current </primary>
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- $\oint \vec{B} . \vec{dl} = \mu_0 I_{enc}$

- I passing the length through an area $\pi R^2$

- Current density = $\frac{I}{\pi R^2}$

- Current enclosed by the path $c_1$

- = current density $\times$ area of path $c_1$

- = $\frac{I}{\pi R^2} \times \pi r^2$

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<footer style = "font-size: 18px">Magnetic Field due to Infinite Wire $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Magnetic Field due to Enclosed Current </span> $\rightarrow$ Magnetic Field due to Enclosed Current $\rightarrow$ Outside the Conductor </footer>

- $I_{enc} = \frac{Ir^2}{R^2}$

- $\oint \vec{B} . {dl} = \oint B r d \phi$

- $= B r \oint d \phi = B r . 2 \pi$

- $B . r . 2 \pi =\mu_0 I_{enc} = \frac{\mu_0 I r^2}{R^2}$

- $B = \frac{\mu_0 I r^2}{R^2} \times \frac{1}{2 \pi r} = \frac{\mu_0 I r}{2 \pi R^2}$

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![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-12-unit-04-chapter-02-generalization-of-amperes-law-and-its-applications-l-2_7-bhcxkuxznrs-12.jpg)

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<footer style = "font-size: 18px">Example $\rightarrow$ Magnetic Field due to Enclosed Current $\rightarrow$ <span style = "color:blue"> Magnetic Field due to Enclosed Current </span> $\rightarrow$ Outside the Conductor $\rightarrow$ Magnetic Field due to Conductor </footer>

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- $\oint \vec{B} . \vec{dl}  =\oint B . r d \phi$

- $ =B r \oint d \phi  = 2 \pi B r$

- $I_{enc} = I$

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

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

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<footer style = "font-size: 18px">Magnetic Field due to Enclosed Current $\rightarrow$ Magnetic Field due to Enclosed Current $\rightarrow$ <span style = "color:blue"> Outside the Conductor </span> $\rightarrow$ Magnetic Field due to Conductor $\rightarrow$ Conical Conductor </footer>

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- $ B =\frac{\mu_0 I}{2 \pi R^2} r =\frac{\mu_1 I}{2 \pi r}$

- $B(R) = \frac{\mu_0 I}{2 \pi R}$

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<footer style = "font-size: 18px">Magnetic Field due to Enclosed Current $\rightarrow$ Outside the Conductor $\rightarrow$ <span style = "color:blue"> Magnetic Field due to Conductor </span> $\rightarrow$ Conical Conductor $\rightarrow$ Magnetic Field due to Circular Current Loop </footer>

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### <primary style="font-size:24px"> Magnetic Field due to Circular Current Loop </primary>
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- $\oint \vec{B} . \vec{dl} = \mu_0 I$

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

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

- a < r < b

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<footer style = "font-size: 18px">Magnetic Field due to Conductor $\rightarrow$ Conical Conductor $\rightarrow$ <span style = "color:blue"> Magnetic Field due to Circular Current Loop </span> $\rightarrow$ Magnetic Field due to Coil $\rightarrow$ Magnetic Field due to Coil </footer>

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### <primary style="font-size:24px"> Magnetic Field due to Coil </primary>
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- $\oint \vec{B} . \vec{dl} = \mu_0 I_{enc}$

- $ \oint \vec{B} . \vec{dA}= 0$

- $\vec{B} . \vec{dA} = B_r . d A$

- $ B_r \int_{\sqrt{3}}^{d A} = 0$

- $B_r . 2 \pi r L=0$

- $\rightarrow B_r=0$

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<footer style = "font-size: 18px">Conical Conductor $\rightarrow$ Magnetic Field due to Circular Current Loop $\rightarrow$ <span style = "color:blue"> Magnetic Field due to Coil </span> $\rightarrow$ Magnetic Field due to Coil $\rightarrow$ Magnetic Field due to Solenoid </footer>

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

- $ B_\phi . 2 \pi r=0$

- $\Rightarrow B_{\phi} = 0$

- $\vec{B}$ has only $B_2$ component

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<footer style = "font-size: 18px">Magnetic Field due to Circular Current Loop $\rightarrow$ Magnetic Field due to Coil $\rightarrow$ <span style = "color:blue"> Magnetic Field due to Coil </span> $\rightarrow$ Magnetic Field due to Solenoid $\rightarrow$ Thank You </footer>

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- $\vec{B} = B \hat{k}$

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

- $\int_a^b \vec{B} . \vec{dl} + \int_b^c \vec{B} . \vec{dl} $

- $+ \int_c^d \vec{B} . \vec{dl} + \int_d^a \vec{B} . \vec{dl}=0$

- $B(r_1)l - B(r_2)l = 0 $

- $\rightarrow B(r_1) = B(r_2)$

- B outside solenoid = 0

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<footer style = "font-size: 18px">Magnetic Field due to Coil $\rightarrow$ Magnetic Field due to Coil $\rightarrow$ <span style = "color:blue"> Magnetic Field due to Solenoid </span> $\rightarrow$ Thank You $\rightarrow$  </footer>