physics-class-12-unit-04-chapter-03-more-applications-of-amperes-law-l-3-7-girkt6sx2sk

### <Success style="font-size:25px"> More Applications Of Amperes Law L-3 </Success>
### <primary style="font-size:24px"> Magnetic Field on a Loop </primary>
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- $\oint \vec{B} . \vec{dl} = \mu I_{enc} = 0$

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

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

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<footer style = "font-size: 18px">Solenoid $\rightarrow$ Magnetic Field due to Solenoid $\rightarrow$ <span style = "color:blue"> Magnetic Field on a Loop </span> $\rightarrow$ Magnetic Field on a Loop $\rightarrow$ Magnetic Field due to Number N Turns </footer>

### <Success style="font-size:25px"> More Applications Of Amperes Law L-3 </Success>
### <primary style="font-size:24px"> Magnetic Field on a Loop </primary>
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- $\int_a^b \vec{B} . \vec{dl} + \int_c^d \vec{B} . \vec{dl} = 0$

- $B(r_1) \int_a^b dl + B(r_2) \int_c^d dl = 0$

- $B(r_1) \int_a^b dl = B(r_2) \int_d^c dl = 0$

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

- For $r_2 \rightarrow \infin $

- $B(r_2) \rightarrow 0$

- B = 0 to point outside the solenoid.

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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-03-more-applications-of-amperes-law-l-3_7-girkt6sx2sk-136-0762.0.jpg)

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<footer style = "font-size: 18px">Magnetic Field due to Solenoid $\rightarrow$ Magnetic Field on a Loop $\rightarrow$ <span style = "color:blue"> Magnetic Field on a Loop </span> $\rightarrow$ Magnetic Field due to Number N Turns $\rightarrow$ Magnetic Field due to Number N Turns </footer>

### <Success style="font-size:25px"> More Applications Of Amperes Law L-3 </Success>
### <primary style="font-size:24px"> Magnetic Field due to Number N Turns </primary>
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- $\oint \vec{B} . \vec{dl} = \mu_a I_{enc}$

- Number of turns per unit length : N

- Number of turns : Nl

- Total current enclosed by the loop

- = N l I

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<footer style = "font-size: 18px">Magnetic Field on a Loop $\rightarrow$ Magnetic Field on a Loop $\rightarrow$ <span style = "color:blue"> Magnetic Field due to Number N Turns </span> $\rightarrow$ Magnetic Field due to Number N Turns $\rightarrow$ Magnetic Field along Axis </footer>

### <Success style="font-size:25px"> More Applications Of Amperes Law L-3 </Success>
### <primary style="font-size:24px"> Magnetic Field due to Number N Turns </primary>
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- $ \oint \vec{B} . \vec{dl} = \mu_0 N I l$

- $B \int_a^b dl = \mu_0 NIl$

- $Bl = \mu_0 NIl$

- $\rightarrow B = \mu _0 NI$

- $\vec{B} = \mu_0 NI \hat{k}$

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<footer style = "font-size: 18px">Magnetic Field on a Loop $\rightarrow$ Magnetic Field due to Number N Turns $\rightarrow$ <span style = "color:blue"> Magnetic Field due to Number N Turns </span> $\rightarrow$ Magnetic Field along Axis $\rightarrow$ Magnetic Field along Axis </footer>

### <Success style="font-size:25px"> More Applications Of Amperes Law L-3 </Success>
### <primary style="font-size:24px"> Magnetic Field along Axis </primary>
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- N = Number of turns per unit length

- $\vec{B}$ along axis

- $ \vec{B} = \frac{\mu_0 I N}{2} \frac{R^2}{(R^2+2^2)^{3 / 2}} \hat{k}$

- Number of turns = Ndz

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<footer style = "font-size: 18px">Magnetic Field due to Number N Turns $\rightarrow$ Magnetic Field due to Number N Turns $\rightarrow$ <span style = "color:blue"> Magnetic Field along Axis </span> $\rightarrow$ Magnetic Field along Axis $\rightarrow$ Magnetic Field along Axis </footer>

### <Success style="font-size:25px"> More Applications Of Amperes Law L-3 </Success>
### <primary style="font-size:24px"> Magnetic Field along Axis </primary>
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- $ d \vec{B} = \frac{\mu_0 I N d z}{2} \frac{a^2}{(a^2 + z^2)^{3 / 2}} \hat{k}$

- $\vec{B} = \frac{\mu_0 N I}{2} a^2 \int_0^L \frac{dz}{(a^2+z^2)^{3 / 2}} \hat{k}$

- $ Z = a \tan \theta $

- $ dz = a \sec ^2 \theta d \theta $

- $ (a^2 + z^2) = a^2 \sec ^2 \theta$

- $\int \frac{d z}{(a^2 + z^2)^{3 / 2}} = \int \frac{a \sec ^2 \theta d \theta}{a^3 \sec ^3 \theta}$

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<footer style = "font-size: 18px">Magnetic Field due to Number N Turns $\rightarrow$ Magnetic Field along Axis $\rightarrow$ <span style = "color:blue"> Magnetic Field along Axis </span> $\rightarrow$ Magnetic Field along Axis $\rightarrow$ Magnetic Field along Axis </footer>

### <Success style="font-size:25px"> More Applications Of Amperes Law L-3 </Success>
### <primary style="font-size:24px"> Magnetic Field along Axis </primary>
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- $ = \frac{1}{a^2} \int \cos \theta d \theta$

- $ = \frac{1}{a^2} \sin \theta |_0 ^{\tan ^{-1} {L / a}}$

- $ = \frac{1}{a^2} \sin [\tan ^{-1}(\frac{L}{a})]$

- $ \vec{B} = \frac{\mu_0 N I}{2} a^2 . \frac{1}{a^2} \sin [\tan ^{-1}(\frac{L}{a})] \hat{a}$

- $ = \frac{\mu_0 N I}{2} \sin [\tan ^{-1}(\frac{L}{a})] \hat{k}$

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<footer style = "font-size: 18px">Magnetic Field along Axis $\rightarrow$ Magnetic Field along Axis $\rightarrow$ <span style = "color:blue"> Magnetic Field along Axis </span> $\rightarrow$ Magnetic Field along Axis $\rightarrow$ Magnetic Field along Axis </footer>

### <Success style="font-size:25px"> More Applications Of Amperes Law L-3 </Success>
### <primary style="font-size:24px"> Toroid </primary>
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- **Path 1:**

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

- $B.2 \pi r_1 = 0 $

- $\rightarrow B = 0$

- **Path 2:**

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

- $ B.2 \pi r_2 = 0$

- B = 0

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<footer style = "font-size: 18px">Magnetic Field along Axis $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Toroid </span> $\rightarrow$ Total Number of Turns in the Torroid $\rightarrow$ Motion of Charged Particles in a Magnetic Fields </footer>

### <Success style="font-size:25px"> More Applications Of Amperes Law L-3 </Success>
### <primary style="font-size:24px"> Motion of Charged Particles in a Magnetic Fields </primary>
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- $ \vec{F}_B = q(\vec{v} \times \vec{B})$

- $ \vec{F} = q\vec{E} + q(\vec{v}\times \vec{B})$

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<footer style = "font-size: 18px">Toroid $\rightarrow$ Total Number of Turns in the Torroid $\rightarrow$ <span style = "color:blue"> Motion of Charged Particles in a Magnetic Fields </span> $\rightarrow$ Centripetal Force $\rightarrow$ Angular Velocity </footer>

### <Success style="font-size:25px"> More Applications Of Amperes Law L-3 </Success>
### <primary style="font-size:24px"> Centripetal Force </primary>
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- Particles will executed a circular motion :-

- $F = |q| v B$

- Centripetal Force $= \frac{mv^2}{R}$

- $ \frac{mv^2}{R} = |q| vB$

- $R =  \frac{mv}{|q|B}$

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<footer style = "font-size: 18px">Total Number of Turns in the Torroid $\rightarrow$ Motion of Charged Particles in a Magnetic Fields $\rightarrow$ <span style = "color:blue"> Centripetal Force </span> $\rightarrow$ Angular Velocity $\rightarrow$ Thomson's charge to mass ratio Experiment </footer>

### <Success style="font-size:25px"> More Applications Of Amperes Law L-3 </Success>
### <primary style="font-size:24px"> Angular Velocity </primary>
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- $w = \frac{v}{R} = \frac{|q|B}{m}$

- Number of revolution per unit time

- $f = \frac{w}{2 \pi} = \frac{|q|B}{2 \pi m}$

- CYCLOTRON FREQUENCY

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<footer style = "font-size: 18px">Motion of Charged Particles in a Magnetic Fields $\rightarrow$ Centripetal Force $\rightarrow$ <span style = "color:blue"> Angular Velocity </span> $\rightarrow$ Thomson's charge to mass ratio Experiment $\rightarrow$ Problem </footer>