physics-class-12-unit-04-chapter-04-magnetic-field-fora-straight-conductor-and-amperes-law-l-4-7-pz4qlp7drig

### <Success style="font-size:25px"> Magnetic Field Fora Straight Conductor And Amperes Law L-4 </Success>
### <primary style="font-size:24px"> Biot - Savart Law </primary>
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- $d\vec{B}=\frac{\mu_0}{4\pi} I\frac{\vec{dl}\times\vec{r}}{r^3}$

- $\mu_0:$ Permeability of free space

- $= 4 \pi\times 10^{-7}$ T.m/A

- $\epsilon_0 \mu_0=\frac{1}{c^2}$

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<footer style = "font-size: 18px">Magnetic Force $\rightarrow$ Magnetic Force $\rightarrow$ <span style = "color:blue"> Biot - Savart Law </span> $\rightarrow$ Magnetic Due to a Loop of Current $\rightarrow$ Magnetic Due to a Loop of Current </footer>

### <Success style="font-size:25px"> Magnetic Field Fora Straight Conductor And Amperes Law L-4 </Success>
### <primary style="font-size:24px"> Magnetic Due to a Loop of Current </primary>
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- $\vec{B}$ due to a circular loop of current

- $\vec{B}$ along the axis

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<footer style = "font-size: 18px">Magnetic Force $\rightarrow$ Biot - Savart Law $\rightarrow$ <span style = "color:blue"> Magnetic Due to a Loop of Current </span> $\rightarrow$ Magnetic Due to a Loop of Current $\rightarrow$ Magnetic Due to a Loop of Current </footer>

- $\vec{d l}=\hat{j} d l$

- $ \vec{r}=-R \hat{i}+z \hat{k}$

- $\vec{dl} \times \vec{r}=\hat{j} d l \times  (-R \hat{i} +z \hat{k})$

- $=Rdl\hat{k}+\hat{i}zdl$

- $d \vec{B}=\mu_0  \frac{\vec{d l} \times \vec{r}}{r^3}$

- $d\vec{B}_1=\frac{\mu_0}{4\pi}I\frac{(Rdl\hat{k}+zdl\hat{i})}{r^3}$

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<footer style = "font-size: 18px">Biot - Savart Law $\rightarrow$ Magnetic Due to a Loop of Current $\rightarrow$ <span style = "color:blue"> Magnetic Due to a Loop of Current </span> $\rightarrow$ Magnetic Due to a Loop of Current $\rightarrow$ Magnetic Due to a Loop of Current </footer>

- $d\hat{B}=\frac{\mu_0}{4\pi}I\frac{\vec{dl}\times\vec{r}}{r^3}$

- $\vec{dl}=-\hat{j}dl$

- $\vec{r}=\hat{k}z+R\hat{i}$

- $\vec{dl}\times\vec{r}=-\hat{j}dl\times(\hat{k}z+\hat{i}R)$

- $=-\hat{i}dlz+\hat{k}Rdl$

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<footer style = "font-size: 18px">Magnetic Due to a Loop of Current $\rightarrow$ Magnetic Due to a Loop of Current $\rightarrow$ <span style = "color:blue"> Magnetic Due to a Loop of Current </span> $\rightarrow$ Magnetic Due to a Loop of Current $\rightarrow$ Magnetic Due to a Loop of Current </footer>

- $ d \vec{B}_2=\frac{\mu_0}{4\pi}I\frac{(-\hat{i}zdl+\hat{k}Rdl)}{r^3}$

- $d\vec{B_1}=\frac{\mu_0}{4\pi}I\frac{(zdl\hat{i}+Rdl\hat{k})}{r^3}$

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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-04-magnetic-field-fora-straight-conductor-and-amperes-law-l-4_7-pz4qlp7drig-05.jpg)

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- $\vec{d l}=\hat{j} d l$

- $ \vec{r}=-R \hat{i}+z \hat{k}$

- $\vec{dl} \times \vec{r}=\hat{j} d l \times  (-R \hat{i} +z \hat{k})$

- $=Rdl\hat{k}+\hat{i}zdl$

- $d \vec{B}=\mu_0  \frac{\vec{d l} \times \vec{r}}{r^3}$

- $d\vec{B}_1=\frac{\mu_0}{4\pi}I\frac{(Rdl\hat{k}+zdl\hat{i})}{r^3}$

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- $ d \vec{B}_2=\frac{\mu_0}{4\pi}I\frac{(-\hat{i}zdl+\hat{k}Rdl)}{r^3}$

- $d\vec{B_1}=\frac{\mu_0}{4\pi}I\frac{(zdl\hat{i}+Rdl\hat{k})}{r^3}$

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- $d\vec{B}=d\vec{B}_1+d\vec{B}_2 = \frac{\mu_0}{4\pi}I\frac{2Rdl}{r^3}\hat{k}$

- $\vec{B} = \frac{\mu_0I}{4 \pi} \frac{2 R}{(R^2+z^2)^{3/2}}$$\hat{k}\int_{semi Circle} dl$

- $=\frac{\mu_0 I 2 R}{4 \pi(R^2+z^2)^{3 / 2}} \pi R \hat{k}$

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

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- $\vec{B}_{..}=\frac{\mu_0 I R^2}{2 R^3}\hat{k}=\frac{\mu_0 I}{2 R} \hat{k}$

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- $\hat{B}=\frac{\mu_0IR^2\hat{k}}{2(R^2+z^2)^{3/2}}$ z >> R

- $\vec{B}=\frac{\mu_0IR^2\hat{R}}{2z^3}$

- $=\frac{\mu_0 I \pi R^2 \hat{k}}{2 \pi z^3}$ $\vec{A}=\pi R^2 \hat{k}$

- $= \frac{\mu_0 I \vec{A}}{2 \pi z^3}$

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<footer style = "font-size: 18px">Magnetic Due to a Loop of Current $\rightarrow$ Magnetic Due to a Loop of Current $\rightarrow$ <span style = "color:blue"> Magnetic Due to a Loop of Current </span> $\rightarrow$ Electric Dipole Moment $\rightarrow$ Electric Dipole </footer>

### <Success style="font-size:25px"> Magnetic Field Fora Straight Conductor And Amperes Law L-4 </Success>
### <primary style="font-size:24px"> Electric Dipole Moment </primary>
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- $\vec{p} = q d \hat{k}$

- **Magnetic dipole moment**

- $\vec{m}=I\vec{A}$

- $\vec{B}=\frac{\mu_0 \vec{m}}{2 \pi z^3}$ $z>>R$

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<footer style = "font-size: 18px">Magnetic Due to a Loop of Current $\rightarrow$ Magnetic Due to a Loop of Current $\rightarrow$ <span style = "color:blue"> Electric Dipole Moment </span> $\rightarrow$ Electric Dipole $\rightarrow$ Electric Dipole </footer>

### <Success style="font-size:25px"> Magnetic Field Fora Straight Conductor And Amperes Law L-4 </Success>
### <primary style="font-size:24px"> Electric Dipole </primary>
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- $\vec{E} = \frac{\vec{p}}{2 \pi \epsilon_0 z^3}$$ z >> a$

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<footer style = "font-size: 18px">Magnetic Due to a Loop of Current $\rightarrow$ Electric Dipole Moment $\rightarrow$ <span style = "color:blue"> Electric Dipole </span> $\rightarrow$ Electric Dipole $\rightarrow$ Infinitely Long Straight Current Carrying Conductor </footer>

### <Success style="font-size:25px"> Magnetic Field Fora Straight Conductor And Amperes Law L-4 </Success>
### <primary style="font-size:24px"> Electric Dipole </primary>
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- Electric dipole

- Magnetic dipole
 
- $\oint \vec{B}.d\vec{A}=0$

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<footer style = "font-size: 18px">Electric Dipole Moment $\rightarrow$ Electric Dipole $\rightarrow$ <span style = "color:blue"> Electric Dipole </span> $\rightarrow$ Infinitely Long Straight Current Carrying Conductor $\rightarrow$ Infinitely Long Straight Current Carrying Conductor </footer>

### <Success style="font-size:25px"> Magnetic Field Fora Straight Conductor And Amperes Law L-4 </Success>
### <primary style="font-size:24px"> Infinitely Long Straight Current Carrying Conductor </primary>
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- $d\vec{B}=\frac{\mu_0}{4 \pi}I\frac{\vec dl \times \vec{r}}{r^3}$

- $\vec dl=dl \hat{j}$

- $\vec{r} = x\vec{i}-y\vec{j}$

- $\vec dl \times \vec r=dl\hat{j}\times (x \hat{i}-y \hat{j})$

- $= -xdl\hat{k}=-xdy\hat{k}$

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<footer style = "font-size: 18px">Electric Dipole $\rightarrow$ Electric Dipole $\rightarrow$ <span style = "color:blue"> Infinitely Long Straight Current Carrying Conductor </span> $\rightarrow$ Infinitely Long Straight Current Carrying Conductor $\rightarrow$ Infinitely Long Straight Current Carrying Conductor </footer>

- $d \vec B=\frac{\mu_0}{4 \pi} I \frac{\vec{dl} \times \vec{r}}{r^3}$

- $ \frac{-\mu_0 I}{4 \pi}I\frac{(-xdy)\hat{k}}{(x^2+y^2)^{(3/2)}}$

- $\vec{B}=\frac{-\mu_0 I}{4 \pi}x \int_{y_1}^{y_2} \frac{dy}{(x^2+y^2)^{3/2}}\hat{k}$

- $y = x \tan \phi$

- $dy=x \sec^2 \phi d \phi$

- $(x^2+y^2)=x^2(1+\tan^2 \phi)=x^2 \sec^2 \phi$

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<footer style = "font-size: 18px">Electric Dipole $\rightarrow$ Infinitely Long Straight Current Carrying Conductor $\rightarrow$ <span style = "color:blue"> Infinitely Long Straight Current Carrying Conductor </span> $\rightarrow$ Infinitely Long Straight Current Carrying Conductor $\rightarrow$ Infinitely Long Straight Current Carrying Conductor </footer>

- $\vec{B}=-\frac{\mu_0 I}{4 \pi}x \int\frac{x \sec^2 \phi d \phi}{x^3 \sec^3 \phi}$

- $=-\frac{\mu_0 I}{4 \pi x} \int_{\phi_1}^{\phi_2}\cos \phi d \phi \hat{k}$

- $=-\frac{\mu_0 I}{4 \pi x}(\sin \phi_2 - \sin \phi_1)\hat{k}$

- $\sin \phi =\frac {y}{(x^2+y^3)^{1/2}}$

- $\sin \phi_1=\frac{y_1}{(x^2+{y_1}^2){1/2}}$

- $\sin \phi_2=\frac{y_2}{(x^2+{y_2}^2){1/2}}$

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<footer style = "font-size: 18px">Infinitely Long Straight Current Carrying Conductor $\rightarrow$ Infinitely Long Straight Current Carrying Conductor $\rightarrow$ <span style = "color:blue"> Infinitely Long Straight Current Carrying Conductor </span> $\rightarrow$ Infinitely Long Straight Current Carrying Conductor $\rightarrow$ Infinitely Long Straight Current Carrying Conductor </footer>

- $\vec{B}=-\frac{\mu_0 I}{4 \pi x} (\frac{y_2}{\sqrt{x^2+{y_2}^2}}-\frac{y_1}{\sqrt{x^2+{y_1}^2}})\hat{k}$

- $\vec{B}=\frac{-\mu_0I}{2 \pi x}\hat{k}$

- r : Perpendicular distance from the conductor

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- $\vec{|B|} = \frac{\mu_0I}{2 \pi r}$

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- $\oint \vec{E}.d\vec{A}=\frac{Q_{enc}}{\epsilon_0}$

- $\oint \vec{B}.d\vec{A}=0$

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<footer style = "font-size: 18px">Infinitely Long Straight Current Carrying Conductor $\rightarrow$ Infinitely Long Straight Current Carrying Conductor $\rightarrow$ <span style = "color:blue"> Infinitely Long Straight Current Carrying Conductor </span> $\rightarrow$ Magnetic Field of Earth $\rightarrow$ Ampere's Law </footer>

### <Success style="font-size:25px"> Magnetic Field Fora Straight Conductor And Amperes Law L-4 </Success>
### <primary style="font-size:24px"> Magnetic Field of Earth </primary>
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- $B=\frac{\mu_0I}{2 \pi r}$

- $=\frac{4 \pi \times 10^{-7}\times 5}{2 \pi \times 0.1} = 10^{-5}T$

- $B(earth)\sim 3 \times 10^{-5}T$

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<footer style = "font-size: 18px">Infinitely Long Straight Current Carrying Conductor $\rightarrow$ Infinitely Long Straight Current Carrying Conductor $\rightarrow$ <span style = "color:blue"> Magnetic Field of Earth </span> $\rightarrow$ Ampere's Law $\rightarrow$ Ampere's Law </footer>

### <Success style="font-size:25px"> Magnetic Field Fora Straight Conductor And Amperes Law L-4 </Success>
### <primary style="font-size:24px"> Ampere's Law </primary>
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- $\vec{|B|}=\frac{\mu_0I}{2 \pi r}$

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

- $=\oint \frac{\mu_I}{2 \pi r}.dl$

- $=\frac{\mu_0I}{2 \pi r}\oint dl=2 \pi r$

- $\oint \vec{B}.\vec{dl}=\mu_0I$

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<footer style = "font-size: 18px">Infinitely Long Straight Current Carrying Conductor $\rightarrow$ Magnetic Field of Earth $\rightarrow$ <span style = "color:blue"> Ampere's Law </span> $\rightarrow$ Ampere's Law $\rightarrow$ Problem </footer>