physics-class-12-unit-08-chapter-03-maxwells-equations-and-electromagnetic-waves-l-3-4-f6dhszqewe4

### <Success style="font-size:25px"> Maxwells Equations And Electromagnetic Waves L-3 </Success>
### <primary style="font-size:24px"> Sinusoidal Electromagnetic Wave </primary>
<hr>
<main>

- $\vec{E} = \hat{i} E_0 \sin (kz - \omega t)$

- $\vec{B} = \hat{j} B_0 \sin(kz-\omega t)$

- $\vec{E} \times \vec{B} \rightarrow \hat{i} \times \hat{j}  = \hat{k}$ propagation along z-direction

</div>
<div style='float: right; width:50%;'>

</main>
<hr>
<footer style = "font-size: 18px">Maxwell's Equations and Electromagnetic Waves $\rightarrow$ Sinusoidal Waves $\rightarrow$ <span style = "color:blue"> Sinusoidal Electromagnetic Wave </span> $\rightarrow$ Free Space $\rightarrow$ Faraday's Law </footer>

### <Success style="font-size:25px"> Maxwells Equations And Electromagnetic Waves L-3 </Success>
### <primary style="font-size:24px"> Free Space </primary>
<hr>
<main>

- <span style="color:green">No charge</span>

- <span style="color:green">No currents</span>

- $\oint \vec{E} \cdot d \vec{A}=0$

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

- $\oint \vec{E} \cdot \vec{dl}=-\frac{d}{d t} \int \vec{B} \cdot d \vec{A}$

- $\oint \vec{B} \cdot \vec{dl}=\mu_0 \epsilon_0 \frac{d}{d t} \int \vec{e} \cdot d \vec{A}$

</div>
<div style='float: right; width:0%;'>

![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-12-unit-08-chapter-03-maxwells-equations-and-electromagnetic-waves-l-3_4-f6dhszqewe4-p21.png)

</main>
<hr>
<footer style = "font-size: 18px">Sinusoidal Waves $\rightarrow$ Sinusoidal Electromagnetic Wave $\rightarrow$ <span style = "color:blue"> Free Space </span> $\rightarrow$ Faraday's Law $\rightarrow$ Faraday's Law </footer>

### <Success style="font-size:25px"> Maxwells Equations And Electromagnetic Waves L-3 </Success>
### <primary style="font-size:24px"> Faraday's Law </primary>
<hr>
<main>

- $ \oint \vec{e} \cdot \vec{dl}=-\frac{d}{d t} \int \vec{B} \cdot d \vec{A}$

- z = z + $\Delta z$

- <span style="color:blue">$\Delta$ z is infintitational</span>

</div>
<div style='float: right; width:50%; max-width:400px;'>

</div>
</main>
<hr>
<footer style = "font-size: 18px">Sinusoidal Electromagnetic Wave $\rightarrow$ Free Space $\rightarrow$ <span style = "color:blue"> Faraday's Law </span> $\rightarrow$ Faraday's Law $\rightarrow$ Faraday's Law </footer>

### <Success style="font-size:25px"> Maxwells Equations And Electromagnetic Waves L-3 </Success>
### <primary style="font-size:24px"> Faraday's Law </primary>
<hr>
<main>

- $ \oint \vec{E} \cdot \vec{dl}$=$\int_P^Q \vec{E} \cdot \vec{dl}$+$\int_Q^R \vec{E} \cdot \vec{dl}$+$\int_R^S \vec{E} \cdot \vec{dl}$+$\int_S^p \vec{E} \cdot \vec{dl}$

- $ \vec{E}=\hat{i} E_0 \sin (k z-\omega t)$

- $ \oint \vec{E} \cdot \vec{dl}$=$\int_P^Q \vec{E} \cdot \vec{dl}$+$\int_R^S \vec{E} \cdot \vec{dl}$

- =$E(z+\Delta z) h-E(z) h $
 
- =$[E(z+\Delta z)-E(z)] R $
 
- $ \frac{d f}{d x}$=$\operatorname{lim}_{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x} $

</div>
<div style='float: right; width:0%;'>

</div>
</main>
<hr>
<footer style = "font-size: 18px">Free Space $\rightarrow$ Faraday's Law $\rightarrow$ <span style = "color:blue"> Faraday's Law </span> $\rightarrow$ Faraday's Law $\rightarrow$ Faraday's Law </footer>

### <Success style="font-size:25px"> Maxwells Equations And Electromagnetic Waves L-3 </Success>
### <primary style="font-size:24px"> Faraday's Law </primary>
<hr>
<main>

- $ \frac{d f}{d x} \simeq \frac{f(x+\Delta x)-f(x)}{\Delta x} $

- $ f(x+\Delta x)=f(x)+\Delta x \cdot \frac{d f}{d x} $

- $ \frac{E(z+\Delta z)-E(z)}{\Delta z}=\frac{d E}{d z}$

- $ E(z+\Delta z)-E(z)=\Delta z \frac{d E}{d z} $

- $ \oint \vec{E} \cdot \vec{dl}$=$\frac{d E}{d z} h \Delta z$=$\frac{d E}{d z} h \Delta z$

</div>
<div style='float: right; width:0%;'>

</main>
<hr>
<footer style = "font-size: 18px">Faraday's Law $\rightarrow$ Faraday's Law $\rightarrow$ <span style = "color:blue"> Faraday's Law </span> $\rightarrow$ Faraday's Law $\rightarrow$ Faraday's Law </footer>

### <Success style="font-size:25px"> Maxwells Equations And Electromagnetic Waves L-3 </Success>
### <primary style="font-size:24px"> Faraday's Law </primary>
<hr>
<main>

- $ \int \vec{B} \cdot d \vec{A} \simeq B(z) h \Delta Z $

- $ \frac{d}{d t} \int \vec{B} \cdot d \vec{A}$=$\frac{\partial B}{\partial t} \cdot h \Delta Z $

- $ \oint \vec{E} \cdot \vec{dl}$=$-\frac{d}{d t} \int \vec{B} \cdot d \vec{A} $

- $ \frac{\partial E}{\partial z} h \Delta z$=$-\frac{\partial R}{\partial t} h \Delta z $

- $ \Rightarrow  \frac{\partial E}{\partial z}$=$-\frac{\partial B}{\partial t}$

</div>
<div style='float: right; width:0%;'>

### <Success style="font-size:25px"> Maxwells Equations And Electromagnetic Waves L-3 </Success>
### <primary style="font-size:24px"> Faraday's Law </primary>
<hr>
<main>

- $\frac{\partial E}{\partial z}$=$-\frac{\partial B}{\partial t} $

- $E=E_0 \sin (kz-\omega t) $

- $\frac{\partial E}{\partial z}$=$k E_0 \cos (kz-\omega t) $

- $B=B_0 \sin (kz-\omega t) $

- $\frac{\partial B}{\partial t}$=$-\omega B_0 \cos (kz-\omega t) $

- $k E_0=\omega B_0$

</div>
<div style='float: right; width:0%;'>

</main>
<hr>
<footer style = "font-size: 18px">Faraday's Law $\rightarrow$ Faraday's Law $\rightarrow$ <span style = "color:blue"> Faraday's Law </span> $\rightarrow$ Ampere's Law $\rightarrow$ Ampere's Law </footer>

### <Success style="font-size:25px"> Maxwells Equations And Electromagnetic Waves L-3 </Success>
### <primary style="font-size:24px"> Ampere's Law </primary>
<hr>
<main>

- $ \oint \vec{B} \cdot \vec{dl}=\int_P^Q \vec{B} \cdot \vec{dl}+\int_Q^R \vec{B} \cdot \vec{dl}+\int_R^S \vec{b} \cdot \vec{dl}+\int_S^P \vec{B} \cdot \vec{dl} $

- =$\int_P^Q \vec{B} \cdot \vec{dl}+\int_R^S \vec{B} \cdot \vec{dl} $

- =$B(z+\Delta z) h-B(z) h $

- =$[B(z+\Delta z)-B(z)] h $

- $ B(z+\Delta z)-B(z) \simeq \Delta z \frac{\partial B}{\partial z} $

- $ \oint \vec{B} \cdot \vec{dl}=\frac{\partial B}{\partial z} h \Delta z$

</div>
<div style='float: right; width:0%;'>

</main>
<hr>
<footer style = "font-size: 18px">Faraday's Law $\rightarrow$ Ampere's Law $\rightarrow$ <span style = "color:blue"> Ampere's Law </span> $\rightarrow$ Ampere's Law $\rightarrow$ Ampere's Law </footer>

### <Success style="font-size:25px"> Maxwells Equations And Electromagnetic Waves L-3 </Success>
### <primary style="font-size:24px"> Ampere's Law </primary>
<hr>
<main>

- $ \int \vec{E} \cdot d \vec{A}$=-$E(z) \cdot h \Delta z $

- $ \mu_0 \epsilon_0 \frac{d}{d t} \int \vec{E} \cdot d \vec{A}$=$-\mu_0 \epsilon_0 \frac{\partial E}{\partial t} \cdot h \Delta z$

- $ \oint \vec{B} \cdot d \vec{R}=\mu_0 \epsilon_0 \frac{d}{d t} \int \vec{E} \cdot d \vec{A} $

- $ \frac{\partial B}{\partial z} h \Delta z$=-$\mu_0 \epsilon_0 \frac{\partial \epsilon}{\partial t} h \Delta z$

- $ \frac{\partial B}{\partial z}$=-$\mu_0 \epsilon_0 \frac{\partial E}{\partial t}$

</div>
<div style='float: right; width:0%;'>

</main>
<hr>
<footer style = "font-size: 18px">Ampere's Law $\rightarrow$ Ampere's Law $\rightarrow$ <span style = "color:blue"> Ampere's Law </span> $\rightarrow$ Ampere's Law $\rightarrow$ Ampere's Law </footer>

### <Success style="font-size:25px"> Maxwells Equations And Electromagnetic Waves L-3 </Success>
### <primary style="font-size:24px"> Ampere's Law </primary>
<hr>
<main>

- $\frac{\partial B}{\partial z}$=-$\mu_0 \epsilon_0 \frac{\partial E}{\partial t} $

- $B=B_0 \sin (k z-\omega t)$

- $\frac{\partial B}{\partial z}$=$k B_0 \cos (k z-\omega t) $

- $E=E_0 \sin (k z-\omega t) $

- $\frac{\partial E}{\partial t}$=-$\omega E_0 \cos (k z-\omega t) $

- $k B_0=\mu_0 \epsilon_0 \omega E_0$

</div>
<div style='float: right; width:0%;'>

### <Success style="font-size:25px"> Maxwells Equations And Electromagnetic Waves L-3 </Success>
### <primary style="font-size:24px"> Ampere's Law </primary>
<hr>
<main>

- $ k \epsilon_0=\omega B_0 $

- $ k B_0=\mu_0 \epsilon_0 \omega E_0 $

- $ k^2 E_0 B_0=\mu_0 \epsilon_0 \omega^2 E_0 B_0 $

- $ k^2=\mu_0 \epsilon_0 \omega^2$

- <span style="color:green">Speed of the wave</span> =$\frac{\omega}{b} $

- =$\frac{1}{\sqrt{\epsilon_0 \mu_0}}$

- = c

</div>
<div style='float: right; width:0%;'>

</main>
<hr>
<footer style = "font-size: 18px">Ampere's Law $\rightarrow$ Ampere's Law $\rightarrow$ <span style = "color:blue"> Ampere's Law </span> $\rightarrow$ Speed of Electromagnetic $\rightarrow$ Energy in Electro Magnetic Waves </footer>

### <Success style="font-size:25px"> Maxwells Equations And Electromagnetic Waves L-3 </Success>
### <primary style="font-size:24px"> Energy in Electro Magnetic Waves </primary>
<hr>
<main>

- $ U_E=\frac{1}{2} \epsilon_0 E^2 $  <span style="color:green">(Electric energy density)</span>

- $ U_B=\frac{1}{2 \mu_0} B^2 $  <span style="color:green">(Magnetic energy density)</span>

- $ k E_0=\omega B_0 $

- $ \Rightarrow B_0=\frac{k}{\omega} E_0=\frac{E_0}{c} $

</div>
<div style='float: right; width:0%;'>

</div>
</main>
<hr>
<footer style = "font-size: 18px">Ampere's Law $\rightarrow$ Speed of Electromagnetic $\rightarrow$ <span style = "color:blue"> Energy in Electro Magnetic Waves </span> $\rightarrow$ Energy in Electro Magnetic Waves $\rightarrow$ Total Energy Density </footer>

### <Success style="font-size:25px"> Maxwells Equations And Electromagnetic Waves L-3 </Success>
### <primary style="font-size:24px"> Energy in Electro Magnetic Waves </primary>
<hr>
<main>

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

- =$\frac{1}{2 \mu_0} B_0^2 \sin ^2(kz-\omega t) $ 
- =$\frac{1}{2 \mu_0} \frac{E_0^2}{c^2} \sin ^2(kz-\omega t) $ 
- =$\frac{1}{2 \mu_0} \epsilon_0 \mu_0 E_0^2 \sin ^2(kz-\omega t)$  
- =$\frac{1}{2} \epsilon_0 E_0^2 \sin ^2(kz-\omega t) $
 
- $U_E  =\frac{1}{2} \epsilon_0 E^2 $
    
- =$\frac{1}{2} \epsilon_0 E_0^2 \sin ^2(kz-\omega t)$

</div>
<div style='float: right; width:0%;'>

</main>
<hr>
<footer style = "font-size: 18px">Speed of Electromagnetic $\rightarrow$ Energy in Electro Magnetic Waves $\rightarrow$ <span style = "color:blue"> Energy in Electro Magnetic Waves </span> $\rightarrow$ Total Energy Density $\rightarrow$ Average Energy </footer>

### <Success style="font-size:25px"> Maxwells Equations And Electromagnetic Waves L-3 </Success>
### <primary style="font-size:24px"> Total Energy Density </primary>
<hr>
<main>

- U = $U_E + U_B$

- = $\epsilon_0 E_0 \sin^2 (kz-\omega t)$
 
- <span style="color:green">Time average</span>

- U'= $\frac{1}{2} \int_0^T U dt$

- = $ \frac{\epsilon_0 E_0}{T} \int_0^T \sin^2 (kz-\omega t)$

- = $\frac{1}{2} \epsilon_0 E_0 ^2$

</div>
<div style='float: right; width:0%;'>

</main>
<hr>
<footer style = "font-size: 18px">Energy in Electro Magnetic Waves $\rightarrow$ Energy in Electro Magnetic Waves $\rightarrow$ <span style = "color:blue"> Total Energy Density </span> $\rightarrow$ Average Energy $\rightarrow$ Example-1 </footer>

### <Success style="font-size:25px"> Maxwells Equations And Electromagnetic Waves L-3 </Success>
### <primary style="font-size:24px"> Average Energy </primary>
<hr>
<main>

* Average energy crossing unit area per unit time  = U c

- = $\frac {1}{2} c \epsilon_0 E_0 ^2$ = Intensity(I)
 
-  I = $\frac{1}{2} c \epsilon_0 E_0 ^2$
 
-  I = $\frac{1}{2} c \epsilon_0 E_0 ^2$

- $E_0 = \sqrt{\frac{2I}{c\epsilon_0}}$

</div>
<div style='float: right; width:50%;'>

</main>
<hr>
<footer style = "font-size: 18px">Energy in Electro Magnetic Waves $\rightarrow$ Total Energy Density $\rightarrow$ <span style = "color:blue"> Average Energy </span> $\rightarrow$ Example-1 $\rightarrow$ Example-2 </footer>

### <Success style="font-size:25px"> Maxwells Equations And Electromagnetic Waves L-3 </Success>
### <primary style="font-size:24px"> Example-2 </primary>
<hr>
<main>

- $B_0 = \frac{E_0}{c} = \frac{870}{3 \times 10^8}$

- $\simeq 3 \times 10^{-6} T$

- <span style="color:green">Voyager-1 (September 1977)</span>

- Current distance $\sim 2 \times 10^{13}$ m
- Transmitted power $\sim 20W$
- Antenna gain $\sim 6.5 \times 10^4$
- Recieves intensity = $\frac{20 \times 6.5 \times 10^4}{4 \pi  \times (2 \times 10^{13})^2}$
 
- $\simeq 2.6 \times 10^{-22} W/m^2$

</div>
<div style='float: right; width:0%;'>

</main>
<hr>
<footer style = "font-size: 18px">Average Energy $\rightarrow$ Example-1 $\rightarrow$ <span style = "color:blue"> Example-2 </span> $\rightarrow$ Laser $\rightarrow$ Thank You </footer>