physics-class-12-unit-02-chapter-03-energy-storedin-capacitors-gausss-lawin-dielectrics-l-3-9-qoau9oftqje

<h3 id="success-stylefont-size25px-energy-stored-in-capacitors-gausss-law-in-dielectrics-l-3-success"><Success style="font-size:25px;"> Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 </Success></h3>
<h3 id="primary-stylefont-size24pxenergy-stored-in-capacitors-gausss-law-in-dielectricsprimary"><primary style="font-size:24px;">Energy Stored in Capacitors Gauss’s Law in Dielectrics</primary></h3>
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<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-12-unit-02-chapter-03-energy-storedin-capacitors-gausss-lawin-dielectrics-l-3_9-qoau9oftqje-014-0027.6.jpg"></p>
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<footer style = "font-size: 18px">  $\rightarrow$  $\rightarrow$ <span style = "color:blue"> Energy Stored in Capacitors Gauss's Law in Dielectrics </span> $\rightarrow$ Electrostatics Energy Stored in a Capacitor $\rightarrow$ Work Done in Moving Charge </footer>

<h4 id="success-stylefont-size25px-energy-stored-in-capacitors-gausss-law-in-dielectrics-l-3-success-1"><Success style="font-size:25px;"> Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 </Success></h4>
<h3 id="primary-stylefont-size24pxelectrostatics-energy-stored-in-a-capacitorprimary"><primary style="font-size:24px;">Electrostatics Energy Stored in a Capacitor</primary></h3>
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<main>
<div style='float: left; width:50%;text-align:left;font-size:30px;'>
<ul>
<li>
<p>Charge : q</p>
</li>
<li>
<p>Potential V= $\frac{q}{C}$</p>
</li>
<li>
<p>Infinite decimal charge dq.</p>
</li>
</ul>
</div>
<div style='float: right; width:50%; max-width:400px'>
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<footer style = "font-size: 18px">  $\rightarrow$ Energy Stored in Capacitors Gauss's Law in Dielectrics $\rightarrow$ <span style = "color:blue"> Electrostatics Energy Stored in a Capacitor </span> $\rightarrow$ Work Done in Moving Charge $\rightarrow$ Work Done in Moving Charge </footer>

<h3 id="success-stylefont-size25px-energy-stored-in-capacitors-gausss-law-in-dielectrics-l-3-success-2"><Success style="font-size:25px;"> Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 </Success></h3>
<h3 id="primary-stylefont-size24pxwork-done-in-moving-chargeprimary"><primary style="font-size:24px;">Work Done in Moving Charge</primary></h3>
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<main>	
<div style='float: left; width:100%;text-align:left;font-size:30px;'>
<ul>
<li>
<p>Work done in moving charge dq = Vdq = $\frac{qdq}{c}$</p>
</li>
<li>
<p>Total work done W= $\int_{0}^{Q}\frac{q}{c}.dq$ = $\frac{1}{c}\int_{0}^{Q} qdq$ = $\frac{Q^2}{2C}$</p>
</li>
<li>
<p>Energy Stored = U = $\frac{Q^2}{2C}$</p>
</li>
</ul>
</div>
<div style='float: right; width:0%;'>
<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-12-unit-02-chapter-03-energy-storedin-capacitors-gausss-lawin-dielectrics-l-3_9-qoau9oftqje-002.jpg"></p>
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<footer style = "font-size: 18px"> Energy Stored in Capacitors Gauss's Law in Dielectrics $\rightarrow$ Electrostatics Energy Stored in a Capacitor $\rightarrow$ <span style = "color:blue"> Work Done in Moving Charge </span> $\rightarrow$ Work Done in Moving Charge $\rightarrow$ Parallel Plate Capacitor </footer>

<h3 id="success-stylefont-size25px-energy-stored-in-capacitors-gausss-law-in-dielectrics-l-3-success-3"><Success style="font-size:25px;"> Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 </Success></h3>
<h3 id="primary-stylefont-size24pxwork-done-in-moving-chargeprimary-1"><primary style="font-size:24px;">Work Done in Moving Charge</primary></h3>
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<main>	
<div style='float: left; width:100%;text-align:left;font-size:30px;'>
<ul>
<li>
<p>$U=\frac{Q^2}{2 C}=\frac{1}{2 C} C^2 v^2=\frac{1}{2} C v^2$</p>
</li>
<li>
<p>$U=\frac{Q}{2 C} c v=\frac{1}{2} Q v$</p>
</li>
<li>
<p>$U=\frac{Q^2}{2 C} ;  U=\frac{1}{2} C v^2 ;  U=\frac{1}{2} Q v$</p>
</li>
</ul>
</div>
<div style='float: right; width:0%;'>
<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-12-unit-02-chapter-03-energy-storedin-capacitors-gausss-lawin-dielectrics-l-3_9-qoau9oftqje-003.jpg"></p>
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<footer style = "font-size: 18px"> Electrostatics Energy Stored in a Capacitor $\rightarrow$ Work Done in Moving Charge $\rightarrow$ <span style = "color:blue"> Work Done in Moving Charge </span> $\rightarrow$ Parallel Plate Capacitor $\rightarrow$ Example </footer>

<h3 id="success-stylefont-size25px-energy-stored-in-capacitors-gausss-law-in-dielectrics-l-3-success-4"><Success style="font-size:25px;"> Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 </Success></h3>
<h3 id="primary-stylefont-size24pxparallel-plate-capacitorprimary"><primary style="font-size:24px;">Parallel Plate Capacitor</primary></h3>
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<ul>
<li>
<p>$U=\frac{Q^2}{2 C}=\frac{1}{2} C v^2$</p>
</li>
<li>
<p>C = $\frac{\epsilon_0 A}{d}$</p>
</li>
<li>
<p>$V=Ed$</p>
</li>
<li>
<p>$U=\frac{1}{2} \frac{\epsilon_0 A}{d} \cdot E^2 d^2$ $=\left(\frac{1}{2} \epsilon_0 E^2\right) \cdot (Ad)$</p>
</li>
<li>
<p>(Ad)= Volume Enclosed</p>
</li>
</ul>
</div>
<div style='float: right; width:40%;'>
<img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/8dae3e7f-2ffc-47df-a7a9-9805c51cfe00/public"  width="100%" >

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</main>
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<footer style = "font-size: 18px"> Work Done in Moving Charge $\rightarrow$ Work Done in Moving Charge $\rightarrow$ <span style = "color:blue"> Parallel Plate Capacitor </span> $\rightarrow$ Example $\rightarrow$ Energy Density </footer>

<h3 id="success-stylefont-size25px-energy-stored-in-capacitors-gausss-law-in-dielectrics-l-3-success-5"><Success style="font-size:25px;"> Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 </Success></h3>
<h3 id="primary-stylefont-size24pxexampleprimary"><primary style="font-size:24px;">Example</primary></h3>
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<ul>
<li>
<p><strong>Spherical Capacitor</strong></p>
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<li>
<p>$C=4 \pi \epsilon_0 \frac{r_a r_b}{\left(r_b-r_a\right)}$</p>
</li>
<li>
<p><strong>Energy Stored</strong></p>
</li>
<li>
<p>$U=\frac{1}{2} \frac{Q^2}{C}=\frac{Q^2\left(r_b-r_a\right)}{8 \pi \epsilon_0 r_a r_b}$</p>
</li>
</ul>
</div>
<div style='float: right; width:50%; max-width:400px;'>
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<footer style = "font-size: 18px"> Work Done in Moving Charge $\rightarrow$ Parallel Plate Capacitor $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Energy Density $\rightarrow$ Total Stored Energy </footer>

<h3 id="success-stylefont-size25px-energy-stored-in-capacitors-gausss-law-in-dielectrics-l-3-success-6"><Success style="font-size:25px;"> Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 </Success></h3>
<h3 id="primary-stylefont-size24pxenergy-densityprimary"><primary style="font-size:24px;">Energy Density</primary></h3>
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<div style='float: left; width:60%;text-align:left;font-size:30px;'>
<ul>
<li>
<p>Energy Density = $\frac{1}{2}\epsilon_0E^2$</p>
</li>
<li>
<p>$4 \pi r^2 E=\frac{Q}{\epsilon_0}$</p>
</li>
<li>
<p>$E=\frac{Q}{4 \pi \epsilon_0 r^2}$</p>
</li>
<li>
<p>$d U=\frac{1}{2} \epsilon_0 \frac{Q^2}{\left(4 \pi \epsilon_0\right)^2 r^4} 4 \pi r^2 d r$ $=\frac{Q^2}{8 \pi \epsilon_0} \frac{d r}{r^2}$</p>
</li>
</ul>
</div>
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<img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/2c19fa3e-9421-45a5-ad1c-15fb95fb0100/public"  width="60%" >

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<footer style = "font-size: 18px"> Parallel Plate Capacitor $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Energy Density </span> $\rightarrow$ Total Stored Energy $\rightarrow$ Dielectrics and Polarization </footer>

<h3 id="success-stylefont-size25px-energy-stored-in-capacitors-gausss-law-in-dielectrics-l-3-success-7"><Success style="font-size:25px;"> Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 </Success></h3>
<h3 id="primary-stylefont-size24pxtotal-stored-energyprimary"><primary style="font-size:24px;">Total Stored Energy</primary></h3>
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<main>	
<div style='float: left; width:100%;text-align:left;font-size:30px;'>
<ul>
<li>U = $\int_{r_a}^{r_b}\frac{Q^2}{8\pi\epsilon_0}\frac{dr}{r^2}$ = $\frac{Q^2}{8\pi\epsilon_0}(\frac{1}{r_a}-\frac{1}{r_b})$ = $\frac{Q^2(r_b-r_a)}{8\pi\epsilon_0r_ar_b}$</li>
</ul>
</div>
<div style='float: right; width:0%;'>
<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-12-unit-02-chapter-03-energy-storedin-capacitors-gausss-lawin-dielectrics-l-3_9-qoau9oftqje-007.jpg"></p>
</main>
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<footer style = "font-size: 18px"> Example $\rightarrow$ Energy Density $\rightarrow$ <span style = "color:blue"> Total Stored Energy </span> $\rightarrow$ Dielectrics and Polarization $\rightarrow$ Bound Surface </footer>

<h3 id="success-stylefont-size25px-energy-stored-in-capacitors-gausss-law-in-dielectrics-l-3-success-8"><Success style="font-size:25px;"> Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 </Success></h3>
<h3 id="primary-stylefont-size24pxdielectrics-and-polarizationprimary"><primary style="font-size:24px;">Dielectrics and Polarization</primary></h3>
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<main>	
<div style='float: left; width:50%;text-align:left;font-size:30px;'>
<ul>
<li>
<p>$\frac{\sigma}{\epsilon_0}=E_0$</p>
</li>
<li>
<p>${\sigma}=\epsilon_0E_0$</p>
</li>
</ul>
</div>
<div style='float: right; width:50%; max-width:400px;'>
<img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/63542a7e-7b2e-43c1-705a-f3631d179a00/public"  width="80%" >

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<hr>
<footer style = "font-size: 18px"> Energy Density $\rightarrow$ Total Stored Energy $\rightarrow$ <span style = "color:blue"> Dielectrics and Polarization </span> $\rightarrow$ Bound Surface $\rightarrow$ Polarization </footer>

<h3 id="success-stylefont-size25px-energy-stored-in-capacitors-gausss-law-in-dielectrics-l-3-success-9"><Success style="font-size:25px;"> Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 </Success></h3>
<h3 id="primary-stylefont-size24pxbound-surfaceprimary"><primary style="font-size:24px;">Bound Surface</primary></h3>
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<div style='float: left; width:50%;text-align:left;font-size:30px;'>
<ul>
<li>
<p>${\sigma_b}:$ Bound Surface</p>
</li>
<li>
<p>Charge Density</p>
</li>
<li>
<p>Polarized</p>
</li>
</ul>
</div>
<div style='float: right; width:50%;'>
<img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/f785e732-7aa9-414a-ab41-fdbd8d2eb200/public"  width="100%" >

</main>
<hr>
<footer style = "font-size: 18px"> Total Stored Energy $\rightarrow$ Dielectrics and Polarization $\rightarrow$ <span style = "color:blue"> Bound Surface </span> $\rightarrow$ Polarization $\rightarrow$ Polarization </footer>

<h3 id="success-stylefont-size25px-energy-stored-in-capacitors-gausss-law-in-dielectrics-l-3-success-10"><Success style="font-size:25px;"> Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 </Success></h3>
<h3 id="primary-stylefont-size24pxpolarizationprimary"><primary style="font-size:24px;">Polarization</primary></h3>
<hr>
<main>
<div style='float: left; width:100%;text-align:left;font-size:30px;'>
<ul>
<li>
<p>Polarization$\vec{P}$: Dipole moment per unit volume.</p>
</li>
<li>
<p>$\vec{P}={\epsilon_0}\chi \vec{E}$</p>
</li>
<li>
<p>Where, $\chi$ = Electric Susceptibility</p>
</li>
</ul>
</div>
<div style='float: right; width:0%;'>
<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-12-unit-02-chapter-03-energy-storedin-capacitors-gausss-lawin-dielectrics-l-3_9-qoau9oftqje-0010.jpg"></p>
</main>
<hr>
<footer style = "font-size: 18px"> Dielectrics and Polarization $\rightarrow$ Bound Surface $\rightarrow$ <span style = "color:blue"> Polarization </span> $\rightarrow$ Polarization $\rightarrow$ Bound Surface Charge Density </footer>

<h3 id="success-stylefont-size25px-energy-stored-in-capacitors-gausss-law-in-dielectrics-l-3-success-11"><Success style="font-size:25px;"> Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 </Success></h3>
<h3 id="primary-stylefont-size24pxpolarizationprimary-1"><primary style="font-size:24px;">Polarization</primary></h3>
<hr>
<main>
<div style='float: left; width:50%;text-align:left;font-size:30px;'>
<ul>
<li>
<p>Polarization = P</p>
</li>
<li>
<p>Volume = Al</p>
</li>
<li>
<p>Dipole moment = PAl</p>
</li>
<li>
<p>Dipole moment = q.l</p>
</li>
<li>
<p>ql = PAl</p>
</li>
<li>
<p>q = PA</p>
</li>
</ul>
</div>
<div style='float: right; width:50%;'>
<img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/336df601-1b59-4397-196e-9f50104dbf00/public"  width="70%" >

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<hr>
<footer style = "font-size: 18px"> Bound Surface $\rightarrow$ Polarization $\rightarrow$ <span style = "color:blue"> Polarization </span> $\rightarrow$ Bound Surface Charge Density $\rightarrow$ Capacitor with a Dielectric </footer>

<h3 id="success-stylefont-size25px-energy-stored-in-capacitors-gausss-law-in-dielectrics-l-3-success-12"><Success style="font-size:25px;"> Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-bound-surface-charge-densityprimary"><primary style="font-size:24px;"> Bound Surface Charge Density</primary></h3>
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<main>
<div style='float: left; width:50%;text-align:left;font-size:30px;'>
<ul>
<li>
<p>$\sigma_b=\frac{q}{A}=p$</p>
</li>
<li>
<p>Area = $\frac{A}{cos{\theta}}$</p>
</li>
<li>
<p>Bound charge Density</p>
</li>
<li>
<p>${\sigma_b}=\frac{q}{A/cos{\theta}}=Pcos{\theta}$</p>
</li>
<li>
<p>$\sigma_b=\vec{P} \cdot \hat{n}$</p>
</li>
</ul>
</div>
<div style='float: right; width:50%;'>
<img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/69c73561-85d7-42b9-6061-378e0e55a100/public"  width="70%" >

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<footer style = "font-size: 18px"> Polarization $\rightarrow$ Polarization $\rightarrow$ <span style = "color:blue"> Bound Surface Charge Density </span> $\rightarrow$ Capacitor with a Dielectric $\rightarrow$ Dielectric Constant </footer>

<h3 id="success-stylefont-size25px-energy-stored-in-capacitors-gausss-law-in-dielectrics-l-3-success-13"><Success style="font-size:25px;"> Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 </Success></h3>
<h3 id="primary-stylefont-size24pxcapacitor-with-a-dielectricprimary"><primary style="font-size:24px;">Capacitor with a Dielectric</primary></h3>
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<main>
<div style='float: left; width:50%;text-align:left;font-size:30px;'>
<ul>
<li>
<p>EA = $\frac{\left(\sigma_f-\sigma_b\right) A}{\epsilon_0}$</p>
</li>
<li>
<p>$\ \epsilon_0 E=\sigma_f-\sigma_b$</p>
</li>
<li>
<p>${\sigma_b}=P={\epsilon_0}{\chi}E$</p>
</li>
<li>
<p>${\epsilon_0}E+{\epsilon_0}{\chi}E={\sigma_f}$</p>
</li>
<li>
<p>${\epsilon_0}(1+{\chi})E={\sigma_f}$</p>
</li>
</ul>
</div>
<div style='float: right; width:50%;'>
<img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/29f40d7d-7d1d-4326-21c2-efb738fe5e00/public"  width="70%" >

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<hr>
<footer style = "font-size: 18px"> Polarization $\rightarrow$ Bound Surface Charge Density $\rightarrow$ <span style = "color:blue"> Capacitor with a Dielectric </span> $\rightarrow$ Dielectric Constant $\rightarrow$ Dielectric Constant </footer>

<h3 id="success-stylefont-size25px-energy-stored-in-capacitors-gausss-law-in-dielectrics-l-3-success-14"><Success style="font-size:25px;"> Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 </Success></h3>
<h3 id="primary-stylefont-size24pxdielectric-constantprimary"><primary style="font-size:24px;">Dielectric Constant</primary></h3>
<hr>
<main>
<div style='float: left; width:100%;text-align:left;font-size:30px;'>
<ul>
<li>
<p>Dielectric Constant K=1+$\chi$</p>
</li>
<li>
<p>Permittivity of the Dielectric ${\epsilon}={\epsilon_0}(1+{\chi})$ = ${\epsilon_0K}$</p>
</li>
<li>
<p>$k \geqslant 1$</p>
</li>
<li>
<p>E=$\frac{\sigma_f}{\epsilon_0k}=\frac{\sigma_f}{\epsilon}$</p>
</li>
</ul>
</div>
<div style='float: right; width:0%;'>
<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-12-unit-02-chapter-03-energy-storedin-capacitors-gausss-lawin-dielectrics-l-3_9-qoau9oftqje-0014.jpg"></p>
</main>
<hr>
<footer style = "font-size: 18px"> Bound Surface Charge Density $\rightarrow$ Capacitor with a Dielectric $\rightarrow$ <span style = "color:blue"> Dielectric Constant </span> $\rightarrow$ Dielectric Constant $\rightarrow$ Example </footer>

<h3 id="success-stylefont-size25px-energy-stored-in-capacitors-gausss-law-in-dielectrics-l-3-success-15"><Success style="font-size:25px;"> Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 </Success></h3>
<h3 id="primary-stylefont-size24pxdielectric-constantprimary-1"><primary style="font-size:24px;">Dielectric Constant</primary></h3>
<hr>
<main>
<div style='float: left; width:100%;text-align:left;font-size:20px;'>
<table>
<thead>
<tr>
<th><ins><strong>Material</strong> </ins></th>
<th><ins><strong>K</strong></ins></th>
</tr>
</thead>
<tbody>
<tr>
<td>Pyrex Glass</td>
<td>4.7</td>
</tr>
<tr>
<td>Polystyrene</td>
<td>2.6</td>
</tr>
<tr>
<td>Paper</td>
<td>3.5</td>
</tr>
<tr>
<td>Porcelain</td>
<td>6.5</td>
</tr>
<tr>
<td>Titanium Ceramic</td>
<td>130</td>
</tr>
<tr>
<td>Strontium Titante</td>
<td>310</td>
</tr>
<tr>
<td>Water</td>
<td>80.4</td>
</tr>
</tbody>
</table>
</div>
<div style='float: right; width:0%; max-width:400px;'>

</main>
<hr>
<footer style = "font-size: 18px"> Capacitor with a Dielectric $\rightarrow$ Dielectric Constant $\rightarrow$ <span style = "color:blue"> Dielectric Constant </span> $\rightarrow$ Example $\rightarrow$ Capacitance </footer>

<h3 id="success-stylefont-size25px-energy-stored-in-capacitors-gausss-law-in-dielectrics-l-3-success-16"><Success style="font-size:25px;"> Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 </Success></h3>
<h3 id="primary-stylefont-size24pxexampleprimary-1"><primary style="font-size:24px;">Example</primary></h3>
<hr>
<main>
<div style='float: left; width:50%;text-align:left;font-size:30px;'>
<ul>
<li>
<p>$E=\frac{\sigma_f}{\epsilon}$</p>
</li>
<li>
<p>$\sigma_f=\frac{Q}{A}$</p>
</li>
<li>
<p>$E=\frac{v}{\alpha}$</p>
</li>
<li>
<p>$\frac{v}{\alpha}=\frac{Q}{A\epsilon}$</p>
</li>
<li>
<p>$Q=cv$</p>
</li>
<li>
<p>$v=Q\frac{\alpha}{A\epsilon}$</p>
</li>
</ul>
</div>
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<footer style = "font-size: 18px"> Dielectric Constant $\rightarrow$ Dielectric Constant $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Capacitance $\rightarrow$ Dielectric Filling Capacitance </footer>

<h3 id="success-stylefont-size25px-energy-stored-in-capacitors-gausss-law-in-dielectrics-l-3-success-17"><Success style="font-size:25px;"> Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 </Success></h3>
<h3 id="primary-stylefont-size24pxcapacitanceprimary"><primary style="font-size:24px;">Capacitance</primary></h3>
<hr>
<main>
<div style='float: left; width:100%;text-align:left;font-size:30px;'>
<ul>
<li>
<p>$c=\frac{\epsilon A}{\alpha}=\frac{\epsilon_0 K A}{\alpha}$</p>
</li>
<li>
<p>Area $=100 \mathrm{~cm}^2$</p>
</li>
<li>
<p>$\alpha= 1{~cm}$</p>
</li>
<li>
<p>(1) Axis separating the plates $\epsilon \simeq \epsilon_0$</p>
</li>
<li>
<p>$c_a=\frac{\epsilon A}{\alpha}=8.85{~p^F}$</p>
</li>
<li>
<p>(2) Dielectric filling ${~K=2.6}$</p>
</li>
<li>
<p>$C_a=23.01{~p^F}$</p>
</li>
</ul>
</div>
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<footer style = "font-size: 18px"> Dielectric Constant $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Capacitance </span> $\rightarrow$ Dielectric Filling Capacitance $\rightarrow$ Gauss's Law in Dielectric </footer>

<h3 id="success-stylefont-size25px-energy-stored-in-capacitors-gausss-law-in-dielectrics-l-3-success-18"><Success style="font-size:25px;"> Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-dielectric-filling-capacitanceprimary"><primary style="font-size:24px;"> Dielectric Filling Capacitance</primary></h3>
<hr>
<main>
<div style='float: left; width:50%;text-align:left;font-size:30px;'>
<ul>
<li>
<p>$E_a=\frac{\sigma_f}{\epsilon_0}$</p>
</li>
<li>
<p>$E_{\alpha}=\frac{\sigma_f}{\epsilon_0 K}$ $=\frac{\sigma_f}{\epsilon}$</p>
</li>
<li>
<p>$E_a>E_\alpha$</p>
</li>
</ul>
</div>
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<footer style = "font-size: 18px"> Example $\rightarrow$ Capacitance $\rightarrow$ <span style = "color:blue"> Delectric Filling Capacitance </span> $\rightarrow$ Gauss's Law in Dielectric $\rightarrow$ Gauss's Law in Dielectric </footer>

<h3 id="success-stylefont-size25px-energy-stored-in-capacitors-gausss-law-in-dielectrics-l-3-success-19"><Success style="font-size:25px;"> Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 </Success></h3>
<h3 id="primary-stylefont-size24pxgausss-law-in-dielectricprimary"><primary style="font-size:24px;">Gauss’s Law in Dielectric</primary></h3>
<hr>
<main>	
<div style='float: left; width:50%;text-align:left;font-size:30px;'>
<ul>
<li>
<p>E A = $({\sigma_f-\sigma_b})\frac{A}{\epsilon_0}$</p>
</li>
<li>
<p>$\epsilon_0 E+\sigma_b=\sigma_f$</p>
</li>
<li>
<p>$\sigma_b=P$</p>
</li>
<li>
<p>$\epsilon_0 E + P = \sigma _f$</p>
</li>
<li>
<p>Displacement vector</p>
</li>
<li>
<p>$\vec{D}=\epsilon_0\vec{E}+\vec{P}$</p>
</li>
</ul>
</div>
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<footer style = "font-size: 18px"> Capacitance $\rightarrow$ Delectric Filling Capacitance $\rightarrow$ <span style = "color:blue"> Gauss's Law in Dielectric </span> $\rightarrow$ Gauss's Law in Dielectric $\rightarrow$ Example </footer>

<h3 id="success-stylefont-size25px-energy-stored-in-capacitors-gausss-law-in-dielectrics-l-3-success-20"><Success style="font-size:25px;"> Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 </Success></h3>
<h3 id="primary-stylefont-size24pxgausss-law-in-dielectricprimary-1"><primary style="font-size:24px;">Gauss’s Law in Dielectric</primary></h3>
<hr>
<main>
<div style='float: left; width:100%;text-align:left;font-size:30px;'>
<ul>
<li>
<p>$ D = \sigma_f$</p>
</li>
<li>
<p>$ D A = \sigma_f A$</p>
</li>
<li>
<p><strong>$\oint \vec{D} \cdot d \vec{A}=Q_{f \text { enc }}$</strong></p>
</li>
<li>
<p><strong>$\oint\epsilon \vec{E} \cdot d \vec{A}=Q_{f \text { enc }}$</strong></p>
</li>
</ul>
</div>
<div style='float: right; width:0%;'>
<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-12-unit-02-chapter-03-energy-storedin-capacitors-gausss-lawin-dielectrics-l-3_9-qoau9oftqje-0025.jpg"></p>
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<hr>
<footer style = "font-size: 18px"> Delectric Filling Capacitance $\rightarrow$ Gauss's Law in Dielectric $\rightarrow$ <span style = "color:blue"> Gauss's Law in Dielectric </span> $\rightarrow$ Example $\rightarrow$ Thankyou </footer>

<h3 id="success-stylefont-size25px-energy-stored-in-capacitors-gausss-law-in-dielectrics-l-3-success-21"><Success style="font-size:25px;"> Energy-Stored in-Capacitors-Gausss-Law-in-Dielectrics L-3 </Success></h3>
<h3 id="primary-stylefont-size24pxexampleprimary-2"><primary style="font-size:24px;">Example</primary></h3>
<hr>
<main>
<div style='float: left; width:50%;text-align:left;font-size:30px;'>
<ul>
<li>
<p>$\oint \vec{D} \cdot d \vec{A}=Q_{\text {f enc }}$</p>
</li>
<li>
<p>D. $4 \pi r^2=Q$</p>
</li>
<li>
<p>$D=\frac{Q}{4 \pi r^2}$</p>
</li>
<li>
<p>$a<r<b \quad D=\frac{Q}{4 \pi r^2}=\in E$</p>
</li>
<li>
<p>$E=\frac{Q}{4 \pi \epsilon r^2}$</p>
</li>
</ul>
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<footer style = "font-size: 18px"> Gauss's Law in Dielectric $\rightarrow$ Gauss's Law in Dielectric $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Thankyou $\rightarrow$  </footer>