physics-class-12-unit-02-chapter-08-problems-in-electromagnetics-electrostatics-l-8-9-q4wchjmjv2i

<h3 id="success-stylefont-size25pxproblems-in-electromagnetics-electrostatics--l-8-success-2"><Success style="font-size:25px;">Problems-in-Electromagnetics-Electrostatics  L-8 </Success></h3>
<h3 id="primary-stylefont-size24pxsolution-1primary"><primary style="font-size:24px;">Solution-1</primary></h3>
<hr>
<main>
<div style='float: left; width:100%;font-size:30px;text-align:left;'>
<ul>
<li>
<p>Electrostatic field satisfy</p>
</li>
<li>
<p>$\oint \vec E \vec{dl} = 0 $</p>
</li>
<li>
<p>$\oint \vec{E} \cdot \vec{dl}=0$</p>
</li>
<li>
<p>$ \oint \vec{E} \cdot \overrightarrow{d l}=\int_A^B \vec{E} \cdot \vec{dl}+\int_B^C \vec{E} \cdot \vec{dl} +\int_C^D \vec{E} \cdot \vec{dl}+\int_D^A \vec{E} \cdot \vec{dl} $</p>
</li>
<li>
<p>$ \int_A^B \vec{E} \cdot \overrightarrow{d l}=\int E_0 x \hat{j} \cdot \hat{\imath} d x = 0 $</p>
</li>
<li>
<p>$ \int_C^D \vec{E} \cdot \vec{dl}=0 $</p>
</li>
<li>
<p>$ \int_{B}^{C} \vec E \cdot \vec{dl} = \int_{0}^{b} E_0 a \hat{j} \cdot \hat{j} d y = \int_{0}^{b} E_0 a d y = E_0 a b $</p>
</li>
</ul>
</div>
<div style='float: right; width:0%;'>
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</main>
<hr>
<footer style = "font-size: 18px"> Problems in Electromagnetics $\rightarrow$ Problem-1 $\rightarrow$ <span style = "color:blue"> Solution-1 </span> $\rightarrow$ Solution-1 $\rightarrow$ Problem-2 </footer>

<h3 id="success-stylefont-size25pxproblems-in-electromagnetics-electrostatics--l-8-success-4"><Success style="font-size:25px;">Problems-in-Electromagnetics-Electrostatics  L-8 </Success></h3>
<h3 id="primary-stylefont-size24pxproblem-2primary"><primary style="font-size:24px;">Problem-2</primary></h3>
<hr>
<main>
<div style='float: left; width:100%;font-size:30px;text-align:left;'>
<ul>
<li>
<p>How about the following fields.</p>
</li>
<li>
<p>$\vec{E}=E_0 x \hat{\imath}$</p>
</li>
<li>
<p>Can they represent an electrostatic fields?</p>
</li>
</ul>
</div>
<div style='float: right; width:0%;'>
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</div>
</main>
<hr>
<footer style = "font-size: 18px"> Solution-1 $\rightarrow$ Solution-1 $\rightarrow$ <span style = "color:blue"> Problem-2 </span> $\rightarrow$ Problem-3 $\rightarrow$ Solution-3 </footer>

<h3 id="success-stylefont-size25pxproblems-in-electromagnetics-electrostatics--l-8-success-5"><Success style="font-size:25px;">Problems-in-Electromagnetics-Electrostatics  L-8 </Success></h3>
<h3 id="primary-stylefont-size24pxproblem-3primary"><primary style="font-size:24px;">Problem-3</primary></h3>
<hr>
<main>
<div style='float: left; width:100%;font-size:30px;text-align:left;'>
<ul>
<li>
<p>Given the following electrostatic potential, calculate the corresponding electric field.</p>
</li>
<li>
<p>$v  =v_0 ; \quad r=\sqrt{x^2+y^2+z^2}<a $</p>
</li>
<li>
<p>$ =\frac{v_0 a}{\sqrt{x^2+y^2+z^2}} ; \quad r=\sqrt{x^2+y^2+z^2}>a $</p>
</li>
</ul>
</div>
<div style='float: right; width:0%;'>
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<hr>
<footer style = "font-size: 18px"> Solution-1 $\rightarrow$ Problem-2 $\rightarrow$ <span style = "color:blue"> Problem-3 </span> $\rightarrow$ Solution-3 $\rightarrow$ Solution-3 </footer>

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<h3 id="primary-stylefont-size24pxsolution-3primary"><primary style="font-size:24px;">Solution-3</primary></h3>
<hr>
<main>
<div style='float: left; width:100%;font-size:30px;text-align:left;'>
<ul>
<li>
<p>$ E_x=-\frac{\partial v}{\partial x}$</p>
</li>
<li>
<p>$ E_y=-\frac{\partial v}{\partial y}$</p>
</li>
<li>
<p>$ E_z=-\frac{\partial v}{\partial z}$</p>
</li>
<li>
<p><strong>A.</strong></p>
</li>
<li>
<p>For $r<a$</p>
</li>
<li>
<p>$E_x=-\frac{\partial v}{\partial x}=0$</p>
</li>
<li>
<p>$E_y=-\frac{\partial v}{\partial y}=0$</p>
</li>
<li>
<p>$E_z=-\frac{\partial v}{\partial z}=0$</p>
</li>
</ul>
</div>
<div style='float: right; width:0%;'>
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</div>
</main>
<hr>
<footer style = "font-size: 18px"> Problem-2 $\rightarrow$ Problem-3 $\rightarrow$ <span style = "color:blue"> Solution-3 </span> $\rightarrow$ Solution-3 $\rightarrow$ Solution-3 </footer>

<h3 id="success-stylefont-size25pxproblems-in-electromagnetics-electrostatics--l-8-success-7"><Success style="font-size:25px;">Problems-in-Electromagnetics-Electrostatics  L-8 </Success></h3>
<h3 id="primary-stylefont-size24pxsolution-3primary-1"><primary style="font-size:24px;">Solution-3</primary></h3>
<hr>
<main>
<div style='float: left; width:100%;font-size:30px;text-align:left;'>
<ul>
<li>
<p><strong>B.</strong> For $x>a$</p>
</li>
<li>
<p>$ E_x  =-\frac{\partial v}{\partial x} $</p>
</li>
<li>
<p>$ =-\frac{\partial}{\partial x}\left[\frac{v_0 a}{\sqrt{x^2+y^2+z^2}}\right]  =-v_0 a \frac{\partial}{\partial x}\left(\frac{1}{\sqrt{x^2+y^2 z^2}}\right) $</p>
</li>
<li>
<p>$ =-v_0 a\left(-\frac{1}{2}\right) \frac{1}{\left(x^2+y^2+z^2\right)^{3 / 2}}(2 x)  =\frac{v_0 a x}{\left(x^2+y^2+z^2\right)^{7 / 2}}=\frac{v_0 a x}{r^7}$</p>
</li>
</ul>
</div>
<div style='float: right; width:0%;'>
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<hr>
<footer style = "font-size: 18px"> Problem-3 $\rightarrow$ Solution-3 $\rightarrow$ <span style = "color:blue"> Solution-3 </span> $\rightarrow$ Solution-3 $\rightarrow$ Problem-4 </footer>

<h3 id="success-stylefont-size25pxproblems-in-electromagnetics-electrostatics--l-8-success-8"><Success style="font-size:25px;">Problems-in-Electromagnetics-Electrostatics  L-8 </Success></h3>
<h3 id="primary-stylefont-size24pxsolution-3primary-2"><primary style="font-size:24px;">Solution-3</primary></h3>
<hr>
<main>
<div style='float: left; width:100%;font-size:30px;text-align:left;'>
<ul>
<li>
<p>$\epsilon_y  =-\frac{\partial v}{\partial y}=\frac{V_0 a y}{r^3} $</p>
</li>
<li>
<p>$\epsilon_z  =-\frac{\partial v}{\partial z}=\frac{v_0 a z}{r^3} $</p>
</li>
<li>
<p>$\vec{E}  = E_x \hat{i}+E_y \hat{\jmath}+E_z \hat{k} $</p>
</li>
<li>
<p>$ = \frac{v_0 a}{r^3}(x \hat{\imath}+y \hat{\jmath}+z \hat{k}) = \frac{v_0 a \vec{r}}{r^3} $</p>
</li>
<li>
<p>$\vec{r}  =x \hat{\imath}+y \hat{\jmath}+z \hat{k}$</p>
</li>
</ul>
</div>
<div style='float: right; width:0%;'>
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<hr>
<footer style = "font-size: 18px"> Solution-3 $\rightarrow$ Solution-3 $\rightarrow$ <span style = "color:blue"> Solution-3 </span> $\rightarrow$ Problem-4 $\rightarrow$ Solution-4 </footer>

<h3 id="success-stylefont-size25pxproblems-in-electromagnetics-electrostatics--l-8-success-9"><Success style="font-size:25px;">Problems-in-Electromagnetics-Electrostatics  L-8 </Success></h3>
<h3 id="primary-stylefont-size24pxproblem-4primary"><primary style="font-size:24px;">Problem-4</primary></h3>
<hr>
<main>
<div style='float: left; width:50%;font-size:30px;text-align:left;'>
<ul>
<li>A pair of charges +2q and -2q are placed at a separation d as shown. The point P is midway between the two charges. What will be the values of</li>
<li>$\int_P^Q \vec{E} \cdot \vec{dl}$ along a  semicircular path of radius d/2.</li>
</ul>
</div>
<div style='float: right; width:50%; max-width:400px'>

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<hr>
<footer style = "font-size: 18px"> Solution-3 $\rightarrow$ Solution-3 $\rightarrow$ <span style = "color:blue"> Problem-4 </span> $\rightarrow$ Solution-4 $\rightarrow$ Problem-5 </footer>

<h3 id="success-stylefont-size25pxproblems-in-electromagnetics-electrostatics--l-8-success-10"><Success style="font-size:25px;">Problems-in-Electromagnetics-Electrostatics  L-8 </Success></h3>
<h3 id="primary-stylefont-size24pxsolution-4primary"><primary style="font-size:24px;">Solution-4</primary></h3>
<hr>
<main>
<div style='float: left; width:100%;font-size:30px;text-align:left;'>
<ul>
<li>
<p>$ \int_P^Q \vec{E} .\vec{dl}=V(P)-V(Q) $</p>
</li>
<li>
<p>$ V(P)=\frac{2 q}{4 \pi E_0 d/2}-\frac{2 q}{4 \pi E_0 d/2}=0$</p>
</li>
<li>
<p>$ V(Q)=\frac{2 q}{4 \pi E_0 \cdot d / 2}-\frac{2 q}{4 \pi E_0 3d/ 2} =\frac{2 q}{3 \pi E_0 d} $</p>
</li>
<li>
<p>$ \int_P^Q \vec{E} \cdot \vec{dl}=V(P)-V(Q)=-\frac{2 q}{3 \pi E_0 d}$</p>
</li>
</ul>
</div>
<div style='float: right; width:0%;'>
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<hr>
<footer style = "font-size: 18px"> Solution-3 $\rightarrow$ Problem-4 $\rightarrow$ <span style = "color:blue"> Solution-4 </span> $\rightarrow$ Problem-5 $\rightarrow$ Solution-5 </footer>

<h3 id="success-stylefont-size25pxproblems-in-electromagnetics-electrostatics--l-8-success-11"><Success style="font-size:25px;">Problems-in-Electromagnetics-Electrostatics  L-8 </Success></h3>
<h3 id="primary-stylefont-size24pxproblem-5primary"><primary style="font-size:24px;">Problem-5</primary></h3>
<hr>
<main>
<div style='float: left; width:100%;font-size:30px;text-align:left;'>
<ul>
<li>Consider an electrostatic field given by $\vec E = (20 \hat i + 30 \hat j)$</li>
<li>v/m . Calculate the potential differnce between the origin and a pont P with corrdinates x=2m, y=2m, z=2m.</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-08-problems-in-electromagnetics-electrostatics-l-8_9-q4wchjmjv2i-10.jpg"></p>
</div>
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<hr>
<footer style = "font-size: 18px"> Problem-4 $\rightarrow$ Solution-4 $\rightarrow$ <span style = "color:blue"> Problem-5 </span> $\rightarrow$ Solution-5 $\rightarrow$ Problem-6 </footer>

<h3 id="success-stylefont-size25pxproblems-in-electromagnetics-electrostatics--l-8-success-12"><Success style="font-size:25px;">Problems-in-Electromagnetics-Electrostatics  L-8 </Success></h3>
<h3 id="primary-stylefont-size24pxsolution-5primary"><primary style="font-size:24px;">Solution-5</primary></h3>
<hr>
<main>
<div style='float: left; width:60%;font-size:30px;text-align:left;'>
<ul>
<li>
<p>$ V=-\int_A^D \vec{E} \cdot \vec{dl}$</p>
</li>
<li>
<p>= $-\int_A^B \bar{E} \cdot \hat{i} d x-\int_B^C \vec{E} \cdot \hat {j} d y-\int_C^D \vec{E} \cdot \hat{k} d z$</p>
</li>
<li>
<p>= $-\int_0^2 20 d x-\int_0^2 30 dy - 0 $</p>
</li>
<li>
<p>= -40-60 = -100 V</p>
</li>
</ul>
</div>
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<footer style = "font-size: 18px"> Solution-4 $\rightarrow$ Problem-5 $\rightarrow$ <span style = "color:blue"> Solution-5 </span> $\rightarrow$ Problem-6 $\rightarrow$ Solution-6 </footer>

<h3 id="success-stylefont-size25pxproblems-in-electromagnetics-electrostatics--l-8-success-13"><Success style="font-size:25px;">Problems-in-Electromagnetics-Electrostatics  L-8 </Success></h3>
<h3 id="primary-stylefont-size24pxproblem-6primary"><primary style="font-size:24px;">Problem-6</primary></h3>
<hr>
<main>
<div style='float: left; width:50%;font-size:30px;text-align:left;'>
<ul>
<li>Two point charge of equal mass m and carrying equal charge q are suspended from a common point by two strings having negligible mass and of length l obtain an expresion relating q and $\theta$ at equilibrium.</li>
</ul>
</div>
<div style='float: right; width:50%;'>

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<footer style = "font-size: 18px"> Problem-5 $\rightarrow$ Solution-5 $\rightarrow$ <span style = "color:blue"> Problem-6 </span> $\rightarrow$ Solution-6 $\rightarrow$ Problem-7 </footer>

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<h3 id="primary-stylefont-size24pxsolution-6primary"><primary style="font-size:24px;">Solution-6</primary></h3>
<hr>
<main>
<div style='float: left; width:50%;font-size:30px;text-align:left;'>
<ul>
<li>
<p>$ T \cos \theta=m g $</p>
</li>
<li>
<p>$ T \sin \theta=F_e=\frac{q^2}{4 \pi E_0(2 l \sin \theta)^2} $</p>
</li>
<li>
<p>$=\frac{q^2}{16 \pi E_0 l^2 \sin ^2 \theta}$</p>
</li>
<li>
<p>$ \tan \theta=\frac{q^2}{16 \pi E_0 l^2 sin^2 \theta}.\frac{1}{m g} $</p>
</li>
<li>
<p>$ q^2=16 \pi E_0 l^2  \sin ^2 \theta \tan^2 \theta$</p>
</li>
</ul>
</div>
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<footer style = "font-size: 18px"> Solution-5 $\rightarrow$ Problem-6 $\rightarrow$ <span style = "color:blue"> Solution-6 </span> $\rightarrow$ Problem-7 $\rightarrow$ Solution-7 </footer>

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<h3 id="primary-stylefont-size24pxproblem-7primary"><primary style="font-size:24px;">Problem-7</primary></h3>
<hr>
<main>
<div style='float: left; width:100%;font-size:30px;text-align:left;'>
<ul>
<li>Free charge is embedded in a linear direction sphere of dielectric constant $k_1$ and radius R . The free charge density $\rho_{f}= \alpha r$. when $\alpha$ is constant and r is the distance from the centre. This sphere is surronded by another spherical shell of radius R and 2R of dielectric constant $k_2$. Calculate the $\vec E$ and the displacement vector - $\vec D$ everywhere.</li>
</ul>
</div>
<div style='float: right; width:0%;'>
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<footer style = "font-size: 18px"> Problem-6 $\rightarrow$ Solution-6 $\rightarrow$ <span style = "color:blue"> Problem-7 </span> $\rightarrow$ Solution-7 $\rightarrow$ Solution-7 </footer>

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<h3 id="primary-stylefont-size24pxsolution-7primary"><primary style="font-size:24px;">Solution-7</primary></h3>
<hr>
<main>
<div style='float: left; width:50%;font-size:30px;text-align:left;'>
<ul>
<li>
<p>$ \oint \vec{D} \cdot d \vec{A}= Q_{F_{enc}}$</p>
</li>
<li>
<p>$ \vec{D}=\epsilon_0 \vec{E}+\vec{P}$</p>
</li>
<li>
<p>$\vec{D}, \vec{E} \ \vec{P}$ will be along radial direction and unit only depends on r .</p>
</li>
</ul>
</div>
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<footer style = "font-size: 18px"> Solution-6 $\rightarrow$ Problem-7 $\rightarrow$ <span style = "color:blue"> Solution-7 </span> $\rightarrow$ Solution-7 $\rightarrow$ Solution-7 </footer>

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<h3 id="primary-stylefont-size24pxsolution-7primary-1"><primary style="font-size:24px;">Solution-7</primary></h3>
<hr>
<main>
<div style='float: left; width:50%;font-size:30px;text-align:left;'>
<ul>
<li>
<p><strong>A.</strong> For  0 < r < R</p>
</li>
<li>
<p>$\oint \vec{D} \cdot d \vec{A}  =\alpha_{f_{enc}} $</p>
</li>
<li>
<p>$\text { D. } 4 \pi r^2  = \int_0^r \alpha r \cdot 4 \pi r^2 d r $</p>
</li>
<li>
<p>$ =4 \pi \alpha \int_0^r r^3 d r  =\pi \alpha r^4 $</p>
</li>
<li>
<p>0 < r < R</p>
</li>
<li>
<p>$\vec{D}  =\frac{\alpha}{4} r^2 \hat{r}$</p>
</li>
</ul>
</div>
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<footer style = "font-size: 18px"> Problem-7 $\rightarrow$ Solution-7 $\rightarrow$ <span style = "color:blue"> Solution-7 </span> $\rightarrow$ Solution-7 $\rightarrow$ Solution-7 </footer>

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<h3 id="primary-stylefont-size24pxsolution-7primary-2"><primary style="font-size:24px;">Solution-7</primary></h3>
<hr>
<main>
<div style='float: left; width:100%;font-size:30px;text-align:left;'>
<ul>
<li>
<p>$ \vec{E}=\frac{\vec{D}}{\epsilon_0 K_i}=\frac{\alpha \hat{k}}{4 \varepsilon_0 k_1} \hat{r} $</p>
</li>
<li>
<p>$ \vec{P}=\vec{D}-\epsilon_0 \vec{E} $</p>
</li>
<li>
<p>$ =\epsilon_0\left(K_1-1\right) \vec{E}  =\frac{\left(K_1-1\right)}{4 K_1} \alpha r^2 \hat{r} $</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-08-problems-in-electromagnetics-electrostatics-l-8_9-q4wchjmjv2i-18.jpg"></p>
</div>
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<hr>
<footer style = "font-size: 18px"> Solution-7 $\rightarrow$ Solution-7 $\rightarrow$ <span style = "color:blue"> Solution-7 </span> $\rightarrow$ Solution-7 $\rightarrow$ Solution-7 </footer>

<h3 id="success-stylefont-size25pxproblems-in-electromagnetics-electrostatics--l-8-success-19"><Success style="font-size:25px;">Problems-in-Electromagnetics-Electrostatics  L-8 </Success></h3>
<h3 id="primary-stylefont-size24pxsolution-7primary-3"><primary style="font-size:24px;">Solution-7</primary></h3>
<hr>
<main>
<div style='float: left; width:100%;font-size:30px;text-align:left;'>
<ul>
<li>
<p>$ R<r<2 R $</p>
</li>
<li>
<p>$ \oint \vec{D} \cdot d \vec{A}=Q_{F_{enc}} $</p>
</li>
<li>
<p>$ 4 \pi r^2 D=\int_0^R \alpha r 4 \pi r^2 d r =\pi \alpha R^4 $</p>
</li>
<li>
<p>$ \vec{D}=\frac{\alpha}{4} \frac{R^4}{r^2} \hat{r} $</p>
</li>
<li>
<p>$ \vec{E}=\frac{\vec{D}}{E_0 K_2}=\frac{\alpha R^4}{4 E_0 K_2 r^2} \hat{r} $</p>
</li>
<li>
<p>$ \vec{P}=\vec{D}-\epsilon_0 \vec{E}$</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-08-problems-in-electromagnetics-electrostatics-l-8_9-q4wchjmjv2i-19.jpg"></p>
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<hr>
<footer style = "font-size: 18px"> Solution-7 $\rightarrow$ Solution-7 $\rightarrow$ <span style = "color:blue"> Solution-7 </span> $\rightarrow$ Solution-7 $\rightarrow$ Solution-7 </footer>

<h3 id="success-stylefont-size25pxproblems-in-electromagnetics-electrostatics--l-8-success-20"><Success style="font-size:25px;">Problems-in-Electromagnetics-Electrostatics  L-8 </Success></h3>
<h3 id="primary-stylefont-size24pxsolution-7primary-4"><primary style="font-size:24px;">Solution-7</primary></h3>
<hr>
<main>
<div style='float: left; width:100%;font-size:30px;text-align:left;'>
<ul>
<li>
<p>$ R<r<2 R $</p>
</li>
<li>
<p>$ \oint \vec{D} \cdot d \vec{A}=Q_{F_{enc}} $</p>
</li>
<li>
<p>$ 4 \pi r^2 D=\int_0^R \alpha r 4 \pi r^2 d r = \pi \alpha R^4 $</p>
</li>
<li>
<p>$ \vec{D}=\frac{\alpha}{4} \frac{R^4}{r^2} \hat{r} $</p>
</li>
<li>
<p>$ \vec{E}=\frac{\vec{D}}{E_0 K_2}=\frac{\alpha R^4}{4 E_0 K_2 r^2} \hat{r} $</p>
</li>
<li>
<p>$ \vec{P}=\vec{D}-\epsilon_0 \vec{E}$</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-08-problems-in-electromagnetics-electrostatics-l-8_9-q4wchjmjv2i-19.jpg"></p>
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<hr>
<footer style = "font-size: 18px"> Solution-7 $\rightarrow$ Solution-7 $\rightarrow$ <span style = "color:blue"> Solution-7 </span> $\rightarrow$ Solution-7 $\rightarrow$ Problem-8</footer>

<h3 id="success-stylefont-size25pxproblems-in-electromagnetics-electrostatics--l-8-success-21"><Success style="font-size:25px;">Problems-in-Electromagnetics-Electrostatics  L-8 </Success></h3>
<h3 id="primary-stylefont-size24pxsolution-7primary-5"><primary style="font-size:24px;">Solution-7</primary></h3>
<hr>
<main>
<div style='float: left; width:100%;font-size:30px;text-align:left;'>
<ul>
<li>
<p><strong>For r > 2R</strong></p>
</li>
<li>
<p>$ \oint \vec{D} \cdot d \vec{A}=Q_{F_{enc}} $</p>
</li>
<li>
<p>$ 4 \pi r^2 D = \pi \alpha R^4 $</p>
</li>
<li>
<p>$ \vec{D}=\frac{\alpha}{4} \frac{R^4}{r^2} \hat{r} $</p>
</li>
<li>
<p>$ \vec{E} $= $ \frac {\vec{D}}{E_0}$ = $\frac{\alpha R^4}{4 E_0 K_2 r^2} \hat{r} $</p>
</li>
<li>
<p>$ \vec{P}=\vec{D}-\epsilon_0 \vec{E}$</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-08-problems-in-electromagnetics-electrostatics-l-8_9-q4wchjmjv2i-20.jpg"></p>
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</main>
<hr>
<footer style = "font-size: 18px"> Solution-7 $\rightarrow$ Solution-7 $\rightarrow$ <span style = "color:blue"> Solution-7 </span> $\rightarrow$ Problem-8 $\rightarrow$ Solution-8 </footer>

<h3 id="success-stylefont-size25pxproblems-in-electromagnetics-electrostatics--l-8-success-22"><Success style="font-size:25px;">Problems-in-Electromagnetics-Electrostatics  L-8 </Success></h3>
<h3 id="primary-stylefont-size24pxproblem-8primary"><primary style="font-size:24px;">Problem-8</primary></h3>
<hr>
<main>
<div style='float: left; width:50%;font-size:30px;text-align:left;'>
<ul>
<li>
<p>A point charge Q is placed at a distance $\frac {a}{2} $ above the centre of a horizontal square surface of edge Q . The electrostatic flux the angle ….. square surface will be .</p>
</li>
<li>
<p>a) $\frac {Q}{E_0}$</p>
</li>
<li>
<p>b) $\frac {Q}{4E_0}$</p>
</li>
<li>
<p>c)$\frac {Q}{6E_0}$</p>
</li>
<li>
<p>d) Zero</p>
</li>
</ul>
</div>
<div style='float: right; width:50%; max-width:400px;'>

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<hr>
<footer style = "font-size: 18px"> Solution-7 $\rightarrow$ Solution-7 $\rightarrow$ <span style = "color:blue"> Problem-8 </span> $\rightarrow$ Solution-8 $\rightarrow$ Problem-9 </footer>

<h3 id="success-stylefont-size25pxproblems-in-electromagnetics-electrostatics--l-8-success-23"><Success style="font-size:25px;">Problems-in-Electromagnetics-Electrostatics  L-8 </Success></h3>
<h3 id="primary-stylefont-size24pxsolution-8primary"><primary style="font-size:24px;">Solution-8</primary></h3>
<hr>
<main>
<div style='float: left; width:50%;font-size:30px;text-align:left;'>
<ul>
<li>$\frac {Q}{E_0}$ : Total flux</li>
</ul>
</div>
<div style='float: right; width:50%; max-width:400px;'>


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<hr>
<footer style = "font-size: 18px"> Solution-7 $\rightarrow$ Problem-8 $\rightarrow$ <span style = "color:blue"> Solution-8</span> $\rightarrow$ Problem-9 $\rightarrow$ Solution-9 </footer>

<h3 id="success-stylefont-size25pxproblems-in-electromagnetics-electrostatics--l-8-success-24"><Success style="font-size:25px;">Problems-in-Electromagnetics-Electrostatics  L-8 </Success></h3>
<h3 id="primary-stylefont-size24pxproblem-9primary"><primary style="font-size:24px;">Problem-9</primary></h3>
<hr>
<main>
<div style='float: left; width:100%;font-size:30px;text-align:left;'>
<ul>
<li>
<p>Consider a set of two point charge $q_1$ abd $q_2$ and a uniformaly charges sphere of radius R with uniform volume charge density $\rho$ obtain the volume of .</p>
</li>
<li>
<p>a) $\oint \vec E. \vec {dA}$ over the closed surface S.</p>
</li>
<li>
<p>b) $\oint \vec E. \vec {dl}$ are curve C.</p>
</li>
<li>
<p>c) For what volume of $\rho$ with $\oint \vec E. \vec {dA}$ over S remain?</p>
</li>
<li>
<p>d) Draw a closed surface triangle what  $\oint \vec E. \vec {dl}$ will be independence of $\rho$</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-08-problems-in-electromagnetics-electrostatics-l-8_9-q4wchjmjv2i-23.jpg"></p>
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</main>
<hr>
<footer style = "font-size: 18px"> Problem-8 $\rightarrow$ Solution-8 $\rightarrow$ <span style = "color:blue"> Problem-9 </span> $\rightarrow$ Solution-9 $\rightarrow$ Problem-10 </footer>

<h3 id="success-stylefont-size25pxproblems-in-electromagnetics-electrostatics--l-8-success-25"><Success style="font-size:25px;">Problems-in-Electromagnetics-Electrostatics  L-8 </Success></h3>
<h3 id="primary-stylefont-size24pxsolution-9primary"><primary style="font-size:24px;">Solution-9</primary></h3>
<hr>
<main>
<div style='float: left; width:50%;font-size:30px;text-align:left;'>
<ul>
<li>
<p>a) $\Rightarrow \oint_S \vec{E} \cdot d \vec{A}=\frac{Q}{E_0}=\frac{1}{E_0}\left[q_2+\frac{4 \pi}{3} R^3 \rho\right]$</p>
</li>
<li>
<p>b) $\oint \vec{E} \cdot \overrightarrow{dl}=0$</p>
</li>
</ul>
</div>
<div style='float: right; width:50%; max-width:400px;'>

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<hr>
<footer style = "font-size: 18px"> Solution-8 $\rightarrow$ Problem-9 $\rightarrow$ <span style = "color:blue"> Solution-9 </span> $\rightarrow$ Problem-10 $\rightarrow$ Problem-11 </footer>

<h3 id="success-stylefont-size25pxproblems-in-electromagnetics-electrostatics--l-8-success-26"><Success style="font-size:25px;">Problems-in-Electromagnetics-Electrostatics  L-8 </Success></h3>
<h3 id="primary-stylefont-size24pxproblem-10primary"><primary style="font-size:24px;">Problem-10</primary></h3>
<hr>
<main>
<div style='float: left; width:100%;font-size:30px;text-align:left;'>
<ul>
<li>
<p>The relationn $\vec D$ = $\epsilon_{0} \vec {E} + \vec {P}$ holds good.</p>
</li>
<li>
<p>a) only in free space</p>
</li>
<li>
<p>b) only inside a dielectric</p>
</li>
<li>
<p>c) only outside a dielectric</p>
</li>
<li>
<p>d) everywhere in space</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-08-problems-in-electromagnetics-electrostatics-l-8_9-q4wchjmjv2i-25.jpg"></p>
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</main>
<hr>
<footer style = "font-size: 18px"> Problem-9 $\rightarrow$ Solution-9 $\rightarrow$ <span style = "color:blue"> Problem-10 </span> $\rightarrow$ Problem-11 $\rightarrow$ Solution-11 </footer>

<h3 id="success-stylefont-size25pxproblems-in-electromagnetics-electrostatics--l-8-success-27"><Success style="font-size:25px;">Problems-in-Electromagnetics-Electrostatics  L-8 </Success></h3>
<h3 id="primary-stylefont-size24pxproblem-11primary"><primary style="font-size:24px;">Problem-11</primary></h3>
<hr>
<main>
<div style='float: left; width:100%;font-size:30px;text-align:left;'>
<ul>
<li>Two non-conducting solid spheres of radius R and 2R having uniform volume charge density $\rho_1$ and $\rho_2$ respectivily touch each other.</li>
<li>The net electric flux at a direction 2R from the centre of the smaller sphere along with line joining the centre is zero obtain $\frac {\rho_1}{\rho_2}$.</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-08-problems-in-electromagnetics-electrostatics-l-8_9-q4wchjmjv2i-26.jpg"></p>
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<hr>
<footer style = "font-size: 18px"> Solution-9 $\rightarrow$ Problem-10 $\rightarrow$ <span style = "color:blue"> Problem-11 </span> $\rightarrow$ Solution-11 $\rightarrow$ Thankyou</footer>

<h3 id="success-stylefont-size25pxproblems-in-electromagnetics-electrostatics--l-8-success-28"><Success style="font-size:25px;">Problems-in-Electromagnetics-Electrostatics  L-8 </Success></h3>
<h3 id="primary-stylefont-size24pxsolution-11primary"><primary style="font-size:24px;">Solution-11</primary></h3>
<hr>
<main>
<div style='float: left; width:50%;font-size:30px;text-align:left;'>
<ul>
<li>
<p>a) $\frac {\rho_1}{\rho_2}$ = $ - \frac {32}{25}$</p>
</li>
<li>
<p>b) $\frac {\rho_1}{\rho_2}$ =  - 4</p>
</li>
</ul>
</div>
<div style='float: right; width:50%; max-width:400px;'>

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<hr>
<footer style = "font-size: 18px"> Problem-10 $\rightarrow$ Problem-11 $\rightarrow$ <span style = "color:blue"> Solution-11 </span> $\rightarrow$ Thankyou $\rightarrow$  </footer>