Question

Draw graphs showing variation in (i) electric field (ii) electric potential due to a

uniformly charged spherical shell of radius r, starting from its centre upto an external

point distant x.​

Answer 1

$\setlength{\unitlength}{1.6mm}\begin{picture}(30,20)\linethickness{0.1mm}\multiput(0,0)(0.5,0){2}{\line(0,-1){45}}\multiput(0,0)(0,-0.5){2}{\line(1,0){43}}\multiput(28,1)(14,0){2}{\line(0,1){4}}\multiput(28,5)(0,-4){2}{\line(1,0){14}}\footnotesize{\put(28.3,2.4){ Date:26/05/20 }}\linethickness{0.01mm}\footnotesize{\put(2,-3){ \blacksquare \underline{ Question }}}\footnotesize{\put(2,-3){ \blacksquare \underline{ Question }}}\footnotesize{\put(2,-6){Uniformly charged spherical shell of radius r , }}\footnotesize{\put(2,-8){starting from its centre upto an external point }}\footnotesize{\put(2,-10){distant x . Draw graphs showing variation in ,}}\footnotesize{\put(2,-12){(i) electric field (ii) electric potential }}\footnotesize{\put(2,-15){ \blacksquare \underline{ SoLutioN }}}\footnotesize{\put(2,-15){ \blacksquare \underline{ SoLutioN }}}\put(7,-23){\circle{10}}\put(7,-23){\line(1,0){30}}\put(7,-23){\line(0,-1){4.5}}\put(22,-22){x}\footnotesize{\put(8,-26){R}}\put(37,-23){\circle*{0.5}}\put(7,-20.5){ \rho }\put(5,-23){Q}\footnotesize{\put(2,-30){consider a sphere of radius R having a uniform }}\footnotesize{\put(2,-32){volume density of charge \rho . Then total charge}}\footnotesize{\put(2,-34){within the sphere is then , }}\footnotesize{\put(18,-39){ Q=\dfrac{4}{3}\pi R^3\rho ........... (i)}}\end{picture}$

$\setlength{\unitlength}{1.6mm}\begin{picture}(30,20)\linethickness{0.1mm}\multiput(0,0)(0.5,0){2}{\line(0,-1){47}}\multiput(0,0)(0,-0.5){2}{\line(1,0){43}}\footnotesize{\put(2,-2.5){\underline{(i) Electric Filed Outside the Sphere :}}}\footnotesize{\put(2,-2.5){\underline{(i) Electric Filed Outside the Sphere :}}}\footnotesize{\put(2,-5){To find E. F. outside the sphere at a distance x}}\footnotesize{\put(2,-7){imagine a spherical surface of radius x }}\footnotesize{\put(2,-9){Concentric with the charged sphere ,}}\footnotesize{\put(2,-11){Now , outward flux through this surface is , }}\footnotesize{\put(14,-14){ \oint \vec{E}.\hat{n}dS=4\pi x^2E }}\footnotesize{\put(2,-19){ \therefore\:\:E=\dfrac{Q}{4\pi\epsilon_o x^2} .............(ii)}}\footnotesize{\put(2,-23){from equation (i) and (ii) , we get ;}}\footnotesize{\put(2,-27){ \implies E=\dfrac{R^3\rho}{3\epsilon_o x^2} }}\footnotesize{\put(2,-31){\underline{(ii) Electric Filed inside the Sphere :}}}\footnotesize{\put(2,-35){By Gaussian law , \:\:\:4\pi x^2E=\dfrac{Q_{int}}{\epsilon_o} }}\footnotesize{\put(2,-39){Now , \:\:\:Q_{int}=\dfrac{4}{3}\pi x^3\rho }}\footnotesize{\put(2,-44){ \therefore\:E_{in} =\dfrac{\rho x}{3\epsilon_o} }}\end{picture}$

$\setlength{\unitlength}{1.6mm}\begin{picture}(30,20)\linethickness{0.1mm}\multiput(0,0)(0.5,0){2}{\line(0,-1){60}}\multiput(0,0)(0,-0.5){2}{\line(1,0){43}}\footnotesize{\put(2,-2.5){ \blacksquare\:\: \underline{Variation of E with x :}}}\footnotesize{\put(2,-2.5){ \blacksquare\:\: \underline{Variation of E with x :}}}\put(10,-35){\vector(1,0){28}}\put(10,-35){\vector(0,1){25}}\put(10,-35){\line(2,3){12}}\qbezier(22,-17)(24,-32)(35,-35)\put(22,-17){\line(-1,0){3}}\put(17,-17){\line(-1,0){3}}\put(22,-35){\line(0,1){18}}\put(20.5,-37){x=R}\put(12,-17){\line(-1,0){2.2}}\footnotesize{\put(6,-17){ \dfrac{\rho R}{3\epsilon_o} }}\put(25,-38){\vector(1,0){2.2}}\put(28.5,-38.5){x}\put(8,-28){\vector(0,1){2.2}}\put(7.6,-25){E}\footnotesize{\put(2,-43){\underline{(iii) Electric Potential outside the Sphere :}}}\footnotesize{\put(2,-46){Electric Potential outside the Sphere is given by ,}}\footnotesize{\put(2,-51){ \phi_{out}=-\int \limits_{\infty}^{x}E_{out}dx=-\dfrac{Q}{4\pi \epsilon_o}\int \limits_{\infty}^{x}\dfrac{dx}{x^2}=\dfrac{Q}{4\pi \epsilon_o x} }}\end{picture}$

$\setlength{\unitlength}{1.6mm}\begin{picture}(30,20)\linethickness{0.1mm}\multiput(0,0)(0.5,0){2}{\line(0,-1){70}}\multiput(0,0)(0,-0.5){2}{\line(1,0){43}}\footnotesize{\put(2,-2.5){\underline{(iii) Electric Potential inside the Sphere :}}}\footnotesize{\put(2,-6){Electric Potential inside the Sphere is given by ,}}\footnotesize{\put(2,-11){ \phi_{int}=\phi_{out}(R)-\int\limits_{x}^{R}E_{int}dx=\dfrac{Q}{4\pi \epsilon_o R}-\dfrac{\rho}{3\epsilon_o}\int\limits_{x}^{R}xdx }}\footnotesize{\put(4,-16){ =\dfrac{Q}{4\pi \epsilon_o R}+\dfrac{\rho}{6\epsilon_o}(R^2-x^2)=\dfrac{R^2\rho}{3\epsilon_o}+\dfrac{R^2\rho}{6\epsilon_o}-\dfrac{x^2\rho}{6\epsilon_o} }}\footnotesize{\put(4,-21){ =\dfrac{\rho}{6\epsilon_o}(3R^2-x^2) }}\footnotesize{\put(2,-26){ \blacksquare\:\: \underline{Variation of \phi with x :}}}\footnotesize{\put(2,-26){ \blacksquare\:\: \underline{Variation of \phi with x :}}}\put(10,-58){\vector(1,0){33}}\put(10,-58){\vector(0,1){25}}\put(28,-60){\vector(1,0){2.5}}\put(8,-50){\vector(0,1){2.5}}\put(32,-60.5){x}\put(7.5,-46){ \phi }\qbezier(10,-38)(18,-38)(23,-45)\qbezier(23,-45)(30,-56)(40,-57)\footnotesize{\put(5,-38){ \dfrac{\rho R^2}{2\epsilon_o} }}\put(22.2,-58){\line(0,1){14}}\put(20.5,-60.5){x=R}\end{picture}$