Question Number 37032 by ajfour last updated on 08/Jun/18
Commented by ajfour last updated on 08/Jun/18
$${Find}\:{Electric}\:{field}\:{at}\:{P}\:\:{due}\:{to} \\ $$$${the}\:{charge}\:{contained}\:{in}\:{the}\: \\ $$$${paraboloid}\:{having}\:{a}\:{uniform} \\ $$$${volume}\:{charge}\:{density}\:\boldsymbol{\rho}. \\ $$
Answered by ajfour last updated on 08/Jun/18
![let z=Ar^2 ā z_0 =AR^2 or A=(z_0 /R^2 ) ā z=((z_0 /R^2 ))r^2 or r^2 = (z/z_0 )R^2 dE at P is (Ļ/(2Īµ_0 ))(1ā(z/( (ā(z^2 +r^2 ))))) due to a disk of charge. here lets choose an elementary disc inside paraboloid at z=z. dE at P (0,0,z_P ) is = ((Ļdz)/(2Īµ_0 ))(1ā((z_P āz)/( (ā((z_P āz)^2 +r^2 ))))) E_P = (Ļ/(2Īµ_0 ))[ā«_0 ^( z_0 ) dzā(1/2)ā«_0 ^( z_0 ) ((2(z_P āz))/( (ā((z_P āz)^2 +(z/z_0 )R^2 )))) dz ..... let u=(z_P āz)^2 +(z/z_0 )R^2 du = (ā2(z_P āz)+(R^2 /z_0 ))dz E_P =(Ļ/(2Īµ_0 ))[z_0 +(1/2)ā«_0 ^( u_0 ) (du/( (āu))) ā (R^2 /(2z_0 ))ā«_0 ^( z_0 ) (dz/( (ā((z_P āz)^2 +((zR^2 )/z_0 ))))) ] =(Ļ/(2Īµ_0 )){z_0 +(ā((z_P āz_0 )^2 +R^2 )) āz_P } ā((ĻR^2 )/(4Īµ_0 z_0 ))ā«_0 ^( z_0 ) (dz/( (ā(z^2 ā2(z_P ā(R^2 /(2z_0 )))+z_P ^2 )))) =(Ļ/(2Īµ_0 )){z_0 +(ā((z_P āz_0 )^2 +R^2 )) āz_P } ā((ĻR^2 )/(4Īµ_0 z_0 ))ā«_0 ^( z_0 ) (dz/( (ā((zāb)^2 +z_P ^2 āb^2 )))) =(Ļ/(2Īµ_0 )){z_0 +(ā((z_P āz_0 )^2 +R^2 )) āz_P } ā((ĻR^2 )/(4Īµ_0 z_0 ))ln ā£zāb+(ā((zāb)^2 +z_P ^2 āb^2 ))ā£_0 ^z_0](https://www.tinkutara.com/question/Q37035.png)
$${let}\:\:\:{z}={Ar}^{\mathrm{2}} \\ $$$$\Rightarrow\:\:\:\:{z}_{\mathrm{0}} ={AR}^{\mathrm{2}} \:\:\:{or}\:\:\:{A}=\frac{{z}_{\mathrm{0}} }{{R}^{\mathrm{2}} } \\ $$$$\Rightarrow\:\:\:\:{z}=\left(\frac{{z}_{\mathrm{0}} }{{R}^{\mathrm{2}} }\right){r}^{\mathrm{2}} \:\:\:{or}\:\:\:\:{r}^{\mathrm{2}} \:=\:\frac{{z}}{{z}_{\mathrm{0}} }{R}^{\mathrm{2}} \\ $$$${dE}\:{at}\:{P}\:\:\:{is}\:\:\:\frac{\sigma}{\mathrm{2}\epsilon_{\mathrm{0}} }\left(\mathrm{1}ā\frac{{z}}{\:\sqrt{{z}^{\mathrm{2}} +{r}^{\mathrm{2}} }}\right) \\ $$$${due}\:{to}\:{a}\:{disk}\:{of}\:{charge}. \\ $$$${here}\:{lets}\:{choose}\:{an}\:{elementary}\:{disc}\: \\ $$$${inside}\:{paraboloid}\:{at}\:{z}={z}. \\ $$$${dE}\:{at}\:\:{P}\:\left(\mathrm{0},\mathrm{0},{z}_{{P}} \right)\:{is} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\rho{dz}}{\mathrm{2}\epsilon_{\mathrm{0}} }\left(\mathrm{1}ā\frac{{z}_{{P}} ā{z}}{\:\sqrt{\left({z}_{{P}} ā{z}\right)^{\mathrm{2}} +{r}^{\mathrm{2}} }}\right) \\ $$$$\:\:\:\:\:{E}_{{P}} \:=\:\frac{\rho}{\mathrm{2}\epsilon_{\mathrm{0}} }\left[\int_{\mathrm{0}} ^{\:\:{z}_{\mathrm{0}} } {dz}ā\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\:\:{z}_{\mathrm{0}} } \frac{\mathrm{2}\left({z}_{{P}} ā{z}\right)}{\:\sqrt{\left({z}_{{P}} ā{z}\right)^{\mathrm{2}} +\frac{{z}}{{z}_{\mathrm{0}} }{R}^{\mathrm{2}} }}\:{dz}\right. \\ $$$$\:\:…..\:\: \\ $$$${let}\:\:\:{u}=\left({z}_{{P}} ā{z}\right)^{\mathrm{2}} +\frac{{z}}{{z}_{\mathrm{0}} }{R}^{\mathrm{2}} \\ $$$$\:\:\:{du}\:=\:\left(ā\mathrm{2}\left({z}_{{P}} ā{z}\right)+\frac{{R}^{\mathrm{2}} }{{z}_{\mathrm{0}} }\right){dz} \\ $$$${E}_{{P}} =\frac{\rho}{\mathrm{2}\epsilon_{\mathrm{0}} }\left[{z}_{\mathrm{0}} +\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\:\:{u}_{\mathrm{0}} } \frac{{du}}{\:\sqrt{{u}}}\right. \\ $$$$\left.\:\:\:\:\:\:\:ā\:\frac{{R}^{\mathrm{2}} }{\mathrm{2}{z}_{\mathrm{0}} }\int_{\mathrm{0}} ^{\:\:{z}_{\mathrm{0}} } \frac{{dz}}{\:\sqrt{\left({z}_{{P}} ā{z}\right)^{\mathrm{2}} +\frac{{zR}^{\mathrm{2}} }{{z}_{\mathrm{0}} }}}\:\right] \\ $$$$\:\:\:\:=\frac{\rho}{\mathrm{2}\epsilon_{\mathrm{0}} }\left\{{z}_{\mathrm{0}} +\sqrt{\left({z}_{{P}} ā{z}_{\mathrm{0}} \right)^{\mathrm{2}} +{R}^{\mathrm{2}} }\:ā{z}_{{P}} \right\} \\ $$$$\:\:\:\:\:\:\:\:\:\:ā\frac{\rho{R}^{\mathrm{2}} }{\mathrm{4}\epsilon_{\mathrm{0}} {z}_{\mathrm{0}} }\int_{\mathrm{0}} ^{\:\:{z}_{\mathrm{0}} } \frac{{dz}}{\:\sqrt{{z}^{\mathrm{2}} ā\mathrm{2}\left({z}_{{P}} ā\frac{{R}^{\mathrm{2}} }{\mathrm{2}{z}_{\mathrm{0}} }\right)+{z}_{{P}} ^{\mathrm{2}} }} \\ $$$$\:\:\:\:=\frac{\rho}{\mathrm{2}\epsilon_{\mathrm{0}} }\left\{{z}_{\mathrm{0}} +\sqrt{\left({z}_{{P}} ā{z}_{\mathrm{0}} \right)^{\mathrm{2}} +{R}^{\mathrm{2}} }\:ā{z}_{{P}} \right\} \\ $$$$\:\:\:\:\:\:\:ā\frac{\rho{R}^{\mathrm{2}} }{\mathrm{4}\epsilon_{\mathrm{0}} {z}_{\mathrm{0}} }\int_{\mathrm{0}} ^{\:\:{z}_{\mathrm{0}} } \frac{{dz}}{\:\sqrt{\left({z}ā{b}\right)^{\mathrm{2}} +{z}_{{P}} ^{\mathrm{2}} ā{b}^{\mathrm{2}} }} \\ $$$$\:\:\:\:=\frac{\rho}{\mathrm{2}\epsilon_{\mathrm{0}} }\left\{{z}_{\mathrm{0}} +\sqrt{\left({z}_{{P}} ā{z}_{\mathrm{0}} \right)^{\mathrm{2}} +{R}^{\mathrm{2}} }\:ā{z}_{{P}} \right\} \\ $$$$\:\:\:\:\:\:\:\:\:ā\frac{\rho{R}^{\mathrm{2}} }{\mathrm{4}\epsilon_{\mathrm{0}} {z}_{\mathrm{0}} }\mathrm{ln}\:\mid{z}ā{b}+\sqrt{\left({z}ā{b}\right)^{\mathrm{2}} +{z}_{{P}} ^{\mathrm{2}} ā{b}^{\mathrm{2}} }\mid_{\mathrm{0}} ^{{z}_{\mathrm{0}} } \\ $$