# Question A point charged particle of 8 μC is held at the center of an insulating spherical shell of inner radius 6 cm and outer radius 9 cm. The shell has a uniform charge density of ρ=2×10−2 C/m3. A) What is the magnitude of the electric field at radial distance r=2cm? Express your answer numerically, to two significant figures, using ϵ0=8.85×10−12C2/(N⋅m2) B)Find

&#12304;General guidance&#12305;The answer provided below has been developed in a clear step by step manner.Step1/4Given that;A point-charged particle $$\mathrm{{q}={8}\mu{C}}$$a the center of the insulating spherical shell of the inner radius $$\mathrm{{a}={6}{c}{m}}$$ and the outer radius $$\mathrm{{b}={9}{c}{m}}$$$$\mathrm{\rho={2}\times{10}^{{-{{2}}}}{C}/{m}^{{3}}}$$ ( uniform charge density)<path transform="" transform-origin="113 46" d=" M 280 46 A 167 167 0 1,0 447 213 A 167 167 0 0,0 280 46 " fill="#FFFFFF" stroke="rgb(51, 51, 51)" stroke-width="2" stroke-dasharray="1 8" stroke-linecap="round" stroke-linejoin="round" stroke-miterlimit="10" dominant-baseline="text-before-edge" vector-effect="non-scaling-stroke" data-id="M_QuoWcrDmSs_gj0npai_"><path transform="" transform-origin="152.25 89.25" d=" M 278.25 89.25 A 126 126 0 1,0 404.25 215.25 A 126 126 0 0,0 278.25 89.25 " fill="#FFFFFF" stroke="rgb(51, 51, 51)" stroke-width="2" stroke-dasharray="0" stroke-linecap="round" stroke-linejoin="round" stroke-miterlimit="10" dominant-baseline="text-before-edge" vector-effect="non-scaling-stroke" data-id="-E-fuoSYkcMEUGhbCqlFF"><path transform="" transform-origin="169.5 103.5" d=" M 279 103.5 A 109.5 109.5 0 1,0 388.5 213 A 109.5 109.5 0 0,0 279 103.5 " fill="#FFFFFF" stroke="rgb(51, 51, 51)" stroke-width="2" stroke-dasharray="1 8" stroke-linecap="round" stroke-linejoin="round" stroke-miterlimit="10" dominant-baseline="text-before-edge" vector-effect="non-scaling-stroke" data-id="ZAqSlX1OxVeW03wN9k7L7"><path transform="" transform-origin="190.75 128.75" d=" M 276.75 128.75 A 86 86 0 1,0 362.75 214.75 A 86 86 0 0,0 276.75 128.75 " fill="#FFFFFF" stroke="rgb(51, 51, 51)" stroke-width="2" stroke-dasharray="0" stroke-linecap="round" stroke-linejoin="round" stroke-miterlimit="10" dominant-baseline="text-before-edge" vector-effect="non-scaling-stroke" data-id="PIQfs-TGWB8F9wv-KZj3g"><path transform="" transform-origin="239 170" d=" M 279.5 170 A 40.5 40.5 0 1,0 320 210.5 A 40.5 40.5 0 0,0 279.5 170 " fill="#FFFFFF" stroke="rgb(51, 51, 51)" stroke-width="2" stroke-dasharray="1 8" stroke-linecap="round" stroke-linejoin="round" stroke-miterlimit="10" dominant-baseline="text-before-edge" vector-effect="non-scaling-stroke" data-id="OwDZOwaIvGtsIW-3-thfd"><path d=" M 277 208 L 350 208 " marker-end="url(#99Rm-wG2-W9oN__aJgqZE-end-arrowhead)" fill="#FFFFFF" stroke="rgb(51, 51, 51)" stroke-width="2" stroke-dasharray="0" stroke-linecap="round" stroke-linejoin="round" stroke-miterlimit="10" dominant-baseline="text-before-edge" vector-effect="non-scaling-stroke" data-id="99Rm-wG2-W9oN__aJgqZE"><path d=" M 276 208 L 311.13344260276 325.41966343554003 " marker-end="url(#vBXfruEjUT2jy-j-SR6NQ-end-arrowhead)" fill="#FFFFFF" stroke="rgb(51, 51, 51)" stroke-width="2" stroke-dasharray="0" stroke-linecap="round" stroke-linejoin="round" stroke-miterlimit="10" dominant-baseline="text-before-edge" vector-effect="non-scaling-stroke" data-id="vBXfruEjUT2jy-j-SR6NQ">abIIIIIIq&#961;Total charge on the insulating spherical shell is given by\begin{align*} \mathrm{{Q}} &= \mathrm{\int_{\tau}\rho{d}\tau} \end{align*}where $$\mathrm{{d}\tau={4}\pi{r}^{{2}}{d}{r}}$$, volume of spherical element.\begin{align*} \mathrm{{Q}} &= \mathrm{{\int_{{a}}^{{b}}}{\left({2}\times{10}^{{-{{2}}}}{C}/{m}^{{3}}\right)}{4}\pi{r}^{{2}}{d}{r}}\\[3pt] &= \mathrm{{\left({8}\pi\times{10}^{{-{{2}}}}{C}/{m}^{{3}}\right)}{{\left[\frac{{r}^{{3}}}{{3}}\right]}_{{a}}^{{b}}}}\\[3pt] &= \mathrm{{8}\times{3.14}\times{10}^{{-{{2}}}}{\left({C}/{m}^{{3}}\right)}{\left[\frac{{\left({9}\times{10}^{{-{{2}}}}{m}\right)}^{{3}}}{{3}}-\frac{{\left({6}\times{10}^{{-{{2}}}}{m}\right)}^{{3}}}{{3}}\right]}}\\[3pt] &= \mathrm{{8}\times{3.14}\times{10}^{{-{{2}}}}{\left({C}/{m}^{{3}}\right)}{\left[\frac{{{729}\times{10}^{{-{{6}}}}}}{{3}}-\frac{{{216}\times{10}^{{-{{6}}}}}}{{3}}\right]}}\\[3pt] &= \mathrm{{25.12}\times{10}^{{-{{2}}}}{\left[{171}\times{10}^{{-{{6}}}}\right]}{C}}\\[3pt] &= \mathrm{{4},{295.52}\times{10}^{{-{{8}}}}{C}}\\[3pt]\mathrm{{Q}} &= \mathrm{{42.955}\times{10}^{{-{{6}}}}{C}} \end{align*}This is the total charge on the spherical shell.Explanation:In the figure, dotted outline surfaces are the gaussian sur ... See the full answer