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<div id="w2wPUip21">Assuming that the charge is distributed evenly, the volume density of charge and surface density of charge are given by</div>
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{tex}\rho=\frac{q}{V}{/tex}and {tex}\sigma=\frac{q}{A}{/tex} respectively, where,
<script type="math/tex" id="MathJax-Element-3">\qquad q</script> is the charge, <nobr aria-hidden="true">V</nobr> is the volume and <nobr aria-hidden="true">A</nobr> is the area.
Let the radii of the two spheres be <nobr aria-hidden="true">r1 </nobr>and <nobr aria-hidden="true">r2 </nobr> respectively.
It is given that the charge on both the spheres is equal.
{tex}\Rightarrow \quad \rho_{1}=\frac{q}{\frac{4}{3} \pi r_{1}^{3}} \text { and } \rho_{2}=\frac{q}{\frac{4}{3} \pi r_{2}^{2}}, \text { and }{/tex}
{tex}\sigma_{1}=\frac{q}{4 \pi r_{1}^{2}} \text { and } \sigma_{2}=\frac{q}{4 \pi r_{1}^{2}}{/tex}
<script type="math/tex" id="MathJax-Element-11">\sigma_2 = \frac{q}{4\pi r_1^2}.</script>
It is given that the ratio of the volume density is <nobr aria-hidden="true">8:64.</nobr>
{tex}\Rightarrow \quad \frac{\rho_{1}}{\rho_{2}}=\frac{8}{64} \quad \Rightarrow \quad \frac{\frac{3 q}{4 \pi_{1}^{3}}}{\frac{3 q}{4 m_{2}^{3}}}=\frac{8}{64}{/tex}
{tex}\Rightarrow \quad\left(\frac{n}{n}\right)^{3}=\frac{8}{\sqrt{4}} \quad \Rightarrow \quad \frac{n}{n}=\frac{1}{2} \quad \Rightarrow \quad\left(\frac{n}{n}\right)^{2}=\frac{1}{4}{/tex}
{tex}\Rightarrow{/tex} The ratio of the surface density of charge is 1 : 4
<script type="math/tex" id="MathJax-Element-15">\Rightarrow \qquad \frac{\sigma_1}{\sigma_2} = \frac{\frac{q}{4\pi r_1^2}}{\frac{q}{4\pi r_2^2}} = \left(\frac{r_2}{r_1}\right)^2 = \frac{1}{4}.</script>
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Sia ? 6 years, 3 months ago
{tex}\rho=\frac{q}{V}{/tex}and {tex}\sigma=\frac{q}{A}{/tex} respectively, where,
<script type="math/tex" id="MathJax-Element-3">\qquad q</script> is the charge, <nobr aria-hidden="true">V</nobr> is the volume and <nobr aria-hidden="true">A</nobr> is the area.
Let the radii of the two spheres be <nobr aria-hidden="true">r1 </nobr>and <nobr aria-hidden="true">r2 </nobr> respectively.
It is given that the charge on both the spheres is equal.
{tex}\Rightarrow \quad \rho_{1}=\frac{q}{\frac{4}{3} \pi r_{1}^{3}} \text { and } \rho_{2}=\frac{q}{\frac{4}{3} \pi r_{2}^{2}}, \text { and }{/tex}
{tex}\sigma_{1}=\frac{q}{4 \pi r_{1}^{2}} \text { and } \sigma_{2}=\frac{q}{4 \pi r_{1}^{2}}{/tex}
<script type="math/tex" id="MathJax-Element-11">\sigma_2 = \frac{q}{4\pi r_1^2}.</script>
It is given that the ratio of the volume density is <nobr aria-hidden="true">8:64.</nobr>
{tex}\Rightarrow \quad \frac{\rho_{1}}{\rho_{2}}=\frac{8}{64} \quad \Rightarrow \quad \frac{\frac{3 q}{4 \pi_{1}^{3}}}{\frac{3 q}{4 m_{2}^{3}}}=\frac{8}{64}{/tex}
{tex}\Rightarrow \quad\left(\frac{n}{n}\right)^{3}=\frac{8}{\sqrt{4}} \quad \Rightarrow \quad \frac{n}{n}=\frac{1}{2} \quad \Rightarrow \quad\left(\frac{n}{n}\right)^{2}=\frac{1}{4}{/tex}
{tex}\Rightarrow{/tex} The ratio of the surface density of charge is 1 : 4
<script type="math/tex" id="MathJax-Element-15">\Rightarrow \qquad \frac{\sigma_1}{\sigma_2} = \frac{\frac{q}{4\pi r_1^2}}{\frac{q}{4\pi r_2^2}} = \left(\frac{r_2}{r_1}\right)^2 = \frac{1}{4}.</script>
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