Solved Problem on Electric Field
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The ratio between the electric charges of two spheres is 3/4, the ratio between its radii is 5/8. Determine the ratio between the densities of electric charge.
Problem data:
- Ratio between charges of the spheres: \( \dfrac{q_{1}}{q_{2}}=\dfrac{3}{4} \);
- Ratio between the radii of the spheres: \( \dfrac{r_{1}}{r_{2}}=\dfrac{5}{8} \).
The surface charge density σ is given by
\[
\begin{gather}
\bbox[#99CCFF,10px]
{\sigma =\frac{Q}{S}} \tag{I}
\end{gather}
\]
where Q is the charge of the body and S the area of its surface, from Geometry,
the area of a sphere is given by
\[
\begin{gather}
\bbox[#99CCFF,10px]
{S=4\pi r^{2}} \tag{II}
\end{gather}
\]
Substituting the expression (II) into (I) and writing the expression for the density of charges of
the spheres 1 and 2, we have
\[
\begin{gather}
\sigma_{1}=\frac{q_{1}}{4\pi r_{1}^{2}} \tag{III-a}
\end{gather}
\]
\[
\begin{gather}
\sigma_{2}=\frac{q_{2}}{4\pi r_{2}^{2}} \tag{III-b}
\end{gather}
\]
To find the ratio between the densities of electric charge, we divide the expression (III-a) by
(III-B)
\[
\begin{gather}
\frac{\sigma_{1}}{\sigma_{2}}=\frac{\dfrac{q_{1}}{\cancel{4\pi} r_{1}^{2}}}{\dfrac{q_{2}}{\cancel{4\pi} r_{2}^{2}}}\\
\frac{\sigma_{1}}{\sigma_{2}}=\frac{q_{1}}{r_{1}^{2}}\frac{r_{2}^{2}}{q_{2}} \tag{IV}
\end{gather}
\]
With the problem data, we can write the charge and radius of sphere 1 as a function of charge and
radius of sphere 2
\[
\begin{gather}
\frac{q_{1}}{q_{2}}=\frac{3}{4}\\
q_{1}=\frac{3}{4}q_{2} \tag{V}
\end{gather}
\]
\[
\begin{gather}
\frac{r_{1}}{r_{2}}=\frac{5}{8}\\
r_{1}=\frac{5}{8}r_{2} \tag{VI}
\end{gather}
\]
substituting expressions (V) and (VI) into the expression (IV), we obtain
\[
\begin{gather}
\frac{\sigma_{1}}{\sigma_{2}}=\frac{\dfrac{3}{4}q_{2}}{\left(\dfrac{5}{8}r_{2}\right)^{2}}\frac{r_{2}^{2}}{q_{2}}\\[5pt]
\frac{\sigma_{1}}{\sigma_{2}}=\frac{\dfrac{3}{4}\cancel{q_{2}}}{\dfrac{25}{64}\cancel{r_{2}^{2}}}\;\frac{\cancel{r_{2}^{2}}}{\cancel{q_{2}}}\\[5pt]
\frac{\sigma_{1}}{\sigma_{2}}=\frac{\dfrac{3}{4}}{\dfrac{25}{64}}\\[5pt]
\frac{\sigma_{1}}{\sigma_{2}}=\frac{3}{4}\times \frac{64}{25}
\end{gather}
\]
\[ \bbox[#FFCCCC,10px]
{\frac{\sigma_{1}}{\sigma_{2}}=\frac{48}{25}}
\]
Note: As
\( \frac{\sigma_{1}}{\sigma_{2}}=1,92 \Rightarrow \sigma_{1}=1,92 \sigma_{2} \)
this means that the density of charge in sphere 1 is 1.92 times higher than the density of charge
of the sphere 2.
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Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .