Question: Q. 9. N spherical droplets, each of radius $r$, have been charged to have a potential $V$ each. If all these droplets were to coalesce to form a single large drop, what would be the potential of this large drop ?
(It is given that the capacitance of a sphere of radius $x$ equals $4 \pi \varepsilon_{0} k r$.)
A] [CBSE SQP 2013]
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Solution:
Ans. Total (initial) charge on all the droplets
$$ =N \times\left(4 \pi \varepsilon_{0} k r V\right) $$
Also
$$ N \times \frac{4}{3} \pi r^{3}=\frac{4}{3} \pi R^{3} $$
$$ \therefore \quad R=N^{1 / 3} r $$
If $V^{\prime}$ is the potential of the large drop, we have
$$ \begin{array}{rlrl} 4 \pi \varepsilon_{0} k R \times V^{\prime} & =N \times 4 \pi \varepsilon_{0} k r \times V & 1 / 2 \ \therefore & V^{\prime} & =\frac{N r}{R} \cdot V=N^{2 / 3} V & 1 / 2 \end{array} $$
[AI Q. 10. Find the IBF. associated with a charge $q$ if it were presenf the point $P$ with respect to the ‘setup’ of two charged spheres, arranged as shown. Here $O$ is the mid-point of the line $O_{1} O_{2}$.
A [CBSE SQP 2013]
Ans.
$$ \begin{aligned} & r_{1}=O_{1} P=\sqrt{r^{2}+(2 a+b)^{2}} \quad 1 / 2 \ & r_{2}=O_{2} P=\sqrt{r^{2}+(a+2 b)^{2}} \quad 1 / 2 \ & \therefore \quad V=\frac{1}{4 \pi \varepsilon_{0}}\left[\frac{Q_{1}}{r_{1}}+\frac{Q_{2}}{r_{2}}\right] \end{aligned} $$
$\therefore$ P.E. of charge, $q$, at $P=q V$
$=\frac{q}{4 \pi \varepsilon_{0}}\left[\frac{Q_{1}}{\left[r^{2}+(2 a+b)^{2}\right]^{1 / 2}}+\frac{Q_{2}}{\left[r^{2}+(a+2 b)^{2}\right]^{1 / 2}}\right]$
