1000 small water drops each of radius r and charge q coalesces together to form one big spherical drop. The potential of the big drop is larger than that of the smaller drop by a factor of:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 100
Two things are conserved when the drops coalesce: the total charge and the total volume. The potential at the surface of a sphere of radius R and charge Q is \( V = \frac{1}{4\pi\varepsilon_0}\frac{Q}{R} \).
Radius of the big drop (volume conserved)
\( \displaystyle \frac{4}{3}\pi R^{3} = 1000\times\frac{4}{3}\pi r^{3} \Rightarrow R = 10r \)
Charge of the big drop: \( Q = 1000q \).
Ratio of potentials
\( \displaystyle \frac{V_{\mathrm{big}}}{V_{\mathrm{small}}} = \frac{Q/R}{q/r} = \frac{1000q/10r}{q/r} = 100 \)
Shortcut: For n identical drops \( Q = nq \) and \( R = n^{1/3}r \), so \( V\propto n/n^{1/3} = n^{2/3} \). For \( n = 1000 \) this is \( 1000^{2/3} = 100 \).