Question 822: Electrostatics & Capacitance

Question

A conducting liquid bubble of radius $$a$$ and thickness $$t(t < < a)$$ is charged to potential V. If the bubble collapses to a droplet, find the potential on the droplet.

Accepted Answer

A liquid-bubble contains air inside and liquid 
in the outer surface. When the bubble collapses 
into a droplet, the volume of liquid remains 
same in this transformation.
Let thickness of liquid in surface = $$t$$
$$	herefore$$ Volume of liquid in bubble $$=4\pi a^2t$$
Volume of the droplet $$=\frac{4}{3}\pi R^3$$
$$	herefore~\frac{4}{3}\pi R^3=4\pi a^2t$$
$$R=(3a^2t)^{1/3}$$ ..... (i)
Let the potential of liquid-bubble = V
$$	herefore \mathrm{V}=\frac{+q}{4 \pi \varepsilon_{0}}$$ where $$q$$ denotes charge on the bubble.
$$	herefore$$ Potential of droplet $$ = {2 \over {4\pi {\varepsilon _0}R}}$$
Potential of droplet $$ = \left( {{{4\pi {\varepsilon _0}aV} \over {4\pi {\varepsilon _0}}}} ight)\left[ {{1 \over {{{(3{a^2}t)}^{1/3}}}}} ight]$$
Potential of droplet $$ = V\left[ {{a \over {{{(3{a^2}t)}^{1/3}}}}} ight]$$
Potential of droplet $$ = V{\left[ {{{{a^3}} \over {3{a^2}t}}} ight]^{1/3}}$$
Potential of droplet $$ = V{\left[ {{a \over {3t}}} ight]^{1/3}}$$

Answer

$$V{\left[ {{2a \over {3t}}} \right]^{1/3}}$$

Answer

$$V{\left[ {{7a \over {3t}}} \right]^{1/3}}$$

Answer

$$V{\left[ {{a \over {3t}}} \right]^{1/3}}$$

Answer

$$V{\left[ {{5a \over {9t}}} \right]^{1/3}}$$

Electrostatics & Capacitance

A conducting liquid bubble of radius $a and thickness t(t < < a)$ is charged to potential V. If the bubble collapses to a droplet, find the potential on the droplet.

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