Question

What is the capacitance of a drop that results when two mercury spheres, each of radius R = 2.00 mmmerge?

Short answer

The capacitance is C' = 0.28 pF

Step-by-step solution

Given

The radius of the mercury sphere is$\mathrm{R}=2.00\mathrm{mm}\left(\frac{{10}^{-3}\mathrm{m}}{1\mathrm{mm}}\right)=2.00\times {10}^{-3}\mathrm{m}$

The two mercury spheres merge.

Determining the concept

Using the equation 25-18 and the formula for the volume of the drop, and assuming the conservation of volume, find the capacitance of the new combined drop.

Formulae are as follows:

$\mathbf{C}\mathbf{=}\mathbf{4}{\mathbf{\tau \tau \epsilon }}_{\mathbf{0}}\mathbf{R}$

Where C is capacitance, R is the radius, and${\mathbf{\epsilon }}_{\mathbf{0}}$ is the permittivity of the medium.

Determining the capacitance of a drop when two mercury spheres merge

By assuming the conservation of volume, find the radius of the combined sphere,

$\mathrm{C}=4{\mathrm{\tau \tau \epsilon }}_0\mathrm{R}$

When the drop combines the volume of the system becomes,

$\mathrm{V}=2\left(\frac{4\mathrm{\tau \tau }}{3}\right){\mathrm{R}}^3$

Thus, the new radiusis given by,

$\frac{4\mathrm{\tau \tau }}{3}{\left(\mathrm{R}\text{'}\right)}^3=2\left(\frac{4\mathrm{\tau \tau }}{3}\right){\mathrm{R}}^3$

This gives,

$\mathrm{R}\text{'}=\left(2^{1/3}\right)\mathrm{R}$

From the equation 25-18, the capacitance of the sphere is,

$\mathrm{C}=4{\mathrm{\tau \tau \epsilon }}_0\mathrm{R}$

Therefore, the new capacitance is,

$\mathrm{C}\text{'}=4{\mathrm{\tau \tau \epsilon }}_0\mathrm{R}\text{'}$

Therefore, this gives,

$$\begin{gathered} \mathrm{C}\text{'}=4{\mathrm{\tau \tau \epsilon }}_0\left(2^{1/3}\right)\mathrm{R}\text{'} \\ =5.04{\mathrm{\tau \tau \epsilon }}_0\mathrm{R} \\ =5.04\times 3.142\times \left(8.85\times {10}^{-12}\mathrm{F}/\mathrm{m}\right)\times \left(2.00\times 10-3\right)\mathrm{m} \\ =2.80\times {10}^{-13}\mathrm{F}\left(\frac{{10}^{12}\mathrm{pF}}{1\mathrm{F}}\right) \\ =0.28\mathrm{pF} \end{gathered}$$

Hence, the capacitance due to merged drop is 0.28 pF

Therefore, by using the formula for the volume of the drop, and assuming the conservation of volume, the capacitance of the new combined drop can be found.