Question

A spherical drop of capacitance 1uF is broken into eight drops of equal radius. The capacity of each small drop is

Answer 1

1/3pir3=1*10-6/8 the capacity of each small drop is

Answer 2

Each small drop will exhibit a capacitance of 0.5μF.

Explanation:

A spherical drop breaks into eight spherical drops of equal radius. So if we consider R as the radius of the big drop or the parent sphere and r as the radius of the newly formed drops or the daughter spheres, the volume of the original sphere tends to be equal to the volume of the eight drops.

Thus,

$\text { Volume of the parent sphere }=8 \times \text { Volume of daughter spheres }$

$\frac{4}{3} \pi R^{3}=8\left(\frac{4}{3} \pi r^{3}\right)$

$\Longrightarrow R^{3}=8 \times r^{3}$

$\Rightarrow R=2 r \rightarrow(1)$

We know that the capacitance of a sphere is,

$C=4 \pi \varepsilon_{0} \times(\text {radius of the sphere})$

So, the capacitance of parent sphere be C,

$C=4 \pi \varepsilon_{0} R \rightarrow(2)$

And capacitance of a daughter sphere be C’. As all the daughter spheres have same radius then finding the capacitance of a single daughter sphere will be equal for all the 7 daughter spheres.

$C^{\prime}=4 \pi \varepsilon_{0} r \rightarrow(3)$

Substitute eqn (1) in eqn (2)

$C=4 \pi \varepsilon_{0} \times 2 r$

$C=2 \times 4 \pi \varepsilon_{0} r$

From eqn (3), we can say that,

$C=2 \times C^{\prime}$

$\therefore C^{\prime}=\frac{C}{2}$

As we know that C=1 \mu F,

$C^{\prime}=\frac{1}{2}=0.5 \mu F$

Thus, each drop will exhibit a capacitance of $0.5\mu F$.