Question 77: Electrostats & Capacitance

Question

If 216 drops of the same size are charged at $$200 \mathrm{~V}$$ each and they combine to form a bigger drop, the potential of the bigger drop will be

Accepted Answer

To determine the potential of the bigger drop formed by the combination of 216 smaller drops, we need to consider the following principles of electrostatics and geometry:
1.
Volume Conservation:
The total volume of the bigger drop will be equal to the sum of the volumes of the 216 smaller drops. If each smaller drop has a radius $$r$$, then the volume of one small drop is:
$$V_{	ext{small}} = \dfrac{4}{3} \pi r^3$$
Since there are 216 such drops, the total volume becomes:
$$V_{	ext{total}} = 216 	imes \dfrac{4}{3} \pi r^3$$
Let $$R$$ be the radius of the bigger drop. The volume of the bigger drop is:
$$V_{	ext{big}} = \dfrac{4}{3} \pi R^3$$
By volume conservation, we have:
$$\dfrac{4}{3} \pi R^3 = 216 	imes \dfrac{4}{3} \pi r^3$$
or,
$$R^3 = 216 r^3$$
Therefore,
$$R = 6r$$
2.
Charge Conservation:
The total charge on the bigger drop will be the sum of the charges on all the smaller drops. If each smaller drop has a potential of $$200 \mathrm{~V}$$ and radius $$r$$, then its charge $$q$$ can be given by:
$$V_{	ext{small}} = \dfrac{k q}{r}$$
Given $$V_{	ext{small}} = 200 \mathrm{~V}$$, we get:
$$q = \dfrac{200 r}{k}$$
For 216 such drops, the total charge $$Q_{	ext{total}}$$ is:
$$Q_{	ext{total}} = 216 	imes \dfrac{200 r}{k} = \dfrac{43200 r}{k}$$
3.
Potential of the Bigger Drop:
The potential of the bigger drop can be calculated using its charge and radius. The formula for the potential of a spherical drop is:
$$V_{	ext{big}} = \dfrac{k Q_{	ext{total}}}{R}$$
Substitute the values:
$$V_{	ext{big}} = \dfrac{k \dfrac{43200 r}{k}}{6r} = \dfrac{43200 r}{6r} = 7200 \mathrm{~V}$$
Thus, the potential of the bigger drop is $$7200 \mathrm{~V}$$.
Answer:
Option B: 7200 V

Answer

2400 V

Answer

7200 V

Answer

1200 V

Answer

8200 V

Electrostats & Capacitance

If 216 drops of the same size are charged at $200 \mathrm{~V}$ each and they combine to form a bigger drop, the potential of the bigger drop will be

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