Question

64 identical spheres each having charge q coalesce to form one big sphere. Find the ratio of the surface charge density of the big sphere to that of small sphere

Hint:

For $n$ coalescing identical drops, general relations are:
Radius: $R = n^{1/3}r$
Charge: $Q = nq$
Potential: $V = n^{2/3}v$
Surface charge density: $\sigma_{\text{big}} = n^{1/3} \sigma_{\text{small}}$ (Here $64^{1/3} = 4$).

Answer 1

Step 1: Understanding the Concept:
When smaller liquid drops or spheres coalesce (merge) into a larger one, two fundamental quantities are conserved:
1. Total charge: The charge of the big sphere is the sum of charges of small spheres.
2. Total volume: The volume of the big sphere is the sum of volumes of small spheres.
Surface charge density ($\sigma$) relates charge to surface area.

Step 2: Key Formula or Approach:
Volume of a sphere: $V = \frac{4}{3}\pi r^3$
Surface charge density: $\sigma = \frac{\text{Total Charge}}{\text{Surface Area}} = \frac{Q}{4\pi r^2}$
Use volume conservation to find the new radius, then compute the new density.

Step 3: Detailed Explanation:
Let $r$ and $q$ be the radius and charge of a small sphere.
Let $R$ and $Q$ be the radius and charge of the big sphere formed from $n=64$ small spheres.
1. Charge conservation:
Total charge $Q = 64 \times q$
2. Volume conservation:
Volume of big sphere = $64 \times$ Volume of small sphere
\[ \frac{4}{3}\pi R^3 = 64 \times \left(\frac{4}{3}\pi r^3\right) \]
\[ R^3 = 64 r^3 \]
Taking the cube root:
\[ R = \sqrt[3]{64} r = 4r \]
3. Surface charge density ratio:
For a small sphere: $\sigma_{\text{small}} = \frac{q}{4\pi r^2}$
For the big sphere: $\sigma_{\text{big}} = \frac{Q}{4\pi R^2}$
Substitute $Q = 64q$ and $R = 4r$:
\[ \sigma_{\text{big}} = \frac{64q}{4\pi (4r)^2} \]
\[ \sigma_{\text{big}} = \frac{64q}{4\pi (16r^2)} \]
\[ \sigma_{\text{big}} = \frac{64}{16} \left(\frac{q}{4\pi r^2}\right) \]
\[ \sigma_{\text{big}} = 4 \times \sigma_{\text{small}} \]
The ratio is $\frac{\sigma_{\text{big}}}{\sigma_{\text{small}}} = \frac{4}{1} = 4:1$.

Step 4: Final Answer:
The ratio is 4:1.