Sixty four identical sphere of change q and capacitance C each are combined to form a large sphere . The charge and capacitance of the large sphere is

Options

A. 64 q, C
B. 16 q, 4 C
C. 64 q, 4 C
D. 16 q , 64 C

Answer

To solve the problem of finding the charge and capacitance of a large sphere formed by combining 64 identical smaller spheres, we can follow these steps:

### Step 1: Understand the Given Information
We have 64 identical spheres, each with a charge \( q \) and capacitance \( C \). We need to find the charge and capacitance of the larger sphere formed by combining these smaller spheres.

### Step 2: Calculate the Total Charge
The total charge \( Q \) on the larger sphere can be calculated by using the principle of conservation of charge. Since there are 64 spheres, each with charge \( q \), the total charge is:
\[ Q = 64q \]

### Step 3: Calculate the Radius of the Large Sphere
Let the radius of each smaller sphere be \( r \). The volume of one smaller sphere is given by:
\[ V_{\text{small}} = \frac{4}{3} \pi r^3 \]
The total volume of the 64 smaller spheres is:
\[ V_{\text{total}} = 64 \times V_{\text{small}} = 64 \times \frac{4}{3} \pi r^3 = \frac{256}{3} \pi r^3 \]
This total volume is equal to the volume of the larger sphere with radius \( R \):
\[ V_{\text{large}} = \frac{4}{3} \pi R^3 \]
Setting the two volumes equal gives:
\[ \frac{256}{3} \pi r^3 = \frac{4}{3} \pi R^3 \]
Cancelling \( \frac{4}{3} \pi \) from both sides, we get:
\[ 256 r^3 = 4 R^3 \]
Dividing both sides by 4:
\[ 64 r^3 = R^3 \]
Taking the cube root of both sides:
\[ R = 4r \]

### Step 4: Calculate the Capacitance of the Large Sphere
The capacitance \( C \) of a sphere is given by:
\[ C = 4 \pi \epsilon_0 R \]
Substituting \( R = 4r \):
\[ C' = 4 \pi \epsilon_0 (4r) = 16 \pi \epsilon_0 r \]
Since the capacitance of the smaller sphere is \( C = 4 \pi \epsilon_0 r \), we can express \( C' \) in terms of \( C \):
\[ C' = 16 \pi \epsilon_0 r = 4C \]

### Final Results
- The charge of the large sphere is \( Q = 64q \).
- The capacitance of the large sphere is \( C' = 4C \).

### Summary
Thus, the charge and capacitance of the large sphere formed by combining 64 identical smaller spheres are:
- Charge: \( 64q \)
- Capacitance: \( 4C \)