A conducting sphere of radius `R` and carrying a charge `Q` is joined to an uncharged conducting sphere of radius `2R`. The charge flowing between them will be

Options

A. `(1)/(4piepsilon_(0)).(Q^(2))/(4R)`
B. `(1)/(4pi epsilon_(0)).(Q^(2))/(2R)`
C. `(1)/(4pi epsilon_(0)).(Q^(2))/(R)`
D. `(2)/(2pi epsilon_(0)).(Q^(2))/(3R)`

Explanation

To solve the problem of charge flow between two conducting spheres, we will follow these steps:

### Step 1: Understand the Problem
We have two conducting spheres:
- Sphere 1 (radius = R, charge = Q)
- Sphere 2 (radius = 2R, charge = 0)

When these two spheres are connected, charge will flow until both spheres reach the same electric potential.

### Step 2: Write the Expression for Electric Potential
The electric potential \( V \) of a conducting sphere is given by:
\[ V = \frac{Q}{4\pi \epsilon_0 R} \]
where \( Q \) is the charge on the sphere and \( R \) is its radius.

For Sphere 1 (with charge \( Q_1 \)):
\[ V_1 = \frac{Q_1}{4\pi \epsilon_0 R} \]

For Sphere 2 (with charge \( Q_2 \)):
\[ V_2 = \frac{Q_2}{4\pi \epsilon_0 (2R)} = \frac{Q_2}{8\pi \epsilon_0 R} \]

### Step 3: Set the Potentials Equal
Since the spheres will reach the same potential when connected, we set \( V_1 = V_2 \):
\[ \frac{Q_1}{4\pi \epsilon_0 R} = \frac{Q_2}{8\pi \epsilon_0 R} \]

### Step 4: Simplify the Equation
We can cancel \( 4\pi \epsilon_0 R \) from both sides:
\[ Q_1 = \frac{Q_2}{2} \]

### Step 5: Use Charge Conservation
The total charge before connecting the spheres is \( Q \) (from Sphere 1) and \( 0 \) (from Sphere 2). Thus, we have:
\[ Q_1 + Q_2 = Q \]

### Step 6: Substitute \( Q_1 \) in Terms of \( Q_2 \)
From the equation \( Q_1 = \frac{Q_2}{2} \), we substitute into the charge conservation equation:
\[ \frac{Q_2}{2} + Q_2 = Q \]
\[ \frac{3Q_2}{2} = Q \]

### Step 7: Solve for \( Q_2 \)
Multiplying both sides by \( \frac{2}{3} \):
\[ Q_2 = \frac{2Q}{3} \]

### Step 8: Find \( Q_1 \)
Now substitute \( Q_2 \) back to find \( Q_1 \):
\[ Q_1 = \frac{Q_2}{2} = \frac{1}{2} \cdot \frac{2Q}{3} = \frac{Q}{3} \]

### Step 9: Calculate Charge Flow
The charge that flows from Sphere 1 to Sphere 2 is:
\[ \text{Charge Flow} = Q - Q_1 = Q - \frac{Q}{3} = \frac{2Q}{3} \]

### Final Answer
The charge flowing between the two spheres is:
\[ \frac{2Q}{3} \]