Two charge each `Q` are released when the distance between is `d` .Then the velocity of each charge of mass `m` each when the distance between them is `2d` is

Options

A. `(Q)/(sqrt(8piepsilon_(0)dm))`
B. `(Q)/(sqrt(4piepsilon_(0)dm))`
C. `(Q)/(sqrt(piepsilon_(0)dm))`
D. `(Q)/(sqrt(2piepsilon_(0)dm))`

Explanation

To solve the problem of finding the velocity of each charge when the distance between them changes from \(d\) to \(2d\), we can use the principle of conservation of energy. Here's a step-by-step solution:

### Step 1: Understand the Initial and Final Conditions
- Initially, we have two charges \(Q\) separated by a distance \(d\).
- When released, they will experience a repulsive force due to their like charges.
- We need to find the velocity of each charge when the distance between them increases to \(2d\).

### Step 2: Calculate the Initial Potential Energy
The initial potential energy \(U_1\) when the charges are at distance \(d\) is given by the formula:
\[ U_1 = \frac{kQ^2}{d} \]
where \(k\) is Coulomb's constant.

### Step 3: Calculate the Final Potential Energy
The final potential energy \(U_2\) when the charges are at distance \(2d\) is:
\[ U_2 = \frac{kQ^2}{2d} \]

### Step 4: Apply Conservation of Energy
According to the conservation of energy, the loss in potential energy equals the gain in kinetic energy. Thus:
\[ \Delta U = U_1 - U_2 = K.E. \]
The change in potential energy is:
\[ \Delta U = \frac{kQ^2}{d} - \frac{kQ^2}{2d} = \frac{kQ^2}{d} \left(1 - \frac{1}{2}\right) = \frac{kQ^2}{2d} \]

### Step 5: Express the Kinetic Energy
The kinetic energy gained by each charge when they move apart is given by:
\[ K.E. = \frac{1}{2}mv^2 + \frac{1}{2}mv^2 = mv^2 \]
where \(v\) is the velocity of each charge.

### Step 6: Set Up the Equation
Equating the change in potential energy to the total kinetic energy:
\[ \frac{kQ^2}{2d} = mv^2 \]

### Step 7: Solve for Velocity
Rearranging the equation to solve for \(v\):
\[ v^2 = \frac{kQ^2}{2md} \]
Taking the square root gives:
\[ v = \sqrt{\frac{kQ^2}{2md}} \]

### Step 8: Substitute for \(k\)
Coulomb's constant \(k\) can be expressed in terms of \(\epsilon_0\) (the permittivity of free space):
\[ k = \frac{1}{4\pi\epsilon_0} \]
Substituting this into the equation for \(v\):
\[ v = \sqrt{\frac{(1/4\pi\epsilon_0)Q^2}{2md}} = \sqrt{\frac{Q^2}{8\pi\epsilon_0 md}} \]

### Final Result
Thus, the velocity of each charge when the distance between them is \(2d\) is:
\[ v = \frac{Q}{\sqrt{8\pi\epsilon_0 md}} \]