If identical charges `(-q)` are placed at each corner of a cube of side `b`, then electric potential energy of charge `(+q)` which is palced at centre of the cube will be

Options

A. `(8sqrt(2)q^(2))/(4pi epsilon_(0)b)`
B. `(-8sqrt(2)q^(2))/(pi epsilon_(0)b)`
C. `(-4sqrt(2)q^(2))/(pi epsilon_(0)b)`
D. `(-4q^(2))/(sqrt(3)pi epsilon_(0)b)`

Explanation

To find the electric potential energy of a charge \( +q \) placed at the center of a cube with identical charges \( -q \) at each corner, we can follow these steps:

### Step 1: Understand the Setup
We have a cube with side length \( b \) and a charge \( +q \) placed at the center of the cube. Each corner of the cube has a charge \( -q \).

### Step 2: Calculate the Distance from the Center to a Corner
The distance from the center of the cube to any corner can be calculated using the diagonal of the cube. The formula for the length of the diagonal \( d \) of a cube with side length \( b \) is given by:
\[ d = b\sqrt{3} \]
Since we need the distance from the center to a corner, we take half of this diagonal:
\[ L = \frac{d}{2} = \frac{b\sqrt{3}}{2} \]

### Step 3: Calculate the Electric Potential Energy
The electric potential energy \( U \) of the charge \( +q \) due to one charge \( -q \) is given by the formula:
\[ U = k \frac{q_1 q_2}{r} \]
where \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the magnitudes of the charges, and \( r \) is the distance between them.

Since there are 8 identical charges \( -q \) at the corners, the total potential energy \( U \) due to all these charges is:
\[ U = \sum_{i=1}^{8} k \frac{(+q)(-q)}{L} \]
This simplifies to:
\[ U = 8 \left( k \frac{(+q)(-q)}{L} \right) = -8k \frac{q^2}{L} \]

### Step 4: Substitute the Value of \( L \)
Now substitute \( L = \frac{b\sqrt{3}}{2} \) into the equation:
\[ U = -8k \frac{q^2}{\frac{b\sqrt{3}}{2}} = -8k \frac{2q^2}{b\sqrt{3}} = -\frac{16kq^2}{b\sqrt{3}} \]

### Step 5: Substitute the Value of \( k \)
Coulomb's constant \( k \) can be expressed as:
\[ k = \frac{1}{4\pi \epsilon_0} \]
Substituting this into the equation gives:
\[ U = -\frac{16 \cdot \frac{1}{4\pi \epsilon_0} \cdot q^2}{b\sqrt{3}} = -\frac{4q^2}{\pi \epsilon_0 b\sqrt{3}} \]

### Final Answer
Thus, the electric potential energy of the charge \( +q \) placed at the center of the cube is:
\[ U = -\frac{4q^2}{\pi \epsilon_0 b\sqrt{3}} \]