The gravitational potential at the centre of a square of side a when four point masses m each are kept at its vertices will be

Options

A. `4sqrt(2)(Gm)/(a)`
B. `-4sqrt(2)(Gm)/(a)`
C. `2sqrt(2)(Gm)/(a)`
D. `-2sqrt(2)(Gm)/(a)`

Answer

To find the gravitational potential at the center of a square of side \( a \) when four point masses \( m \) are placed at its vertices, we can follow these steps:

### Step 1: Understanding Gravitational Potential
The gravitational potential \( V \) at a point due to a point mass \( m \) is given by the formula:
\[ V = -\frac{Gm}{r} \]
where \( G \) is the gravitational constant and \( r \) is the distance from the mass to the point where the potential is being calculated.

### Step 2: Calculate the Distance from Each Mass to the Center
For a square of side \( a \), the distance from each vertex (where the masses are located) to the center of the square can be calculated using the Pythagorean theorem. The center of the square is at coordinates \( \left(\frac{a}{2}, \frac{a}{2}\right) \), and the vertices are at \( (0, 0) \), \( (a, 0) \), \( (0, a) \), and \( (a, a) \).

The distance \( r \) from any vertex to the center is:
\[ r = \sqrt{\left(\frac{a}{2}\right)^2 + \left(\frac{a}{2}\right)^2} = \sqrt{\frac{a^2}{4} + \frac{a^2}{4}} = \sqrt{\frac{a^2}{2}} = \frac{a}{\sqrt{2}} \]

### Step 3: Calculate the Gravitational Potential from Each Mass
Since there are four masses \( m \) located at the vertices, the gravitational potential at the center due to each mass is:
\[ V_i = -\frac{Gm}{\frac{a}{\sqrt{2}}} = -\frac{Gm \sqrt{2}}{a} \]

### Step 4: Total Gravitational Potential at the Center
The total gravitational potential \( V \) at the center due to all four masses is the sum of the potentials from each mass:
\[ V = 4 \times V_i = 4 \left(-\frac{Gm \sqrt{2}}{a}\right) = -\frac{4Gm \sqrt{2}}{a} \]

### Final Answer
Thus, the gravitational potential at the center of the square is:
\[ V = -\frac{4Gm \sqrt{2}}{a} \]

---