Each of the four corners of a square with edge a is occupied by a point mass m. There is a fifth mass, also m, at the center of the square. To remove the mass from the center to a point far away the work that must be done by an external agent is given by:

Options

A. `4Gm^2//a`
B. `-4Gm^2//a`
C. `4sqrt2Gm^2//a`
D. `-4sqrt2Gm^2//a`

Explanation

To solve the problem of calculating the work done by an external agent to remove the mass from the center of a square to a point far away, we can follow these steps:

### Step 1: Understand the Configuration
We have a square with side length \( a \) and point masses \( m \) at each of the four corners (let's label them A, B, C, and D) and one mass \( m \) at the center (O) of the square.

### Step 2: Calculate the Distance from the Center to the Corners
The distance from the center (O) to any corner (e.g., A) can be calculated using the Pythagorean theorem. The diagonal of the square is given by:
\[ d = \sqrt{a^2 + a^2} = a\sqrt{2} \]
Since O is the midpoint of the diagonal, the distance from O to A (or any corner) is:
\[ r = \frac{a\sqrt{2}}{2} = \frac{a}{\sqrt{2}} \]

### Step 3: Calculate the Initial Potential Energy
The gravitational potential energy \( U \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by:
\[ U = -\frac{G m_1 m_2}{r} \]
In our case, the mass at the center (O) interacts with each of the four corner masses (A, B, C, D). The initial potential energy \( U_{\text{initial}} \) when the mass is at the center is:
\[ U_{\text{initial}} = U_{OA} + U_{OB} + U_{OC} + U_{OD} \]
Since all distances are equal, we can express this as:
\[ U_{\text{initial}} = 4 \left(-\frac{G m^2}{\frac{a}{\sqrt{2}}}\right) = -\frac{4G m^2 \sqrt{2}}{a} \]

### Step 4: Calculate the Final Potential Energy
When the mass is moved to a point far away (at infinity), the gravitational potential energy approaches zero:
\[ U_{\text{final}} = 0 \]

### Step 5: Calculate the Work Done
The work done by the external agent is equal to the change in potential energy:
\[ W = U_{\text{final}} - U_{\text{initial}} = 0 - \left(-\frac{4G m^2 \sqrt{2}}{a}\right) = \frac{4G m^2 \sqrt{2}}{a} \]

### Final Answer
Thus, the work done by the external agent to remove the mass from the center to a point far away is:
\[ W = \frac{4G m^2 \sqrt{2}}{a} \]