Two particles, each of mass m, are a distance d apart. To bring a third particle, also having mass m, from far away to the point midway between the two particles an external agent does work given by:

Options

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

Explanation

To solve the problem, we need to calculate the work done by an external agent to bring a third particle of mass \( m \) from far away to the midpoint between two other particles, each of mass \( m \), which are separated by a distance \( d \).

### Step-by-Step Solution:

1. **Identify the Initial Configuration**:
- We have two particles, each of mass \( m \), separated by a distance \( d \).
- The gravitational potential energy \( U \) between two masses is given by the formula:
\[ U = -\frac{G m_1 m_2}{r} \]
- For our two particles, the initial potential energy \( U_{\text{initial}} \) is:
\[ U_{\text{initial}} = -\frac{G m m}{d} = -\frac{G m^2}{d} \]

2. **Calculate the Final Configuration**:
- When the third particle of mass \( m \) is brought to the midpoint between the two particles, the distances from the third particle to each of the other two particles are \( \frac{d}{2} \).
- The potential energy contributions from the third particle to each of the two particles are:
\[ U_{1} = -\frac{G m m}{\frac{d}{2}} = -\frac{2G m^2}{d} \]
- Since there are two such contributions (one from each particle), the total potential energy \( U_{\text{final}} \) when the third particle is at the midpoint is:
\[ U_{\text{final}} = -\frac{G m^2}{d} + U_{1} + U_{2} = -\frac{G m^2}{d} - \frac{2G m^2}{d} = -\frac{5G m^2}{d} \]

3. **Calculate the Work Done by the External Agent**:
- The work done \( W \) by the external agent is equal to the change in potential energy:
\[ W = U_{\text{final}} - U_{\text{initial}} \]
- Substituting the values we found:
\[ W = -\frac{5G m^2}{d} - \left(-\frac{G m^2}{d}\right) = -\frac{5G m^2}{d} + \frac{G m^2}{d} = -\frac{4G m^2}{d} \]

### Final Answer:
The work done by the external agent is:
\[ W = -\frac{4G m^2}{d} \]