Three particles, each having a charge of `10 muC`, are. placed at the vertices of an equilateral triangle of side. 10 cm. Find the work done by a person in pulling them apart to infinite separations.

Options

A. `27 J`
B. `2.7 J`
C. `-2.7 J`
D. `-27 J`

Explanation

To find the work done by a person in pulling apart three particles, each with a charge of \(10 \, \mu C\), placed at the vertices of an equilateral triangle with a side of \(10 \, cm\), we can follow these steps:

### Step 1: Understand the System
We have three identical charges \(Q = 10 \, \mu C = 10 \times 10^{-6} \, C\) placed at the vertices of an equilateral triangle with a side length \(R = 10 \, cm = 0.1 \, m\).

### Step 2: Calculate the Potential Energy of the System
The potential energy \(U\) of a system of point charges can be calculated using the formula:
\[ U = k \sum \frac{Q_i Q_j}{r_{ij}} \]
where \(k\) is Coulomb's constant (\(k = 9 \times 10^9 \, N \cdot m^2/C^2\)), \(Q_i\) and \(Q_j\) are the charges, and \(r_{ij}\) is the distance between the charges.

For three charges, the potential energy can be expressed as:
\[ U = U_{12} + U_{13} + U_{23} \]
where \(U_{ij} = k \frac{Q_i Q_j}{R}\).

Since all charges are equal, we can simplify this to:
\[ U = 3 \cdot k \frac{Q^2}{R} \]

### Step 3: Substitute the Values
Now we substitute the values into the equation:
\[ U = 3 \cdot \left(9 \times 10^9 \, N \cdot m^2/C^2\right) \cdot \frac{(10 \times 10^{-6} \, C)^2}{0.1 \, m} \]

Calculating \(Q^2\):
\[ Q^2 = (10 \times 10^{-6})^2 = 100 \times 10^{-12} = 10^{-10} \, C^2 \]

Now substituting:
\[ U = 3 \cdot 9 \times 10^9 \cdot \frac{10^{-10}}{0.1} \]
\[ = 3 \cdot 9 \times 10^9 \cdot 10^{-9} \]
\[ = 3 \cdot 9 = 27 \, J \]

### Step 4: Work Done in Pulling Apart the Charges
The work done by a person in pulling the charges apart to infinite separation is equal to the negative of the potential energy of the system, since the potential energy at infinity is zero:
\[ W = -U \]
Thus,
\[ W = -27 \, J \]

### Final Answer
The work done by a person in pulling the three charges apart to infinite separation is \(-27 \, J\).

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