Three point charges, which initially are infinitely far apart, are placed at the corners of an equilateral triangle with sides d. Two of the point charges are identical and have charge q.If zero net work is required to place the three charges at the corners of the triangles, what must the value of the third charge be?

Explanation

To solve the problem of finding the value of the third charge when three point charges are placed at the corners of an equilateral triangle with sides \(d\) and zero net work is required to place them, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two identical charges \(q\) placed at two corners of an equilateral triangle, and we need to find the value of the third charge \(Q\) such that the total work done in bringing the charges from infinity to the corners of the triangle is zero.

2. **Initial Potential Energy**:
Initially, when the charges are infinitely far apart, the potential energy \(U_i\) is zero:
\[ U_i = 0 \]

3. **Final Potential Energy Calculation**:
When the charges are brought to the corners of the triangle, the final potential energy \(U_f\) can be calculated using the formula for potential energy between point charges:
\[ U_f = k \left( \frac{q \cdot q}{d} + \frac{q \cdot Q}{d} + \frac{Q \cdot q}{d} \right) \]
Here, \(k\) is Coulomb's constant.

4. **Expanding the Final Potential Energy**:
The expression simplifies to:
\[ U_f = k \left( \frac{q^2}{d} + \frac{qQ}{d} + \frac{qQ}{d} \right) = k \left( \frac{q^2}{d} + 2 \frac{qQ}{d} \right) \]
This can be rewritten as:
\[ U_f = \frac{k}{d} \left( q^2 + 2qQ \right) \]

5. **Setting Work Done to Zero**:
Since the work done \(W\) is equal to the change in potential energy, we set:
\[ W = U_f - U_i = U_f - 0 = U_f \]
For zero work done, we need:
\[ U_f = 0 \]
Thus, we have:
\[ \frac{k}{d} \left( q^2 + 2qQ \right) = 0 \]

6. **Solving for \(Q\)**:
Since \(k/d\) is not zero, we can set the term in parentheses to zero:
\[ q^2 + 2qQ = 0 \]
Rearranging gives:
\[ 2qQ = -q^2 \]
Dividing both sides by \(2q\) (assuming \(q \neq 0\)):
\[ Q = -\frac{q}{2} \]

7. **Final Result**:
The value of the third charge \(Q\) must be:
\[ Q = -2q \]

### Summary:
The third charge \(Q\) must be \(-2q\) for the net work required to place the three charges at the corners of the triangle to be zero.