Three charges `2q, -q, and -q` are located at the vertices of an equilateral triangle. At the center of the triangle,

Options

A. the field is zero but potential is non-zero
B. the field is non-zero but potential is zero
C. Both field and potential are zero
D. Both field and potential are non-zero

Answer

To solve the problem of finding the electric field and potential at the center of an equilateral triangle formed by three charges \(2q\), \(-q\), and \(-q\), we can follow these steps:

### Step 1: Understand the Configuration
We have three charges located at the vertices of an equilateral triangle. Let's denote the vertices as \(A\), \(B\), and \(C\):
- Charge at vertex \(A\) = \(2q\)
- Charge at vertex \(B\) = \(-q\)
- Charge at vertex \(C\) = \(-q\)

### Step 2: Determine the Center of the Triangle
The center of an equilateral triangle (also known as the centroid) is the point where the medians intersect. This point is equidistant from all three vertices.

### Step 3: Calculate the Electric Field at the Center
The electric field \(E\) due to a point charge is given by the formula:
\[ E = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r^2} \]
where \(Q\) is the charge and \(r\) is the distance from the charge to the point where the field is being calculated.

1. **Electric Field due to Charge \(2q\)**:
- The electric field at the center \(O\) due to charge \(2q\) will point away from the charge since it is positive.

2. **Electric Field due to Charge \(-q\) at Vertex \(B\)**:
- The electric field at \(O\) due to charge \(-q\) will point towards the charge because it is negative.

3. **Electric Field due to Charge \(-q\) at Vertex \(C\)**:
- Similar to charge at \(B\), the electric field at \(O\) due to charge \(-q\) will also point towards the charge.

Since the two \(-q\) charges are symmetrically located with respect to the center, their electric fields will have components that cancel each other in the horizontal direction (if we assume the triangle is oriented with one side horizontal). The vertical components will add up.

### Step 4: Resultant Electric Field
The resultant electric field at the center \(O\) will not be zero because the contributions from the two \(-q\) charges will add up, while the contribution from \(2q\) will also point in the same direction. Therefore, the electric field at the center is non-zero.

### Step 5: Calculate the Electric Potential at the Center
The electric potential \(V\) due to a point charge is given by:
\[ V = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r} \]

1. **Potential due to Charge \(2q\)**:
\[ V_{2q} = \frac{1}{4 \pi \epsilon_0} \frac{2q}{r} \]

2. **Potential due to Charge \(-q\) at Vertex \(B\)**:
\[ V_{-q} = \frac{1}{4 \pi \epsilon_0} \frac{-q}{r} \]

3. **Potential due to Charge \(-q\) at Vertex \(C\)**:
\[ V_{-q} = \frac{1}{4 \pi \epsilon_0} \frac{-q}{r} \]

### Step 6: Total Electric Potential
The total potential \(V\) at the center \(O\) will be:
\[ V = V_{2q} + V_{-q} + V_{-q} = \frac{1}{4 \pi \epsilon_0} \left( \frac{2q}{r} - \frac{q}{r} - \frac{q}{r} \right) = \frac{1}{4 \pi \epsilon_0} \left( \frac{2q - q - q}{r} \right) = 0 \]

### Conclusion
- The electric field at the center of the triangle is **non-zero**.
- The electric potential at the center of the triangle is **zero**.

Thus, the correct answer is that the electric field is non-zero but the potential is zero.