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Let's analyze the problem.

We have three charges in a triangular arrangement, where:
- The two bottom charges are negative, \(-q = -3.1 \, \text{nC}\), and are positioned symmetrically around point A, separated by a distance \(d = 12.0 \, \text{mm}\).
- The top charge is positive, \(+q = +3.1 \, \text{nC}\), located a distance \(2d\) above point A.
  
We need to find the electric field at point A.

### Steps:
1. **Electric Field Due to a Point Charge**:
   The electric field at a distance \(r\) from a point charge \(q\) is given by:
   \[
   E = \frac{k |q|}{r^2}
   \]
   where \(k = 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2\).

2. **Electric Fields Due to Negative Charges**:
   Both negative charges, being symmetrically placed, will create electric fields that point toward each charge. At point A (the midpoint between the two charges), the horizontal components of these fields will cancel each other out, while the vertical components will add up.

   The distance from each negative charge to point A is \(d = 12.0 \, \text{mm} = 0.012 \, \text{m}\).

   The electric field due to each negative charge is:
   \[
   E_{-q} = \frac{k |q|}{d^2}
   \]
   where \(|q| = 3.1 \, \text{nC} = 3.1 \times 10^{-9} \, \text{C}\).

3. **Resulting Electric Field from the Two Negative Charges**:
   Since only the vertical components add up:
   \[
   E_{\text{total, negative}} = 2E_{-q} \sin(\theta)
   \]
   where \(\sin(\theta)\) is determined by the geometry of the right triangle with hypotenuse \(d\) and vertical height equal to \(d/2\), giving \(\sin(\theta) = 1/2\).

4. **Electric Field Due to the Positive Charge**:
   The electric field from the positive charge at point A will point downward, directly opposite the net electric field from the two negative charges. The distance to point A from the positive charge is \(2d = 2 \times 0.012 = 0.024 \, \text{m}\).

   The electric field due to the positive charge is:
   \[
   E_{+q} = \frac{k |q|}{(2d)^2}
   \]

5. **Net Electric Field**:
   The total electric field at point A will be the vector sum of the contributions from the negative and positive charges. Since the fields from the negative charges and the positive charge are in opposite directions, we subtract the electric field from the positive charge from the total field of the two negative charges:
   \[
   E_{\text{net}} = E_{\text{total, negative}} - E_{+q}
   \]

Let me calculate these values.The net electric field at point A is approximately \( 1.45 \times 10^5 \, \text{N/C} \), which does not match any of the provided answer choices exactly. However, the closest option is:

- \( 5.2 \times 10^4 \, \text{N/C} \), which is answer **(c)**.

This discrepancy might arise from a possible rounding or approximation in the problem, but based on the calculations, the correct magnitude is approximately \( 1.45 \times 10^5 \, \text{N/C} \).

Would you like more details on any of the steps, or have any questions? Here are some related questions for further exploration:

1. How does the direction of the electric field change if the charge values are swapped?
2. What would happen to the electric field at point A if the distance \( d \) is doubled?
3. How would you calculate the electric potential at point A instead of the electric field?
4. What happens to the net electric field if one of the negative charges is removed?
5. How does the magnitude of the electric field depend on the distance from the charges?

**Tip**: Always remember that the superposition principle helps simplify the computation of electric fields by allowing vector addition of individual contributions.

In the figure below, all the charges are point charges of magnitude 3.1 nC, and the base of the triangle has a value of d = 12.0 mm. What is the magnitude of the electric field at point A (halfway between the two negative charges)?

Calculate the magnitude of the electric field at a point between charges of 3.1 nC arranged in a triangular configuration with a base of 12 mm. Explore the use of Coulomb's Law and the Superposition Principle for vector addition of electric fields.

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