Question

Three point charges, 1 pC each, are kept at the vertices of an equilateral triangle of side 10 cm. Find the net electric field at the centroid of the triangle.

Hint:

For symmetrically placed charges in an equilateral triangle, the resultant electric field at the centroid can be found by calculating the field due to one charge and using the symmetry to sum the components.

Answer 1

Given: Charges at the vertices: \( q_1 = q_2 = q_3 = 1 \, \text{pC} = 1 \times 10^{-12} \, \text{C} \)
Side of the equilateral triangle: \( a = 10 \, \text{cm} = 0.1 \, \text{m} \)

Step 1: Calculate the distance from the centroid to a vertex.
For an equilateral triangle, the distance (\( r \)) from the centroid to any vertex is:
\[ r = \frac{a}{\sqrt{3}} = \frac{0.1}{\sqrt{3}} \, \text{m} \] 

Step 2: Calculate the electric field due to each charge.
The electric field (\( E \)) due to a point charge is given by:
\[ E = \frac{k \cdot q}{r^2} \] where \( k = 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \). Electric fields due to charges at vertices:
1. **Electric field due to \( q_1 \)**:
\[ E_1 = \frac{9 \times 10^9 \cdot 1 \times 10^{-12}}{\left(\frac{0.1}{\sqrt{3}}\right)^2} \] \[ E_1 = 2.7 \, \text{N/C} \] The direction of \( E_1 \) is along the line from the centroid to \( q_1 \). 2. **Electric field due to \( q_2 \)**:
\[ E_2 = 2.7 \, \text{N/C} \] The direction of \( E_2 \) is along the line from the centroid to \( q_2 \). 3. **Electric field due to \( q_3 \)**:
\[ E_3 = 2.7 \, \text{N/C} \] The direction of \( E_3 \) is along the line from the centroid to \( q_3 \). Step 3: Resolve the electric fields into components.
Due to the symmetry of the equilateral triangle, the electric fields \( E_1 \), \( E_2 \), and \( E_3 \) are at \( 120^\circ \) to each other. The horizontal and vertical components of these fields will cancel out. Step 4: Calculate the net electric field.
Since the electric fields are symmetrically distributed and their components cancel out, the **net electric field at the centroid is zero**. Final Answer: \[ \boxed{\text{The net electric field at the centroid is } 0 \, \text{N/C.}} \]

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