Calculating Electric Field Intensity at the Centroid of an Equilateral Triangle

In this guide, we will explore the process of calculating the electric field intensity at the centroid of an equilateral triangle when charges are placed at each corner. Understanding the electric field is crucial in various scientific and engineering applications, particularly in electrostatics.

Step-by-Step Calculation

Let us consider an equilateral triangle ABC, where each vertex has a charge q. We are interested in finding the electric field intensity at the centroid O of the triangle. The following steps outline the necessary calculations:

Step 1: Understand the Configuration

Let A, B, and C be the vertices of the equilateral triangle, and let O be the centroid. The distance d from the centroid to any vertex is given by:

``` d frac{a}{sqrt{3}} ```

where a is the side length of the equilateral triangle.

Step 2: Calculate the Electric Field Due to One Charge

The electric field E due to a point charge q at a distance r is given by:

``` E frac{k cdot q}{r^2} ```

In our case, r d frac{a}{sqrt{3}}. Therefore, the electric field at O due to one charge q is:

``` E_A frac{k cdot q}{left(frac{a}{sqrt{3}}right)^2} frac{k cdot q cdot 3}{a^2} ```

Step 3: Determine the Direction of the Electric Fields

Each electric field E_A, E_B, and E_C due to the charges at points A, B, and C will point away from the charges.

Step 4: Resolve the Electric Fields into Components

Due to the symmetry of the equilateral triangle, the horizontal components (x-components) will cancel out, and we only need to consider the vertical components (y-components).

The y-component of the electric field due to one charge is given by:

``` E_{Ay} E_A cdot sin 60^circ E_A cdot frac{sqrt{3}}{2} ```

Therefore, the total y-component of the electric field at O due to all three charges is:

``` E_{total} 3 cdot E_{Ay} 3 cdot left(frac{k cdot q cdot 3}{a^2} cdot frac{sqrt{3}}{2}right) ```

Finally, combining everything together, we obtain:

``` E_{total} frac{9kqsqrt{3}}{2a^2} ```

Final Expression for Electric Field Intensity at O

The electric field intensity at the centroid O of the equilateral triangle ABC, with charges q at each corner, is:

``` E_{total} frac{9kqsqrt{3}}{2a^2} ```

This electric field points upwards away from the triangle.

Conclusion

In summary, by following the steps outlined in this guide, we have derived the expression for the electric field intensity at the centroid of an equilateral triangle when charges are placed at each corner. This result is valuable for a range of electrostatics applications, from theoretical physics to practical engineering problems.