Electric field due to charges located at the vertices of an equilateral triangle

Problem statement:

Three point charges q1, q2 and q3 lie at the vertices of an equilateral triangle with a side length a as shown in the figure below. Calculate:

  1. The electric field due to q1 and q2 at point P where q3 is located.
  2. The electric force experimented by q3.

Givens: q1 = q2 = 4 μC; q3 = -2 μC; a = 0.5 m; k = 9 109 Nm2/C2

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Solution:

First we are going to draw the electric field vectors due to each one of the point charges at point P.

To find the direction of the electric field vector due to a point charge at any point of space we perform a “thought experiment” that consist in placing a positive test charge at this point. The direction of the electric field is the direction of the force the positive test charge would experience.

A positive test charge located at point P would experience a repulsive force as q1 is positive, and therefore E1 goes outwards q1. Recall that positive charges are sources of electric field lines. We deduce the direction of vector E2 by performing the same “thought experiment” for q2. We deduce the net field E with the help of the parallelogram law.

As you can see in the previous figure, the electric field at point P depends only on the source charges q1 and q2 and not on the target charge we will place later at this location.

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The net field at point P is oriented towards the positive direction of the y-axis, because the horizontal components of E1 and E2 cancel out.

Furthermore, the vertical projections of E1 y E2 are equal (in light green in the figure), and their value is:

Therefore, the resultant of vectors E1 y E2 expressed in vector form is:

The magnitude of E1 given by Coulomb’s law is:

Where r1 is the distance between q1 and point P. In this problem, r1 is equal to the equilateral triangle side a. The angle α is equal to 600. After substituting them and the problem givens in the net field expression, we obtain the net field at point P:


When a test charge (in this case q3) is placed in a region of space where an electric field exists, the electrostatic force it experiments is given by:

You will find how to calculate the electric field at the centroid of an equilateral triangle in this problem.

You can see the names of the multiples and submultiples of the SI units in the units of measurement page.

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