There particles each with charge Q or Q form an equilateral

There particles, each with charge Q or -Q, form an equilateral triangle, with each side of length a as seen at right. a) What is the net electric field produced at the midpointt of the bottom side of the triangle? b) What is the electric field at the top charge due to the bottom two charges? c) What is the net force on the top charge due to the bottom two charges?

Solution

a. the height of the equilateral triangle is,

       h = 3a/2

the electric field due to other ccharges at midpoint of the bottom side charges is,

E = E1 - E2 +E3

     = kQ/(a/2)² + kQ/(a/2)² - kQ/(3a/2)²

     = 8kQ/a² - 4kQ/3a²

     = kQ/a² [ 8- (4/3)]

     = 20kQ/3a²

     = (1/4*pi*₀)20Q/3a²

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b.

the net electric field at the top charge along the x-axis is,

   Ex = E₁cos60 + E₂cos60

        = [kQ/a²]cos60 + [kQ/a²] E2cos60

        = 2 [kQ/a²]cos60

        = kQ/a²

the net electric field along the y-direction is,

      Ey = E₁sin60 - E₂sin60 = 0

therefore, the electric field at the top charge is,

     E = kQ/a²

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c.

the force is calculated as follows:

   F = QE = kQ²/a²