+Q charge situate at center of equilateral triangle of side 'a'. If +q charge situate at each corner of triangle. Then find the direction and magnitude of resultant force.
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To find the direction and magnitude of the resultant force acting on the charge placed at the center of an equilateral triangle, with charges at each corner, we can use the principle of superposition.
First, let's calculate the magnitude of the resultant force.
Each of the three charges at the corners of the equilateral triangle exerts an electrostatic force on the charge at the center. The magnitude of this force can be calculated using Coulomb's Law:
F = k * (q1 * q2) / r^2
Here, k is the electrostatic constant (9 x 10^9 Nm^2/C^2), q1 and q2 are the magnitudes of the charges, and r is the distance between the charges.
Since the charges at each corner are of magnitude +q, the magnitude of this force on the charge at the center is:
F1 = k * (q * q) / a^2 -- (1)
Since the triangle is equilateral, the distance between the charge at the center and each corner is equal to the length of the sides of the triangle (side 'a').
Let's define the direction facing upward from the charge at the center as the positive y-axis. The electrostatic force acting on the charge at the center from any corner will be radially inward along the line connecting the center and the corner.
Considering the symmetry of the triangle, we can conclude that the forces from any two opposite corners will cancel each other out in the x-axis component, while the forces from all three corners will cancel each other out in the y-axis component.
Therefore, the resultant force acting on the charge at the center will have only a y-component.
The total y-component of the resultant force (Fr) can be found by taking the sum of the y-components of the individual forces:
Fr = F1 * sin(60 degrees) + F1 * sin(60 degrees) + F1 * sin(60 degrees)
Since sin(60 degrees) = √(3) / 2, the equation becomes:
Fr = 3 * F1 * √(3) / 2
Substituting the value of F1 from equation (1), we get:
Fr = 3 * (k * q^2 / a^2) * √(3) / 2
Simplifying further:
Fr = (3 * √(3) * k * q^2) / (2 * a^2)
So, the magnitude of the resultant force acting on the charge at the center of the equilateral triangle is (3 * √(3) * k * q^2) / (2 * a^2).
The direction of the resultant force is upward along the positive y-axis since all the individual forces acting on the charge at the center are inward and cancel each other out in the x-axis.
To summarize:
- Magnitude of the resultant force: (3 * √(3) * k * q^2) / (2 * a^2)
- Direction of the resultant force: Positive y-axis, pointing upward.