three point charges each of magnitude +q are located at the vertices of an equilateral triangle of side L. what magnitude of charge should be placed at the centroid of triangle to keep all the three charges in equilibrium?
To find the magnitude of the charge to be placed at the centroid of the triangle, we need to consider the concept of equilibrium in an electrostatic system.
In an electrostatic equilibrium, the net force acting on any charge in the system should be zero. This means that the sum of the forces due to the other charges must cancel out.
Let's denote the charge at the centroid as Q. According to Coulomb's Law, the electrostatic force between two point charges q1 and q2 separated by a distance r is given by:
F = (k * q1 * q2) / r^2
where k is the electrostatic constant. In this case, we can consider the forces acting on each charge individually.
Charge 1 (at the centroid) experiences forces from Charges 2 and 3. If Charges 2 and 3 are symmetrical to each other, the forces will have the same magnitude but opposite directions. The net force on Charge 1 will be zero only if the magnitudes of the charges are equal, i.e., |q2| = |q3|.
Similarly, Charge 2 experiences forces from Charges 1 and 3, and Charge 3 experiences forces from Charges 1 and 2. For equilibrium, we also need |q1| = |q3| and |q1| = |q2|.
Since all three charges have the same magnitude (+q), we can denote q1 = q2 = q3 = q.
Now, we need to calculate the magnitude of the centroid charge (Q) such that the net force on this charge is zero.
For Charges 2 and 3, the forces due to Charge 1 can be resolved into horizontal and vertical components. The vertical components will cancel each other, while the horizontal components will add up to zero because the triangle is equilateral.
Using trigonometry, we can determine that the horizontal component of the force will be given by F_horizontal = (k * q^2) / (4 * L^2 * sin(60)).
With F_horizontal = 0, we have:
(k * q^2) / (4 * L^2 * sin(60)) = 0.
sin(60) = sqrt(3)/2, so we can simplify the equation to:
q^2 = 0.
Therefore, the net force acting on the centroid charge is zero regardless of its magnitude. This means that the magnitude of the charge at the centroid of the triangle could be anything and still keep all three charges in equilibrium.