2 PARTICLES [FREE TO MOVE] WITH CHARGES +q and +4q are a distance L APART. A 3RD CHARGE IS PLACED SO THAT THE ENTIRE SYSTEM IS IN EQ2UILIBRIUM.FIND THE LOCATION,MAGNITUDE AND SIGN OF THE 3RD CHARGE
The third charge must be placed between the q and 4q charges, so that there are equal and opposite forces along the line between the charges. The distance from the q charge must be half the distance from the 4q charge, so that Q/d^2 is the same for both.
Therefore, the third charge must be 1/3 of the way from q to 4q. That will be L/3 from charge q.
To find the location, magnitude, and sign of the third charge in order for the entire system to be in equilibrium, we can use the principle of electrostatics that states the net force on a charged object should be zero when it is in equilibrium.
Let's analyze the forces acting on the third charge, which we'll call q3.
1. Force due to charge +q: The force between the charges +q and q3 can be calculated using Coulomb's law: F1 = (k * |q3 * q|) / (L/2)^2, where k is the electrostatic constant.
2. Force due to charge +4q: The force between the charges +4q and q3 can be calculated using Coulomb's law: F2 = (k * |q3 * 4q|) / (L/2)^2.
In equilibrium, the net force acting on q3 should be zero. Therefore, the forces due to the charges +q and +4q must cancel each other out.
F1 + F2 = 0
Using the equations for F1 and F2 mentioned above, we can solve for q3.
(k * |q3 * q|) / (L/2)^2 + (k * |q3 * 4q|) / (L/2)^2 = 0
Simplifying the equation, we get:
q3/q + 4(q3/4q) = 0
q3/q + q3/q = 0
2q3/q = 0
2q3 = 0
q3 = 0
Thus, the magnitude of the third charge, q3, is 0. This means that the third charge has no charge (neutral). Since it does not have any charge, it doesn't matter where the third charge is located in the system.