Two negative charges each of unit magnitude and a positive charge q are placed along a straight line. At what value of q will the system be in equilibrium? Check whether it is stable, unstable or neutral equilibrium.
If q = 1/4 "units", and the positive charge q is placed midway between the negative charges, there will be stable equilibrium for the positive charge.
To find the value of q at which the system is in equilibrium, we need to consider the forces on each charge.
Let's assume that the two negative charges are located at positions A and B, and the positive charge q is located between them at position C.
Since like charges repel each other, the negative charge at A will exert a force away from it on q. Similarly, the negative charge at B will exert a force away from it on q. These forces will be in the same direction.
Now, let's analyze the forces on q:
1. The force exerted by the negative charge at A on q:
We can calculate this force using Coulomb's Law: F1 = k*q1*q/(r1)^2, where k is the electrostatic constant, q1 is the magnitude of the negative charge, q is the magnitude of the positive charge q, and r1 is the distance between A and C.
2. The force exerted by the negative charge at B on q:
We can calculate this force using Coulomb's Law: F2 = k*q2*q/(r2)^2, where q2 is the magnitude of the negative charge, and r2 is the distance between B and C.
For the system to be in equilibrium, the net force on q must be zero. In other words, F1 + F2 = 0.
Now, substitute the expressions for F1 and F2:
k*q1*q/(r1)^2 + k*q2*q/(r2)^2 = 0
Since the magnitudes q1, q2, r1, and r2 are given as unit magnitude and the electrostatic constant k is a constant, we can simplify the equation to:
q/(r1)^2 + q/(r2)^2 = 0
To determine the value of q at which the system is in equilibrium, we need to know the values of r1 and r2.
Now, let's analyze the stability of the equilibrium:
The system will be in stable equilibrium if any slight displacement of q from its initial position causes a restoring force that brings q back to its equilibrium position. This can be determined by analyzing the second derivative of the potential energy function.
If the second derivative is positive, then the equilibrium is stable. If it is negative, the equilibrium is unstable. If the second derivative is zero, the equilibrium is neutral.
To analyze the stability mathematically, we need to calculate the potential energy of the system and find its second derivative.
However, qualitatively, we can infer that the equilibrium will be unstable because the forces on q are in the same direction due to the like charges at A and B.
In conclusion, the system will be in equilibrium when the value of q is such that q/(r1)^2 + q/(r2)^2 = 0. However, this equilibrium is likely to be unstable. To determine the exact value of q and whether the equilibrium is stable, unstable, or neutral, you need to know the specific values of q, r1, and r2.