To solve the problem of finding the position and nature of the charge Q that keeps the system in equilibrium, we start by considering the forces acting on Q due to the two identical point charges, q.
Understanding the Forces
Since the charges q are identical and separated by a distance of 2 m, they will exert equal and opposite forces on charge Q. For Q to be in equilibrium, the net force acting on it must be zero.
Position of Charge Q
Let’s denote the distance from charge q1 (one of the identical charges) to charge Q as x. Consequently, the distance from charge Q to charge q2 will be (2 - x). The forces acting on Q due to q1 and q2 can be expressed as:
- Force due to q1: F1 = k * |q * Q| / x²
- Force due to q2: F2 = k * |q * Q| / (2 - x)²
Here, k is Coulomb's constant. For equilibrium, these forces must be equal in magnitude:
F1 = F2
Setting Up the Equation
Substituting the expressions for F1 and F2 gives:
k * |q * Q| / x² = k * |q * Q| / (2 - x)²
Since k and |q * Q| are common on both sides, they can be canceled out (assuming Q is not zero). This simplifies to:
1/x² = 1/(2 - x)²
Solving for x
Cross-multiplying leads to:
(2 - x)² = x²
Expanding and rearranging gives:
4 - 4x + x² = x²
Thus, simplifying further results in:
4 - 4x = 0
From this, we find:
x = 1 m
Nature of Charge Q
Now, to determine the nature of charge Q, we need to consider the forces. If Q is positive, it will be repelled by both q charges, leading to an imbalance. Conversely, if Q is negative, it will be attracted to both q charges, which can balance the forces. Therefore, for equilibrium:
Charge Q must be negative.
Summary
In conclusion, the charge Q should be placed 1 meter from either of the identical charges, and it must have a negative sign to maintain equilibrium in the system.
