Two points charges + 9e and +e are kept 16 cm apart to each other. Where a third charge q is placed between them so that the system is in the equilibrium state
in simple
In order for the system to be in equilibrium, the net force on the third charge q must be zero. This means that the electric force on q due to the charge +9e must be exactly opposite and equal in magnitude to the electric force on q due to the charge +e.
The electric force between two charges is given by Coulomb's law:
F = k * (|q1| * |q2|) / r^2
where F is the electric force, k is the Coulomb's constant, q1 and q2 are the magnitudes of the charges, and r is the distance between the charges.
Let's calculate the forces on q due to the charges +9e and +e.
Force due to +9e:
F1 = k * (|9e| * |q|) / r1^2
Force due to +e:
F2 = k * (|e| * |q|) / r2^2
Since the distances r1 and r2 are not given, we can't calculate the exact values of the forces. However, we can derive a relationship between r1 and r2 that results in a net force of zero on q.
Since the system is in equilibrium, the magnitudes of the forces must be equal:
F1 = F2
k * (|9e| * |q|) / r1^2 = k * (|e| * |q|) / r2^2
Simplifying the equation gives:
9e / r1^2 = e / r2^2
Cross multiplying gives:
e * r1^2 = 9e * r2^2
Simplifying further:
r1^2 = 9 * r2^2
Taking the square root of both sides gives:
r1 = 3 * r2
Therefore, in order for the system to be in equilibrium, the distance of the third charge q from the charge +9e must be three times the distance of q from the charge +e.
To find the position at which the third charge q should be placed for the system to be in equilibrium, we need to consider the electrostatic forces acting on the third charge.
Let's analyze the forces acting on q:
1. The force between q and the charge +9e:
According to Coulomb's Law, the force between two charges is given by:
F = k*q1*q2 / r^2
where F is the force, k is the electrostatic constant, q1 and q2 are the charges, and r is the distance between the charges.
The force between q and the charge +9e will be attractive since the charges have opposite signs, so the force will be directed towards the +9e charge.
2. The force between q and the charge +e:
Similarly, the force between q and the charge +e will also be attractive, directed towards the +e charge.
For the system to be in equilibrium, the net force acting on the third charge q should be zero. This means that the magnitudes of the attractive forces from both charges should be equal.
Let's calculate the equilibrium position:
1. The force between q and the charge +9e:
F1 = k*(q)*(9e) / r1^2
2. The force between q and the charge +e:
F2 = k*(q)*(e) / r2^2
Since the charges are positive, the forces will be attractive, so we can equate the magnitudes:
|F1| = |F2|
k*(q)*(9e) / r1^2 = k*(q)*(e) / r2^2
Simplifying:
(9e) / r1^2 = (e) / r2^2
Cross-multiplying:
(9e)*(r2^2) = (e)*(r1^2)
9*(r2^2) = r1^2
Taking the square root of both sides:
3*r2 = r1
Given that the distance between the charges +9e and +e is 16 cm, we can substitute r1 = 16 cm into the equation:
3*r2 = 16
r2 = 16 / 3
So, the third charge q should be placed at a distance of 16 / 3 cm from the charge +e (counting towards the charge +9e) for the system to be in equilibrium in a simple case.
To determine the position of the third charge q such that the system is in equilibrium, we can calculate the net force on q and set it to zero.
The forces exerted on the third charge q by the other two charges are electrostatic forces given by Coulomb's law:
F1 = (k * |9e| * |q|) / r1^2
F2 = (k * |e| * |q|) / r2^2
where k is the electrostatic constant (9 × 10^9 Nm^2/C^2), r1 is the distance between q and the charge +9e, and r2 is the distance between q and the charge +e.
Since the system is in equilibrium, the net force on q is zero. Thus, we can set up an equation:
F1 + F2 = 0
Substituting the values, we get:
(k * |9e| * |q|) / r1^2 + (k * |e| * |q|) / r2^2 = 0
Simplifying further:
(9e * q) / r1^2 + (e * q) / r2^2 = 0
Multiplying through by r1^2 * r2^2:
(9e * q * r2^2) + (e * q * r1^2) = 0
We can solve this equation for the unknown charge q to find its position between the two given charges +9e and +e.