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-

Answer 1

To find the position of the third charge q where the system is in equilibrium, we need to determine the net force acting on it and set it to zero.

Let's assume the positions of the charges are as follows:
Charge +9e is at point A
Charge +e is at point B
Charge q is at point C

Given:
Distance between A and B = 16 cm = 0.16 m

The force between two charges is given by Coulomb's Law:
F = k * (|q1 * q2| / r^2)

Where:
F is the force between the charges
k is the Coulomb's constant (9 x 10^9 Nm^2/C^2)
q1 and q2 are the magnitudes of the charges
r is the distance between the charges

The net force on charge q is the vector sum of the forces acting on it due to charges +9e and +e. Since the charges have the same sign, the forces will repel.

So, the net force on charge q is:
F_net = F1 + F2

Since the system is in equilibrium, F_net = 0. Therefore we have:
F1 + F2 = 0

Now let's calculate the magnitudes of F1 and F2.

F1 = k * (|9e * q| / r1^2)----(1)
F2 = k * (|e * q| / r2^2)------(2)

Where:
r1 is the distance between charges +9e and q
r2 is the distance between charges +e and q

Since q is placed between charges +9e and +e, we have:
r1 + r2 = 0.16 m ---(3)

Substituting the values of F1 and F2 from equations (1) and (2) into the equilibrium equation (F1 + F2 = 0) and rearranging, we get:

k * (|9e * q| / r1^2) + k * (|e * q| / r2^2) = 0

Multiplying both sides by r1^2 * r2^2, we have:

k * (|9e * q| * r2^2) + k * (|e * q| * r1^2) = 0

Since r2 = 0.16 - r1 (from equation 3), we can rewrite the equation as:

k * (|9e * q| * (0.16 - r1)^2) + k * (|e * q| * r1^2) = 0

Now we have a quadratic equation in terms of r1. We can simplify this equation, solve for r1, and then find r2 using equation 3.

Let's solve the equation for r1:

k * (|9e * q| * (0.16 - r1)^2) + k * (|e * q| * r1^2) = 0

Since k is common in both terms, we can cancel it out to simplify:

|9e * q| * (0.16 - r1)^2 + |e * q| * r1^2 = 0

Expanding the squares:

|9e * q| * (0.0256 - 0.32r1 + r1^2) + |e * q| * r1^2 = 0

Now we can solve for r1:

|9e * q| * 0.0256 - |9e * q| * 0.32r1 + |9e * q| * r1^2 + |e * q| * r1^2 = 0

Combining like terms:

(|9e * q| + |e * q|) * r1^2 - |9e * q| * 0.32r1 + |9e * q| * 0.0256 = 0

This is a quadratic equation in r1. We can solve it using the quadratic formula:

r1 = (-b +/- sqrt(b^2 - 4ac)) / (2a)

Where:
a = (|9e * q| + |e * q|)
b = -|9e * q| * 0.32
c = |9e * q| * 0.0256

Plugging in the values, we can solve for r1. Once we find r1, we can substitute it into equation 3 to find r2.

However, without knowing the value of the charge q or fixing its sign, it is not possible to determine the exact position of the third charge q between the two points charges for equilibrium.