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-
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.
To find the position where the third charge q should be placed for the system to be in equilibrium, we need to consider the forces acting on it.
Let's assume that the third charge q is placed at a distance x from the charge +9e.
The force between the third charge q and the charge +9e is given by Coulomb's law:
F1 = k * (9e) * q / (16 + x)^2, where k is the Coulomb's constant.
The force between the third charge q and the charge +e is given by:
F2 = k * (e) * q / (16 - x)^2
For the system to be in equilibrium, the net force on the third charge q should be zero. Therefore, F1 and F2 should cancel each other out:
F1 + F2 = 0,
k * (9e) * q / (16 + x)^2 + k * (e) * q / (16 - x)^2 = 0.
We can solve this equation to find the value of x.
Multiplying both sides by (16 + x)^2 * (16 - x)^2:
(9e) * (16 - x)^2 + (e) * (16 + x)^2 = 0.
Simplifying the equation:
144(16) - 32x + x^2 + 16(16) + 32x + x^2 = 0.
Expanding and combining like terms:
2304 + 2x^2 = 0.
Dividing both sides by 2:
x^2 = -1152.
Since we cannot have a negative value for x, it means that there is no position where the third charge q can be placed between the charges +9e and +e to achieve equilibrium.
To find the position of the third charge (q) where the system is in equilibrium, we need to consider the net electrostatic force on q due to the other two charges.
First, let's assign some values:
- Charge 1 (q1) = +9e
- Charge 2 (q2) = +e
- Distance between charges = 16 cm = 0.16 m
We know that the electrostatic force between two charges can be calculated using Coulomb's law:
F = k * |q1*q2| / r^2
Where:
- F is the electrostatic force
- k is the electrostatic constant (k = 9 * 10^9 N*m^2/C^2)
- q1 and q2 are the charges
- r is the distance between the charges
Now, let's calculate the force exerted on the third charge (q) by the +9e charge:
F1 = k * |q * 9e| / r1^2
Next, let's calculate the force exerted on the third charge (q) by the +e charge:
F2 = k * |q * e| / r2^2
Since the system is in equilibrium, the net force on charge q should be zero:
F1 + F2 = 0
Now, let's substitute the values into the equation and solve for the position of charge q:
k * |q * 9e| / r1^2 + k * |q * e| / r2^2 = 0
Simplifying the equation:
9 * |q * e| / r1^2 + |q * e| / r2^2 = 0
Now, let's substitute the values of r1 and r2:
9 * |q * e| / (0.16^2) + |q * e| / (0.16^2) = 0
Combining the terms with q:
9 * 0.16^-2 * |q * e| + 0.16^-2 * |q * e| = 0
Multiplying through by the common denominator:
(9 + 1) * 0.16^-2 * |q * e| = 0
Simplifying further:
10 * 0.16^-2 * |q * e| = 0
0.16^-2 * |q * e| = 0
Since 0.16^-2 is a positive non-zero number, |q * e| must be zero for the equation to hold true. This means that the charge q must be zero (q = 0) for the system to be in equilibrium.
Therefore, the third charge (q) must be placed at the center point between the two charges (+9e and +e) to achieve equilibrium.