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 steps
To find the position of the third charge q where the system is in equilibrium, we can consider the forces acting on it.
1. The force between the charge +9e and charge q:
The force between two point charges is given by Coulomb's law: F = k * (q1 * q2) / r^2
Here, k is the electrostatic constant, q1 is the charge of the first point charge (+9e), q2 is the charge of the third point charge (q), and r is the distance between them.
Let's denote the force between +9e and q as F1.
2. The force between the charge +e and charge q:
Similar to the first case, let's denote the force between +e and q as F2.
When the system is in equilibrium, the net force on q will be zero. So, F1 + F2 = 0.
3. The equilibrium condition:
Since the force between two point charges is attractive or repulsive depending on the sign of the charges, we need to consider two possible scenarios for equilibrium:
a) If q is positive:
In this case, F1 will be attractive and F2 will be repulsive. To make their sum zero, F1 (attractive) should be equal in magnitude to F2 (repulsive).
So, F1 = F2
Using Coulomb's law, we have:
k * ((9e) * q) / (16^2) = k * ((e) * q) / ((16 - x)^2)
where x is the distance between q and the charge +e.
Simplifying this equation, we get:
9 * q / 256 = q / (256 - 32x + x^2)
9 * (256 - 32x + x^2) = 256
9x^2 - 288x + 2304 = 0
We can solve this quadratic equation to find the possible values of x. The solution will give the position of q between the two charges.
b) If q is negative:
In this case, F1 will be repulsive and F2 will be attractive. Similar to the previous case, we can set up the equation and solve it for the position of q.
By solving the respective equations, we can find the position of the charge q for the system to be in equilibrium.
To determine the location of the charge q where the system is in equilibrium, you can follow these steps:
Step 1: Calculate the net electrostatic force between the two charges +9e and +e.
- The electrostatic force between two charges can be determined using Coulomb's law: F = k * |q1 * q2| / r^2, where k is the electrostatic constant, q1 and q2 are the magnitudes of the two charges, and r is the distance between them.
- Plugging in the values, the net force between the two charges is F_net = k * |9e * e| / (0.16)^2.
Step 2: Determine the magnitude of the force between +9e and the charge q.
- The distance between +9e and q is unknown, so let's call it d.
- The electrostatic force between +9e and q is F_9q = k * |9e * q| / d^2.
Step 3: Determine the magnitude of the force between +e and the charge q.
- The distance between +e and q is also unknown, so let's call it x.
- The electrostatic force between +e and q is F_eq = k * |e * q| / x^2.
Step 4: Set up the equilibrium condition.
- In equilibrium, the net force acting on the charge q is zero.
- This means that the net force F_net = F_9q + F_eq is equal to zero.
Step 5: Solve the equation for the unknown distances d and x.
- Substitute the calculated values from Step 1 into the equilibrium condition equation from Step 4.
- Simplify and solve the equation for d and x.
By following these steps, you can determine the distances d and x, which will tell you the location of the charge q where the system is in equilibrium.
To find the position of the third charge q that will result in an equilibrium state, follow these simple steps:
Step 1: Identify the forces acting on the third charge q:
- The first point charge (+9e) exerts a repulsive force on q.
- The second point charge (+e) also exerts a repulsive force on q.
Step 2: Analyze the forces using Coulomb's law:
- The magnitude of the repulsive force between two point 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 × 10^9 N m^2/C^2), q1 and q2 are the magnitudes of the charges, and r is the distance between the charges.
- Let's assume the third charge q is placed at a distance of x cm from the +9e charge and at a distance of (16 - x) cm from the +e charge.
Step 3: Equate the forces:
- The force exerted by the +9e charge on q is: F1 = (9e * q) / (x * 10^-2)^2
- The force exerted by the +e charge on q is: F2 = (e * q) / ((16 - x) * 10^-2)^2
- For equilibrium, the magnitudes of the forces F1 and F2 should be equal: F1 = F2.
Step 4: Solve the equation for x:
- Set F1 = F2 and solve the equation for x to find the position of the third charge q.
This calculation can be done either by simplifying the equation or by using algebraic methods such as cross-multiplication, substitution, or factorization.
Note: The charges are given in terms of e, which represents the elementary charge. You can substitute the value for e (1.6 × 10^-19 C) to calculate the exact forces and position of q.