A certain charge Q is divided into two parts q anD Q-q WHICH ARE SEPERATED by certain distance what must q be in terms of Q TO MAXIMISE electrostatic repulsion between two charges?

To determine the value of charge \( q \) that maximizes the electrostatic repulsion between two charges \( q \) and \( Q - q \), we can use Coulomb's law, which describes the force between two point charges. The electrostatic force \( F \) between two charges is given by the formula:

Understanding the Forces Involved

The formula for the electrostatic force is:

F = k * |q1 * q2| / r²

Where:

• F is the electrostatic force between the charges.
• k is Coulomb's constant (approximately \( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \)).
• q1 and q2 are the magnitudes of the charges.
• r is the distance between the charges.

Setting Up the Problem

In our case, we have two charges: \( q \) and \( Q - q \). The electrostatic force of repulsion \( F \) can be expressed as:

F = k * |q * (Q - q)| / r²

To maximize this force, we need to find the optimal value of \( q \) in terms of \( Q \).

Maximizing the Force

To find the maximum force, we can treat \( F \) as a function of \( q \) and differentiate it with respect to \( q \). First, we can rewrite the force equation:

F(q) = k * q * (Q - q) / r²

Next, we differentiate \( F \) with respect to \( q \):

dF/dq = k * (Q - 2q) / r²

Finding Critical Points

To find the critical points, we set the derivative equal to zero:

0 = Q - 2q

Solving for \( q \), we get:

2q = Q

q = Q/2

Verifying Maximum Condition

To ensure that this value of \( q \) indeed maximizes the force, we can check the second derivative:

d²F/dq² = -2k / r²

Since this second derivative is negative, it confirms that the function \( F(q) \) has a maximum at \( q = Q/2 \).

Conclusion

Thus, to maximize the electrostatic repulsion between the two charges, \( q \) should be equal to half of the total charge \( Q \). This means that if you divide the charge \( Q \) into two equal parts, each charge will be \( Q/2 \), leading to the maximum force of repulsion between them.