Maximizing Electrostatic Force Between Two Charge Fractions

In the realm of electrostatics, understanding the distribution of charges is crucial for optimizing the force between them. This article delves into a specific problem where we need to find the optimal division of a total charge into two parts to maximize the electrostatic force between them.

The Problem

Consider a charge q divided into two parts, q_1 and q_2 q - q_1. The electrostatic force between these two parts when placed at a given distance is given by the formula:

F kq_1(q - q_1)

where k is a proportionality constant. Our goal is to find the ratio q / q_1 that maximizes this force.

Steps to Solve the Problem

To find the maximum force, we need to take the first derivative of F with respect to q_1 and set it to zero. This will lead us to the critical points, from which we can determine the maximum force.

First Derivative and Solution

The force is given by:

F kq_1q - kq_1^2

Now, take the first derivative of F with respect to q_1 to find the critical points:

frac{dF}{dq_1} kq - 2kq_1

Set the derivative equal to zero to find the critical point:

kq - 2kq_1 0

Solving for q_1:

q_1 frac{q}{2}

This solution indicates that the maximum force occurs when the charge q is equally divided between the two parts, i.e., q_1 q / 2. Therefore, the ratio of the total charge to the first part of the charge is:

frac{q}{q_1} 2

Conclusion

In conclusion, dividing the total charge q into two equal parts maximizes the electrostatic force between them. This result can be verified through both algebraic manipulation and a deeper understanding of the underlying physics. The division of the total charge in half is the optimal strategy for maximizing the electrostatic force.

Additional Resources:

This problem is part of a series of electrostatics problems available on my YouTube channel, the RCM Science Channel. I have posted a pair of videos—one addressing the problem statement and the other providing the solution. These videos are part of a collection of interesting physics problems and their solutions, making it easier to understand and apply the concepts in real-world scenarios.

Forward these videos to others who may find them useful, and share your thoughts in the comments section below!