Introduction

Understanding the behavior of electric fields around charges is a fundamental concept in electromagnetism. This article explores the scenario where two point charges, with charge q and 4q, are placed at a certain separation distance d. Specifically, we will determine the location where the total electric field due to both charges is zero. This problem requires a deep dive into Coulomb's law and the principles of vector addition, making it a comprehensive example of applying these concepts.

Setting Up the Scenario

The first step is to visualize and mathematically establish the position of the charges. Assume a coordinate system where the charge q is placed at x 0, and the charge 4q is at x d. The total electric field at any point x along the line joining the charges is the vector sum of the individual electric fields due to each charge. This setup allows us to analyze the electric field in a straightforward manner.

Electric Field Contributions

The electric field due to a point charge Q at a distance r is given by Coulomb's law:

[ E frac{kQ}{r^2} ]

where k is Coulomb's constant.

Zero Electric Field Point Location

For the total electric field to be zero, the contributions from both charges must cancel each other out. Let's denote the distance from the charge q to a point P where the electric field is zero as x. Then, the distance from the charge 4q to the same point is d - x.

Between the Charges

When the zero field point P is between the charges, the electric field contributions can be described as follows:

The electric field due to charge q is: [ E_q frac{k q}{x^2} ] The electric field due to charge 4q is: [ E_{4q} frac{k 4q}{(d - x)^2} ]

For the total electric field to be zero:

[ E_q -E_{4q} ]

Substituting the expressions for the electric fields:

[ frac{k q}{x^2} -frac{k 4q}{(d - x)^2} ]

Dividing both sides by common factors k and q:

[ frac{1}{x^2} frac{4}{(d - x)^2} ]

Taking the square root of both sides:

[ frac{1}{x} 2 cdot frac{1}{d - x} ]

Cross-multiplying:

[ d - x 2x ]

Solving for x gives:

[ d 3x ]

[ x frac{d}{3} ]

Thus, the electric field due to the charges q and 4q is zero at a distance x d/3 from charge q and 2d/3 from charge 4q.

Conclusion

The zero electric field point for the given charges is located at a position that is one-third of the total distance between the charges. This result provides a clear and precise answer to the initial problem.