Understanding Coulomb's Law and Its Applications in Electrostatic Forces

In physics, the force between two charged spheres can be described by Coulomb's Law. This law is fundamental in electrostatics and helps us understand how the force between charged objects changes with distance. Let's delve into the details of this fascinating topic, including how the force changes when the distance between two charged spheres is doubled or tripled.

Introduction to Coulomb's Law

Coulomb's Law is a principle in electrostatics that describes the force between two point charges in a vacuum. The law states that the magnitude of the electrostatic force between two point charges (q_1) and (q_2) separated by a distance (R) is given by:

[ F k frac{q_1 q_2}{R^2} ]

Where:

F is the magnitude of the electrostatic force between the charges, k is Coulomb's constant (approximately (8.99 times 10^9 , text{N m}^2/text{C}^2)), q_1 and q_2 are the magnitudes of the charges, R is the distance between the charges.

Effect of Doubling the Distance

Let's consider what happens to the force when the distance (R) between two charged spheres is doubled. If the distance (R) is doubled, i.e., (R) becomes (2R), we can substitute this into Coulomb's Law:

[ F k frac{q_1 q_2}{(2R)^2} k frac{q_1 q_2}{4R^2} frac{F}{4} ]

Thus, the new force (F) is one-fourth of the original force (F). This demonstrates that the force decreases by a factor of four when the distance between the charges is doubled, illustrating the inverse square nature of Coulomb's Law.

It is important to note that this relationship holds true as long as the spheres are considered as point charges or the distance is measured relative to the center of the spheres.

Effect of Tripling the Distance

Now, consider what happens if the distance between two charges is tripled, i.e., the distance becomes (3R). According to Coulomb's Law of Electrostatics:

[ F_e k frac{q_1 q_2}{d^2} ]

where (d) is the distance between the charges. If the distance (d) is tripled (i.e., (3R)), the new force (F_e) becomes:

[ F_e k frac{q_1 q_2}{(3R)^2} k frac{q_1 q_2}{9R^2} frac{F}{9} ]

This means that the force decreases by a factor of nine when the distance is tripled, adhering to the inverse square law. For example, if the previous value of (F_e 0.9 , text{N}), the new value becomes 0.1 , text{N}.

Radii of Spheres and Conducting vs. Insulating Surfaces

It's important to recognize that the behavior of the force changes depending on whether the spheres are insulating or conducting. For insulating spheres with even charge distribution, the force behaves like that of point charges at the center, and the inverse square law holds true.

However, when dealing with conducting spherical surfaces, the charges concentrate on the near or far ends, depending on the nature of the charges. To calculate the force, the distance (R) might need to be adjusted by adding or subtracting the radii of the spheres (r). This leads to the following formulae:

[ F frac{1}{R-2r^2} text{ for attractive force} ]

and

[ F frac{1}{R 2r^2} text{ for repulsive force} ]

These formulas reduce to the regular inverse square law when the radius (r) is zero, i.e., when the spheres are point charges.

Conclusion

The relationship described by Coulomb's Law is consistent and can be applied to various scenarios, whether the spheres are insulating or conducting. Understanding this law is crucial in many fields, including electric circuits, electromagnetism, and even quantum mechanics.

In summary, the force between two charged spheres is inversely proportional to the square of the distance between them. When the distance is doubled, the force decreases by a factor of four, and when tripled, it decreases by a factor of nine. These relationships are key principles in electrostatics and play a vital role in many scientific and engineering applications.