How Does Tripling the Distance Affect the Electric Force Between Two Charged Objects?

In the realm of classical electromagnetism, understanding the behavior of electric forces between charged objects is crucial. This article delves into the impact of tripling the distance between two charged objects and the underlying principles of electric force, electric field, and the inverse square law.

Relevant Formulas and Concepts

The electric force, F, is a fundamental concept in electromagnetism and is described by Coulomb's law. Mathematically, it is expressed as:

F kQQ/r2

Where:

k is the Coulomb constant (approximately 8.99 × 109 N m2 C?2). Q1 and Q2 are the charges of the two objects. r is the distance between the charges.

Effect of Tripling the Distance on Electric Force

When the distance between two charged objects is tripled, the impact on the electric force can be understood through the inverse square law. Let's explore this relationship in more detail.

Electric Field and Voltage

Before diving into the electric force, it's helpful to understand the electric field and voltage. The electric field, E, and voltage, V, are related to the electric force and energy. The electric field is given by:

E kQ/r2

The voltage, V, between two points is given by:

V kQQ/r

When the distance is tripled (r' 3r), the electric field and voltage are affected as follows:

E' k2Q/(3r)3 k2Q/27r3 2/27 E

V' kQ2Q/(3r)2 k4Q2/9r2 4/9 V

Effect on Electric Force

Now, let's consider the electric force directly. If we assume the charge remains constant and the distance is tripled, the electric force decreases by a factor of (1/9).

Let's derive this using Coulomb's law:

F kQ1Q2/r2

When the distance is doubled (initially r, now 3r):

F' kQ1Q2/ (3r)2 kQ1Q2/ 9r2 1/9 F

The derivation for tripling the distance is similar:

F' kQ1Q2/ (3r)2 kQ1Q2/ 9r2 1/9 F

Application and Examples

To better understand this concept, let's consider an example:

Initially, let's assume the charges are Q1 2 C and Q2 3 C, and the initial distance is r 4 m.

Using Coulomb's law:

F (8.99 × 109) × 2 × 3 / (4)2 (8.99 × 109) × 6 / 16 3.372 × 109 N

Now, if the distance is tripled (3 × 4 12 m):

F' (8.99 × 109) × 2 × 3 / (12)2 (8.99 × 109) × 6 / 144 3.372 × 107 N

The new force is (1/9) of the initial force.

Conclusion

The inverse square law is a fundamental principle in physics, explaining how the electric force between two charged objects changes with distance. By tripling the distance, the electric force decreases by a factor of (1/9), demonstrating the powerful predictive nature of Coulomb's law and the inverse square relationship.

Key Takeaways

The electric force between two charged objects is inversely proportional to the square of the distance between them. Tripling the distance results in the electric force decreasing by a factor of (1/9). Coulomb's law provides a basis for understanding the relationship between electric force, charge, and distance. The inverse square law has numerous applications in physics and engineering, making it a crucial concept to grasp.

Related Keywords

electric force, inverse square law, Coulomb's law