Question

Approximate the change in the magnitude of the electrostatic force between two charges when the distance between them increases from \(r=20 \mathrm{m}\) to \(r=21 \mathrm{m}\left(F(r)=0.01 / r^{2}\right)\).

Short answer

Answer: The change in the magnitude of the electrostatic force is approximately -0.000002325 N (Newtons).

Step-by-step solution

Find the initial force

To find the initial force, substitute the initial distance (r = 20m) into the formula: \(F(20) = 0.01 / 20^2\) Compute the force: \(F(20) = 0.01 / (20 \times 20)\) \(F(20) = 0.01 / 400\) \(F(20) = 0.000025\) The initial force is 0.000025 N (Newtons).

Find the final force

To find the final force, substitute the final distance (r = 21m) into the formula: \(F(21) = 0.01 / 21^2\) Compute the force: \(F(21) = 0.01 / (21 \times 21)\) \(F(21) = 0.01 / 441\) \(F(21) = 0.000022675\) The final force is approximately 0.000022675 N (Newtons).

Find the change in force

To find the change in the magnitude of the electrostatic force, subtract the initial force from the final force: \(\Delta F = F(21) - F(20)\) Compute the change in force: \(\Delta F = 0.000022675 - 0.000025\) \(\Delta F = -0.000002325\) The change in the magnitude of the electrostatic force is approximately -0.000002325 N (Newtons). This negative sign indicates a decrease in the magnitude of the force as the distance between the charges increases.

Key concepts

Coulomb's Law

Coulomb's Law is a fundamental principle that describes the electrostatic interaction between electrically charged particles. It was named after French physicist Charles-Augustin de Coulomb, who introduced the law in the 1780s. The law states that the magnitude of the electrostatic force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.

The mathematical formula for Coulomb's Law is expressed as:\[F = k \frac{{|q_1q_2|}}{{r^2}}\]Here, \( F \) is the magnitude of the force between the charges, \( q_1 \) and \( q_2 \) are the amounts of the charges, \( r \) is the distance between the centers of the two charges, and \( k \) is Coulomb's constant. This constant depends on the medium between the charges and the system of units that is being used; in a vacuum, its value is approximately \( 8.988 \times 10^9 \) N m²/C².

Understanding Coulomb's Law is crucial for calculating forces in electrostatic situations and forms the basis of many electrical theories and applications.

Inverse Square Law

The inverse square law is a principle used in physics that relates the intensity of an effect to the inverse square of the distance from the cause. This law applies to various physical phenomena including gravity, light, sound, and, relevant to this context, electrostatic forces. In electrostatics, the inverse square law is reflected in Coulomb's Law where the force between two charges diminishes with the square of the distance between them.

In mathematical terms, if the distance is doubled, the force is reduced to a quarter of its original value; if the distance is tripled, the force drops to a ninth, and so on. Therfore, even a small increase in distance can lead to a substantial decrease in force, which is crucial to understand when calculating the interaction between charges. The exercise illustrated how the force changes when the distance between charges goes from \( 20m \) to \( 21m \), a move that, while seemingly minor, still impacts the electrostatic force significantly due to this inverse square relationship.

Force Between Charges

The concept of the force between charges is central to understanding electrostatic interactions. This force can be either attractive or repulsive depending on the nature of the charges involved; like charges repel each other, while opposite charges attract. This fundamental electrostatic phenomenon is the reason why charged particles move and interact the way they do.

In the context of the provided exercise, we're analyzing the electrostatic force between two charges as they change their positions. The formula derived from Coulomb's Law enables us to calculate this force precisely. The exercise's step-by-step solution begins by calculating the initial force when the charges are at a distance of \( 20m \) and then the force at the new distance of \( 21m \). By comparing these two calculated forces, one can see how sensitive the force is to changes in distance, reinforcing the concept that the electrostatic force is a key player in the dance of charged particles.

Change in Magnitude of Force

The change in magnitude of force is a measurement of how the electrostatic force varies when the variables, such as the distance between the charges or the amount of charge, are altered. In electrostatics, this typically involves observing how the force decreases as the charges move further apart, aligning with the inverse square law. Such changes can have significant consequences in physical systems where electrostatic forces are at work.

In our specific exercise, the calculation involves finding the difference between the initial and final forces acting between the charges. As demonstrated, even a small change in distance resulted in a change in the force magnitude, denoted by a negative value in the provided answer, which signals a decrease in force. As students learn to apply these calculations, they gain insights into how sensitive electrostatic forces are to spatial changes and why precision is crucial in fields that depend on electrostatic principles, like electronics and materials science.