Problem 53 - Fundamentals of Physics

Question

What would be the magnitude of the electrostatic force between two \(1.00 \mathrm{C}\) point charges separated by a distance of (a) \(1.00 \mathrm{~m}\) and (b) \(1.00 \mathrm{~km}\) if such point charges existed (they do not) and this configuration could be set up?

Step-by-Step Solution

Answer

(a) The force is \(8.99 \times 10^9 \mathrm{~N}\). (b) The force is \(8.99 \times 10^3 \mathrm{~N}\).

Step 1: Identify the relevant formula

The magnitude of the electrostatic force between two point charges is given by Coulomb's Law, which is: \( F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \). Here, \( k \) is Coulomb's constant \( 8.99 \times 10^9 \mathrm{~N~m^2/C^2} \), \( q_1 \) and \( q_2 \) are the charges, each \( 1.00 \mathrm{~C} \), and \( r \) is the separation distance.

Step 2: Calculate the force for distance 1 meter

Using the formula from Step 1 with \( r = 1.00 \mathrm{~m} \), substitute in the given values: \( F = \frac{8.99 \times 10^9 \cdot |1.00 \cdot 1.00|}{(1.00)^2} = 8.99 \times 10^9 \mathrm{~N} \).

Step 3: Calculate the force for distance 1 kilometer

Convert the distance to meters: \( 1.00 \mathrm{~km} = 1000 \mathrm{~m} \). Use the formula with \( r = 1000 \mathrm{~m} \): \( F = \frac{8.99 \times 10^9 \cdot |1.00 \cdot 1.00|}{(1000)^2} = \frac{8.99 \times 10^9}{10^6} = 8.99 \times 10^3 \mathrm{~N} \).

Key Concepts

Coulomb's Law

Coulomb's Law is at the core of understanding the electrostatic force between two point charges. It helps us quantify the attractive or repulsive force due to electric charges. The law is expressed mathematically as:\[ F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \]where:

• \( F \) is the magnitude of the force between the charges.
• \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \mathrm{~N~m^2/C^2} \).
• \( q_1 \) and \( q_2 \) are the values of the two charges in Coulombs.
• \( r \) is the distance between the centers of the two charges in meters.

This formula shows that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. Thus, any change in the magnitudes of the charges or the distance between them greatly affects the force experienced.

Point Charges

Point charges are idealized models used in physics to simplify complex electric interactions. When dealing with real-world problems, point charges help focus on the basic principles of electrostatics without the complexity of considering the physical size or shape of the objects. These charges are treated as if all their charge is concentrated at a single point in space, making calculations for electrostatic forces more straightforward. The use of point charges assumes no spatial extent, streamlining the application of Coulomb's Law. In actuality, such high-magnitude or isolated charges do not exist due to electrostatic limits, but the concept serves as a crucial stepping stone in mastering electric force computations and understanding ion interaction at a more abstract level.

Distance and Force Relationship

The distance between two point charges is critical in determining the magnitude of the electrostatic force between them. In Coulomb's Law, the force \( F \) is inversely proportional to the square of the distance \( r \) between the charges. This relationship is expressed in the formula as:\[ F \propto \frac{1}{r^2} \]This means that as the distance \( r \) increases, the force \( F \) decreases rapidly. If the distance is doubled, the force becomes one-fourth of its original value. Conversely, if the distance is halved, the force increases by four times.Understanding this relationship is essential for grasping the effects of distance on electrostatic interactions, highlighting how electrical forces diminish with increasing separation. The exercise exemplifies this when comparing the forces at distances of 1 meter and 1 kilometer, illustrating how dramatically the force decreases over larger distances.