Problem 7 - Physics

Question

Two charges attract each other with a force of \(1.5 \mathrm{~N}\). What will be the force if the distance between them is reduced to one-ninth of its original value?

Step-by-Step Solution

Answer

The new force is 121.5 N.

Step 1: Understanding Coulomb's Law

Coulomb's Law states that the force between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them. It is given by the formula: \( F = k \frac{q_1 q_2}{r^2} \), where \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between the charges.

Step 2: Initial Force Expression

Let's consider the initial force: \( F_1 = k \frac{q_1 q_2}{r^2} = 1.5 \text{ N} \). This expression gives us the force exerted at the original distance \( r \).

Step 3: New Distance Calculation

The problem states that the distance is reduced to one-ninth of its original value, so the new distance \( r' = \frac{r}{9} \).

Step 4: New Force Expression

Substituting \( r' \) into Coulomb's Law, the new force \( F_2 = k \frac{q_1 q_2}{(r/9)^2} \). Simplifying this gives \( F_2 = k \frac{q_1 q_2}{r^2/81} = 81 \times k \frac{q_1 q_2}{r^2} = 81F_1 \).

Step 5: Calculate the New Force

Given that \( F_1 = 1.5 \text{ N} \), the new force \( F_2 = 81 \times 1.5 \text{ N} = 121.5 \text{ N} \).

Key Concepts

Electric Force

Electric force is a fundamental concept in electromagnetism, describing the interaction between two charged objects. It arises due to the presence of charges and is either attractive or repulsive, depending on the nature of the charges involved. Coulomb's Law helps us quantify this force. When two charges with the same sign repel each other, the force is pushing them apart. Conversely, when their signs differ, they attract each other, pulling them closer.

The electric force plays a crucial role in chemical bonding, atomic structure, and even in everyday phenomena like static electricity. Its magnitude and direction rely on the respective charges involved and their separation distance.

Point Charges

In physics, a point charge is considered to be a charged object that is small enough such that its size does not affect the analysis of the problem at hand. Essentially, a point charge is an idealized model where all the charge of an object is concentrated at a single point.

Point charges are crucial when using Coulomb's Law to calculate electric forces because they simplify the geometry of the situation. When dealing with point charges, attention is focused purely on the magnitude of the charges and the distance separating them, without worrying about the physical dimensions or shape of the objects.

Inverse Square Law

The inverse square law is a principle stating that a specified quantity or intensity is inversely proportional to the square of the distance from the source of that physical quantity. In the context of Coulomb's Law, the electric force between two point charges decreases with the square of the distance between them.

This means if the distance between the charges is doubled, the electric force becomes one fourth as strong. Likewise, reducing the distance brings a significant increase in force. This law illustrates why interactions such as gravitational, acoustic, and electric forces weaken as distance increases.

• The further apart two charges are, the weaker the force.
• The closer they are, the stronger the attractive or repulsive force.

Distance and Force Relationship

Understanding the relationship between distance and force is vital in problems involving Coulomb's Law. This relationship is mathematically modeled using the inverse square law, as previously discussed. In applied problems, changing the distance between charges directly affects the magnitude of the electric force.

If the distance between two point charges is reduced to one-ninth of its original value, the force becomes dramatically stronger. Specifically, under these conditions, the force increases by a factor of 81, as the exercise example showcases. This direct relationship helps predict and understand the effects of spatial changes in charge interactions.

• Reducing distance increases force exponentially.
• Increasing distance decreases force exponentially.