Problem 28 - College Physics

Question

\(\bullet$$\bullet\) Two unequal charges repel each other with a force \(F .\) If both charges are doubled in magnitude, what will be the new force in terms of \(F ?\)

Step-by-Step Solution

Answer

The new force is \( 4F \).

Step 1: Identify Given Information

We are given two charges which repel each other with a force \( F \). These charges are unequal, and both charges are doubled.

Step 2: Recall the Formula for Electrostatic Force

The formula for the force between two charges is given by Coulomb's law: \[ F = k \frac{q_1 q_2}{r^2} \]where \( F \) is the force, \( q_1 \) and \( q_2 \) are the charges, \( r \) is the distance between the charges, and \( k \) is Coulomb's constant.

Step 3: Determine the Effect of Doubling the Charges

Since both charges are doubled, the new charges are \( 2q_1 \) and \( 2q_2 \). Substitute into Coulomb's law:\[ F' = k \frac{(2q_1)(2q_2)}{r^2} = k \frac{4q_1 q_2}{r^2} \]

Step 4: Simplify the Expression

The expression for the new force becomes:\[ F' = 4 \left(k \frac{q_1 q_2}{r^2}\right) \]Since \( k \frac{q_1 q_2}{r^2} = F \), the new force \( F' \) is:\[ F' = 4F \]

Step 5: Conclusion

Doubling each charge results in the force being multiplied by 4.

Key Concepts

Electrostatic Force

Electrostatic force is the force of attraction or repulsion between electrically charged objects. The strength and direction of this force depend on the nature of the charges and their separation. We use Coulomb's Law to calculate it, given by the equation:
\[ F = k \frac{q_1 q_2}{r^2} \]
This force can either be attractive (if charges are opposite) or repulsive (if charges are similar).

• Attractive Force: When one charge is positive and the other is negative.
• Repulsive Force: When both charges are either positive or both are negative.

The magnitude of this force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. So, if you increase the distance, the force decreases. Understanding this concept helps predict how objects with different charges will interact.

Charges

Charges are properties of matter that cause them to experience a force when placed in an electromagnetic field. They come in two types: positive and negative. Each charge influences the other through an invisible field around them. This is known as the electric field.

• Positive Charge: Is typically represented by a plus sign (+).
• Negative Charge: Is represented by a minus sign (-).
• Neutral: When there's no net charge.

In our exercise, we considered two charges that were initially unequal. When both charges are doubled, their influence on each other grows. According to Coulomb's Law, the electrostatic force between them will then increase by a factor of four, assuming their distance remains unchanged.

Coulomb's Constant

Coulomb's constant, denoted as \( k \), is an essential part of Coulomb's Law. It helps to calculate the force between two point charges. Its value in vacuum is approximately \( 8.99 \times 10^9 \) N m²/C².

• Denotes strength of the electrostatic force.
• A higher constant means a stronger interaction between charges per unit distance.

The constant provides a way to quantify the dielectric properties of the medium between charges. In other words, it helps account for how the medium influences the force. By knowing \( k \), you can predict how two charges within a given medium would interact.