91Ó°ÊÓ

In electrostatics, understanding how charged particles interact is key to solving problems related to forces and motion. One of the foundational principles governing these interactions is Coulomb's Law. This law describes the force between two point charges. The formula is given by:

• \( F = \frac{k |q_1 q_2|}{d^2} \)

Where:

• \( F \) is the magnitude of the electrostatic force.
• \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \frac{\text{N m}^2}{\text{C}^2} \).
• \( q_1 \) and \( q_2 \) are the magnitudes of the charges.
• \( d \) is the distance between the centers of the two charges.

Coulomb's Law tells us that the force is directly proportional to the magnitude of the charges and inversely proportional to the square of the distance between them. This means if you double the distance, the force becomes one-fourth as strong. Using this principle allows us to calculate how strong the force will be and how it will affect each charge. Forces can be attractive (between opposite charges) or repulsive (between like charges), which impacts the direction of the force vector.

To calculate the electrostatic force between two charges, you first need to determine the distance between them, which can be done using the coordinate distance formula:

• \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

Once you have the distance, plug it into Coulomb's Law to find the force. In the given exercise, you have two charges: \( q_1 = +3.0 \mu C \) and \( q_2 = -4.0 \mu C \), with coordinates for each particle. After calculating the distance as approximately 5.57 cm, you apply Coulomb's Law:

• \( F = \frac{8.99 \times 10^9 \times |3.0 \times 10^{-6} \, \text{C} \times (-4.0 \times 10^{-6} \, \text{C})|}{(0.0557)^2} \)

After solving the equation, the magnitude of the force comes out to about 34.77 N. Remember, since the forces are vector quantities, they have both magnitude and direction, with the direction determined by the nature of the charges (attraction in this case because one charge is positive and the other negative). This direction can be expressed in terms of an angle relative to a reference axis, often requiring some trigonometric calculations to resolve fully.

Coordinate geometry, also known as analytic geometry, plays a vital role in understanding and solving physics problems involving spatial relationships, such as distances and angles between points. In physics, especially in electrostatic problems, we frequently need to calculate relative positions and directions of forces between particles.The basic tool we use is the distance formula derived from the Pythagorean theorem:

• \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

This formula allows us to find the straight-line distance between any two points in a plane. Knowing this distance is essential for applying Coulomb’s Law when charges are not placed along a straight line.In the presented exercise, coordinate geometry aids in not only calculating the distance between charges but also in determining angles for direction using trigonometric functions. For instance, the angle \( \theta \) with respect to the positive \( x \)-axis is found using the arctangent function:

• \( \theta = \tan^{-1}\left(\frac{y_2 - y_1}{x_2 - x_1}\right) \)

Such calculations are crucial in ensuring that the computed direction aligns correctly with the problem's geometry. Understanding these concepts ensures a comprehensive approach to solving physics problems involving multiple dimensions.