91Ó°ÊÓ

Coulomb's Law is a fundamental principle that describes the force between two point charges. It states that the electric force between two charged objects is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. This law can be mathematically expressed as:

• Formula: \( F = \frac{k \cdot |Q_1| \cdot |Q_2|}{r^2} \)
• Where \( F \) is the force between the charges, \( Q_1 \) and \( Q_2 \) are the magnitudes of the charges, \( r \) is the distance between the charges, and \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \text{ N m}^2/\text{C}^2 \).

Coulomb's Law not only applies to the forces between the charges but is also instrumental in calculating the electric field generated by a point charge. The electric field due to a point charge can be found using a similar formulation:

• Formula: \( E = \frac{k \cdot |Q|}{r^2} \)
• Where \( E \) is the electric field, \( Q \) is the charge, and \( r \) is the distance from the charge.

This principle is pivotal in solving the exercise as it allows us to determine the influence each charge exerts on a point in space—namely, the third vertex of the equilateral triangle. The electric field from each point charge acts as a vector, which means its direction and magnitude must be considered to find the resultant field when multiple charges are involved.

An equilateral triangle is a triangle in which all three sides are of equal length, and all three internal angles are equal, each being 60°. These properties make equilateral triangles quite symmetrical, which simplifies calculations in physics problems involving geometry, like the one in this exercise.In the context of electric fields, the symmetry of an equilateral triangle makes it easier to break down vector components and perform vector addition. For example, when calculating the net electric field at a vertex of the triangle where charges are placed at the other two vertices, the symmetry ensures that the electric fields from the charges are of the same magnitude, provided the charges are identical.The angles of 60° play a critical role when resolving vectors into components. Because these angles are equal, it's easy to predict the behavior of the vector components:

• The x-component of an electric field vector at the third vertex is found by multiplying the field magnitude by \( \cos(60^{\circ}) \).
• The y-component is found by multiplying the field magnitude by \( \sin(60^{\circ}) \).

This consistency allows physicists and students alike to rely on geometric symmetry to simplify otherwise complex vector mathematics.

Point charges are a theoretical concept used to represent charged objects that are small enough so that their size does not significantly affect the electric field or the force calculations. In essence, we consider the whole charge to be concentrated at a single point in space.The exercise involving two point charges at vertices of an equilateral triangle calls upon this concept. By treating the charged particles as point charges, we can directly apply formulas and principles such as Coulomb's Law to find the electric field at any given point away from the charges.

• Features of point charges include:
  • Defined by their charge magnitude (e.g., 12 nC in the exercise).
  • Their influence (electric field) diminishes with distance, specifically following an inverse square law calculated by \( E = \frac{k \cdot |Q|}{r^2} \).
• Applications:
  • Modeling atoms in a molecule where electrons and protons are treated as point charges.
  • Understanding the distribution of electric charge in conductors and insulators.

In the exercise scenario, viewing the particles as point charges allows us to deduce the electric field at the third vertex by calculating the contributions from each charge independently, and then using vector addition to find the net field influenced by both charges.