To solve this problem, we need to analyze the forces acting on a point charge placed at different locations relative to three other point charges positioned at the corners of a square. The square has a side length of 2 cm, and each corner has a charge of 3 × 10^-6 C. We will calculate the resultant force on a charge of 1 × 10^-6 C placed at both the center of the square and at the vacant corner. Let's break this down step by step.
Understanding the Forces Between Charges
The force between two point charges can be calculated using Coulomb's Law, which states:
F = k * |q1 * q2| / r²
Where:
- F is the magnitude of the force between the charges.
- k is Coulomb's constant (approximately 8.99 × 10^9 N m²/C²).
- q1 and q2 are the magnitudes of the charges.
- r is the distance between the charges.
Case (a): Charge at the Center of the Square
When the 1 × 10^-6 C charge is placed at the center of the square, it is equidistant from all three corner charges. The distance from the center to a corner of the square can be calculated using the Pythagorean theorem:
r = √((2 cm)² + (2 cm)²) / 2 = √(8) / 2 = √2 cm ≈ 1.41 cm = 0.0141 m
Now, we can calculate the force exerted on the central charge by each of the three corner charges:
F = (8.99 × 10^9 N m²/C²) * |(3 × 10^-6 C) * (1 × 10^-6 C)| / (0.0141 m)²
This simplifies to:
F ≈ (8.99 × 10^9) * (3 × 10^-6) * (1 × 10^-6) / (0.00019881) ≈ 135.5 N
Since all three forces will be directed away from the corner charges (because they are all positive), we can find the resultant force by vector addition. The forces from the two corner charges adjacent to the center will have components that cancel each other out in the horizontal direction, while the vertical components will add up.
Thus, the resultant force will be directed straight up towards the vacant corner, and its magnitude will be:
Resultant Force ≈ 135.5 N in the direction towards the vacant corner.
Case (b): Charge at the Vacant Corner of the Square
Now, if we place the 1 × 10^-6 C charge at the vacant corner, it will only experience forces from the two adjacent corner charges. The distance between the charge at the vacant corner and each of the adjacent corner charges is simply the side length of the square, which is 2 cm or 0.02 m.
Using Coulomb's Law again for each of the two forces:
F = (8.99 × 10^9 N m²/C²) * |(3 × 10^-6 C) * (1 × 10^-6 C)| / (0.02 m)²
This gives:
F ≈ (8.99 × 10^9) * (3 × 10^-6) * (1 × 10^-6) / (0.0004) ≈ 67.425 N
Both forces from the adjacent corner charges will act away from their respective charges, creating a net force that can be calculated by vector addition. The angle between these two forces is 90 degrees, so we can use the Pythagorean theorem to find the resultant force:
Resultant Force = √(F₁² + F₂²) = √(67.425² + 67.425²) ≈ 95.2 N
The direction of this resultant force will be at a 45-degree angle away from the line connecting the two adjacent corner charges, directed towards the center of the square.
Summary of Results
In summary, the forces acting on the 1 × 10^-6 C charge are:
- At the center of the square: Resultant force of approximately 135.5 N directed towards the vacant corner.
- At the vacant corner: Resultant force of approximately 95.2 N directed at a 45-degree angle towards the center of the square.
This analysis illustrates how the arrangement of charges and their positions significantly influence the resultant forces acting on a charge in an electric field. Understanding these principles is crucial in fields such as electrostatics and electrical engineering.
