91Ó°ÊÓ

Achieving equilibrium involves a balance of forces where the net force acting on an object is zero. For particle B to stay in equilibrium on the inclined plane, the forces acting on it must be perfectly balanced. This concept is crucial in many physics problems, ensuring that the object remains stationary and doesn't accelerate.

• Gravitational Force Down the Incline: This force tries to pull particle B downwards along the slope of the incline. It's calculated using the equation: \[ F_{\text{gravity, down}} = mg \sin(\theta) \]
• Electrostatic Force: This is the force of repulsion between the two charged particles, A and B, trying to push them apart. It's determined using Coulomb's law.

To achieve equilibrium, these two forces must be equal in magnitude but opposite in direction. By setting these two forces equal, you derive the necessary condition to keep B stationary at a specified point on the incline.

Gravitational force acts on every mass and is directed towards the center of the Earth. When dealing with an object on an inclined plane, such as particle B, this force has two components:

• Parallel to the Incline: This component attempts to slide the object downslope. Calculated as: \[ F_{\text{gravity, down}} = mg \sin(\theta) \]
• Perpendicular to the Incline: Helps keep the object pressed against the surface and is calculated using: \[ mg \cos(\theta) \]

In our case, we focus on the parallel component since it directly influences equilibrium by counteracting the electrostatic force. For particle B, the gravitational component along the incline was calculated to be 0.49 N, which influences how far up the slope the particle will need to be positioned to remain in balance.

Coulomb's law describes the force of attraction or repulsion between two point charges. It is one of the fundamental principles used to understand interactions between charged objects.
This law tells us that the electrostatic force between two charges is:
- Directly proportional to the product of the magnitude of the charges- Inversely proportional to the square of the distance between them
The formula for calculating this force is:\[ F_{\text{electrostatic}} = \frac{k |q_1 q_2|}{r^2} \]where:

• \( k \) is Coulomb's constant \((8.99 \times 10^9 \text{ Nm}^2/\text{C}^2)\)
• \( q_1 \text{ and } q_2 \) are the charges in coulombs
• \( r \) is the distance between the charges

In the given problem, this formula helps calculate the electrostatic force that must balance the gravitational force component to achieve equilibrium. By equating this force to the gravitational force along the incline, we find the precise distance at which particle B needs to be placed.