To find the angle of the thread with the vertical when particle B is in equilibrium, we need to analyze the forces acting on particle B due to its charge and weight. Since both particles A and B have the same charge, they will repel each other, creating a force that affects the angle of the thread. Let's break this down step by step.
Understanding the Forces at Play
When particle B is suspended, it experiences two main forces:
- Gravitational Force (Weight): This acts downward and can be calculated using the formula W = mg, where m is the mass of particle B and g is the acceleration due to gravity (approximately 9.81 m/s²).
- Electrostatic Force: This is the repulsive force between the two charged particles, which can be calculated using Coulomb's Law: F = k * |q1 * q2| / r², where k is Coulomb's constant (approximately 8.99 x 10^9 N m²/C²), q1 and q2 are the charges, and r is the distance between the two charges.
Calculating the Gravitational Force
First, let's calculate the weight of particle B:
Given:
- Mass of particle B, m = 100 g = 0.1 kg
- Acceleration due to gravity, g = 9.81 m/s²
Now, using the formula for weight:
W = mg = 0.1 kg * 9.81 m/s² = 0.981 N
Calculating the Electrostatic Force
Next, we need to find the electrostatic force between particles A and B. Both particles have a charge of q = 5.0 x 10^-7 C. The distance r between them is equal to the length of the thread, which is 30 cm or 0.3 m.
Using Coulomb's Law:
F = k * |q1 * q2| / r²
Substituting the values:
F = (8.99 x 10^9 N m²/C²) * (5.0 x 10^-7 C * 5.0 x 10^-7 C) / (0.3 m)²
Calculating this gives:
F = (8.99 x 10^9) * (25 x 10^{-14}) / 0.09
F = (8.99 x 10^9 * 25 x 10^{-14}) / 0.09 = 2.49 x 10^{-3} N
Finding the Angle of the Thread
In equilibrium, the forces acting on particle B must balance. The gravitational force acts vertically downward, while the electrostatic force acts horizontally away from particle A. The angle θ can be found using trigonometry:
Using the tangent function:
tan(θ) = F / W
Substituting the forces we calculated:
tan(θ) = (2.49 x 10^{-3} N) / (0.981 N)
Calculating this gives:
tan(θ) ≈ 0.00254
Now, to find the angle θ, we take the arctangent:
θ = arctan(0.00254)
Using a calculator, we find:
θ ≈ 0.145°
Final Thoughts
The angle of the thread with the vertical when particle B is in equilibrium is approximately 0.145°. This small angle indicates that the gravitational force is much greater than the electrostatic force, which is typical in scenarios where the mass of the object is significant compared to the forces due to electric charges.
