Askiitians Tutor Team
Last Activity: 4 Months ago
To analyze the situation of a charged ball hanging from a silk thread near a positively charged conducting sheet, we need to consider the forces acting on the ball and how they relate to the angle θ formed by the thread. The key to solving this problem lies in understanding the relationship between the electric field produced by the charged sheet and the forces acting on the ball.
Understanding the Forces at Play
When the charged ball is suspended, it experiences two main forces: the gravitational force acting downward and the electric force acting due to the electric field created by the positively charged conducting sheet. The electric field (E) produced by an infinite sheet of charge with surface charge density σ is given by:
E = σ / (2ε₀)
where ε₀ is the permittivity of free space. This electric field points away from the sheet since it is positively charged.
Analyzing the Geometry
As the ball hangs at an angle θ, we can break down the forces into components. The tension in the thread provides a force that can be resolved into two components: one that balances the gravitational force and another that balances the electric force. The tension (T) can be expressed as:
- T cos(θ) balances the gravitational force (mg).
- T sin(θ) balances the electric force (qE).
Establishing the Relationship
From the balance of forces, we can write:
T sin(θ) = qE
T cos(θ) = mg
Dividing these two equations gives:
tan(θ) = (qE) / (mg)
Substituting the expression for the electric field, we have:
tan(θ) = (q(σ / (2ε₀))) / (mg)
Finding the Surface Charge Density
Rearranging this equation allows us to express the surface charge density σ in terms of the angle θ:
σ = (2ε₀mg / q) tan(θ)
This shows that the surface charge density σ is directly proportional to tan(θ). Thus, the correct answer to your question is option 4: tan(θ).
Why the Other Options Are Incorrect
- 1. cos(θ): This would imply that the surface charge density decreases as the angle increases, which contradicts the observed behavior of the system where an increase in angle leads to a greater electric force and thus a greater charge density.
- 2. cot(θ): This is the reciprocal of tan(θ) and would suggest that as the angle increases, the charge density decreases, which is not consistent with our derived relationship.
- 3. sin(θ): This option would imply a linear relationship with the vertical component of the tension, which does not account for the electric field's influence effectively. The electric force depends on the tangent of the angle, not the sine.
In summary, the relationship between the angle θ and the surface charge density σ is best described by the tangent function, making option 4 the correct choice. Understanding these relationships helps clarify how electric fields interact with charged objects in various configurations.
