A charged ball B hangs from a silk thread S, which makes an angle θ with a large charged conducting sheet P, as shown in the figure. The surface charge density σ of the sheet is proportional to

• A. sin θ
• B. tan θ
• C. cos θ
• D. cot θ

To determine the relationship between the surface charge density (σ) of the conducting sheet and the angle (θ) that the silk thread makes with the sheet, we need to analyze the forces acting on the charged ball B.

Forces Acting on the Charged Ball

The ball experiences two main forces:

• Gravitational Force (Weight): This acts downward and is equal to mg, where m is the mass of the ball and g is the acceleration due to gravity.
• Electrostatic Force: This force arises due to the electric field created by the charged conducting sheet.

Understanding the Angle θ

The angle θ affects the components of the forces acting on the ball. The electrostatic force can be expressed in relation to the angle:

• The vertical component of the tension in the thread balances the weight of the ball.
• The horizontal component of the tension relates to the electrostatic force acting on the ball.

Relationship Between σ and θ

The electric field (E) due to the charged sheet is related to the surface charge density by the equation:

E = σ / (2ε₀), where ε₀ is the permittivity of free space.

As the angle θ increases, the horizontal component of the tension increases, which is proportional to the electrostatic force. This leads us to conclude that:

Final Answer

The surface charge density σ is proportional to tan θ. Therefore, the correct answer is:

B. tan θ