a small sphere whose m mass is 1.0 mg carries a charge q of 2.0*10^(-8)C. It hangs from a silk thread that makes an angle of 30
degrees with a large charged nonconducting sheet as in figure. Calculate the uniform charge density σ for the sheet
1. A small sphere carries a charge of magnitude -32 µC , ( a ) Find the number of electrons forming this charge , and ( b ) Is this number need to be gained or lost ?
To calculate the uniform charge density (σ) for the sheet, we need to consider the electrostatic equilibrium of the small sphere hanging from the silk thread.
In this case, the electrostatic force acting on the small sphere is balanced by the gravitational force and the tension in the silk thread.
Let's break down the problem step-by-step:
1. Start by drawing a free body diagram of the small sphere. Label the forces acting on it.
- The weight of the small sphere (mg) acts vertically downward.
- The electrostatic force (Fe) is attractive and acts along the line of the electric field.
- The tension in the silk thread (T) acts along the thread.
2. Find the components of the electrostatic force.
- The electrostatic force (Fe) can be resolved into two components:
- Fe⊥, the component perpendicular to the sheet.
- Fe∥, the component parallel to the sheet.
3. Calculate the components of the electrostatic force.
- Fe⊥ is balanced by the perpendicular component of the tension force (T⊥), using the equation T⊥ = Fe⊥.
- Fe∥ has no balancing force since the thread is insulating and cannot exert a force in that direction.
4. Calculate the tension force.
- In the equilibrium condition, the tension force (T) is equal to the sum of the components:
- T⊥ = mg
- T∥ = Fe∥
5. Calculate the force component.
- The electrostatic force component parallel to the sheet (Fe∥) can be calculated using Coulomb's law:
- Fe∥ = k * q1 * q2 / r^2, where k is the electrostatic constant, q1 and q2 are the charges, and r is the distance between them.
6. Calculate the charge density (σ).
- The charge density (σ) is defined as σ = Q / A, where Q is the total charge on the sheet and A is the area.
- The total charge on the sheet is given by Q = σ * A, where σ is the uniform charge density and A is the area of the sheet.
By following these steps and plugging in the given values (mass m, charge q, angle θ), you can calculate the uniform charge density σ for the sheet.