Electrostatics

The charge of one electron is .
When electrons are removed from a neutral metal plate, the charge on the plate becomes positive because the removal of negatively charged particles leads to an excess of positive charge.
The total charge removed from the metal plate can be calculated by: where:
- is the total charge,
- is the number of electrons removed,
- is the charge of one electron. Substituting the values:
Thus, the charge on the plate is . Therefore, the correct answer is:

The surface charge density of a sphere is defined as: Where is the charge and is the radius. Initially, the two spheres have the same surface charge density, so we have: Let and be the charges on the spheres initially. Since the surface charge densities are the same: Simplifying this: So, . When the spheres are brought into contact, the total charge is shared between the two spheres. The charge will distribute according to their radii:
Thus, the new surface charge densities are: Therefore, the ratio of the new surface charge densities is: Thus, the new surface charge density ratio is 4 : 1.

The problem involves three charges: two charges and one charge , with the charge displaced slightly from the equilibrium position.
The forces acting on the charge due to the two fixed charges will create an electric field, and if we displace the charge by a small distance , it will experience a restoring force.
We calculate the force acting on the charge due to the electric field of the fixed charges. Using Coulomb’s law and the approximation for small displacements, the force on is: Where is the effective spring constant related to the electric force.
For the oscillations to occur, the time period of the charge is given by: Since is related to the charges and the separation distance, the expression for becomes: Thus, the time period of oscillation for the charge is .