1-What is the shape of the trajectory of a charged particle moving in a uniform electric field if its initial velocity is null?

2-What is the shape of the trajectory of a charged particle moving in a uniform electric field if its initial velocity is perpendicular to the field?
3-What is the expression of the acceleration of a charged particle of mass m and charge q moving between two parallel plates a distance d apart between which a potential difference U is applied?

1- If a charged particle has an initial velocity of null (zero), and it moves in a uniform electric field, the shape of its trajectory will be a straight line. In this case, the electric field will accelerate the particle, causing it to gain momentum and move in a straight line. The force acting on the particle will be proportional to its charge and the electric field strength.

To get this answer, you can utilize the knowledge of the behavior of charged particles in electric fields. By understanding that an electric field exerts a force on a charged particle, you can deduce that a null initial velocity will result in the particle moving in a straight line.

2- If a charged particle has an initial velocity perpendicular to a uniform electric field, the shape of its trajectory will be a semicircle or a part of a circle. In this scenario, the electric field will cause a force on the charged particle, resulting in a curved path.

To arrive at this conclusion, you can consider the fact that a particle moving perpendicular to an electric field will experience a force that is always perpendicular to its velocity. This perpendicular force will cause the particle's path to curve, resulting in a semicircular or circular trajectory.

3- The expression for the acceleration (a) of a charged particle moving between two parallel plates with a potential difference (U) applied can be found using the equation:

a = (q * U) / (m * d)

where:
- a is the acceleration of the charged particle
- q is the charge of the particle
- m is the mass of the particle
- U is the potential difference applied across the plates
- d is the distance between the plates.

To obtain this expression, you can use the formula for the electric field between the charged plates, which is given by E = U/d. Then, you can use Newton's second law of motion, F = ma, where the force exerted on the particle is F = qE. By substituting F = qE into F = ma and rearranging the equation, you will arrive at the expression for acceleration.