A small sphere of mass m and charge +q with zero velocity falls under the influence of gravity (acceleration = g from height h onto an infinite uniformly charged plane with positive charge density sigma as shown below. What will the velocity of the sphere be when it hits the plane? Note that the electric field above an infinite plane of charge is constant everywhere, pointing away from the plane (for a positive charged plane) and equal to sigma/2*episilon . make the downward direction positive (acceleration due to gravity positive) and the repuslive electrostatic force up the negative direction

Question

A small sphere of mass m and charge +q with zero velocity falls under the influence of gravity (acceleration = g from height h onto an infinite uniformly charged plane with positive charge density sigma as shown below. What will the velocity of the sphere be when it hits the plane? Note that the electric field above an infinite plane of charge is constant everywhere, pointing away from the plane (for a positive charged plane) and equal to sigma/2*episilon . make the downward direction positive (acceleration due to gravity positive) and the repuslive electrostatic force up the negative direction

Answer 1

To determine the velocity of the sphere when it hits the plane, we need to consider the forces acting on it.

1. Gravitational Force: The sphere experiences a downward force due to gravity. The magnitude of the gravitational force is given by F_gravity = m * g, where m is the mass of the sphere and g is the acceleration due to gravity (positive value in the downward direction).

2. Electrostatic Force: The sphere is also subject to an upward electrostatic force due to the charged plane. The magnitude of the electrostatic force is given by F_electrostatic = q * E, where q is the charge of the sphere and E is the electric field above the charged plane. For a positively charged plane, the electric field points away from the plane and has a magnitude of E = sigma / (2 * epsilon), where sigma is the charge density of the plane and epsilon is the permittivity of the medium.

Since the electrostatic force and gravitational force act in opposite directions, they will balance each other when the sphere reaches a terminal velocity. This occurs when the net force on the sphere becomes zero.

Setting the gravitational force equal to the electrostatic force, we have:

m * g = q * E

Rearranging the equation, we get:

q * E = m * g

Now, let's solve for the velocity of the sphere using kinematic equations:

1. Initial velocity (u) = 0 (sphere is initially at rest)
2. Final velocity (v) = ?
3. Acceleration (a) = g (acceleration due to gravity is acting)

Using the equation v^2 = u^2 + 2 * a * s, where s is the distance fallen (h), we have:

v^2 = 0^2 + 2 * g * h
v^2 = 2gh

To find v, we take the square root of both sides:

v = sqrt(2gh)

Therefore, the velocity of the sphere when it hits the plane is v = sqrt(2gh).