An electrostatic paint sprayer has a 0.200-m-diameter metal sphere

at a potential of 25.0 kV that repels paint droplets onto a grounded object.
What charge must a 0.100-mg
drop of paint have to arrive at the object with a speed of 10.0 m/s?

To find the charge on the paint droplet, we can use the principle of conservation of energy. The electric potential energy gained by the charged droplet as it moves from the metal sphere to the grounded object will be converted into kinetic energy.

The electric potential energy gained by the droplet can be calculated using the formula:

Potential energy = charge x potential difference

In this case, the potential energy gained is equal to the kinetic energy of the droplet when it reaches the grounded object. The kinetic energy can be calculated using the formula:

Kinetic energy = 0.5 x mass x velocity^2

Now, let's use the given information to calculate the charge on the paint droplet.

1. Convert the diameter of the metal sphere to radius:
Radius = diameter / 2 = 0.200 m / 2 = 0.100 m

2. Calculate the electric potential energy gained by the droplet:
Potential difference = 25.0 kV = 25,000 V
Potential energy = charge x potential difference

3. Calculate the kinetic energy of the droplet:
Mass = 0.100 mg = 0.100 × 10^(-3) g = 0.100 × 10^(-9) kg
Velocity = 10.0 m/s
Kinetic energy = 0.5 x mass x velocity^2

4. Equate the potential energy gained by the droplet to its kinetic energy:
Potential energy = Kinetic energy

Now, by setting up the equation and solving for charge, we can determine the charge on the paint droplet.