A 3.2 cm diameter plastic sphere is charged to -19.0 nC. What orbital frequency must a proton have to maintain a circular orbit around the sphere 2.4 mm above its surface?

To find the orbital frequency of a proton around the charged plastic sphere, we can use the principles of electrostatics and circular motion.

First, we need to calculate the electric field intensity due to the charged plastic sphere at a distance of 2.4 mm above its surface. The electric field intensity at a distance r from a charged sphere can be calculated using the formula:

E = k * |Q| / r^2

where:
- E is the electric field intensity
- k is Coulomb's constant (8.99 x 10^9 N m^2/C^2)
- |Q| is the magnitude of the charge on the sphere
- r is the distance from the center of the sphere to the proton's orbit

Given:
- Diameter of the plastic sphere = 3.2 cm, so radius (r_sphere) = 3.2 cm / 2 = 1.6 cm = 0.016 m
- Charge on the sphere (Q) = -19.0 nC = -19.0 x 10^-9 C
- Distance from the sphere's surface to the proton's orbit (r_proton) = 2.4 mm = 2.4 x 10^-3 m

Now, let's calculate the electric field intensity (E) at the proton's orbit:

E = (8.99 x 10^9 N m^2/C^2) * |-19.0 x 10^-9 C| / (0.016 m + 0.0024 m)^2

E = (8.99 x 10^9 N m^2/C^2) * (19.0 x 10^-9 C) / (0.0184 m)^2

E ≈ 815.8 N/C

Next, we need to find the force exerted on the proton (F) by the electric field. The force experienced by a charged particle in an electric field can be calculated using the formula:

F = q * E

where:
- F is the force
- q is the charge of the particle
- E is the electric field intensity

Given:
- Charge of the proton (q) = 1.6 x 10^-19 C

F = (1.6 x 10^-19 C) * 815.8 N/C

F ≈ 1.3 x 10^-16 N

Finally, to find the orbital frequency (f) of the proton, we can equate the gravitational force (F_gravitational) acting on the proton with the centripetal force (F_circular) required to maintain a circular orbit. The centripetal force is given by:

F_circular = (m * v^2) / r

where:
- F_circular is the centripetal force
- m is the mass of the proton
- v is the orbital velocity of the proton
- r is the radius of the proton's orbit

The gravitational force acting on the proton is given by:

F_gravitational = (G * m * M) / R^2

where:
- F_gravitational is the gravitational force
- G is the gravitational constant (6.67430 x 10^-11 N m^2/kg^2)
- m is the mass of the proton
- M is the mass of the plastic sphere
- R is the distance from the center of the plastic sphere to the proton's orbit

The gravitational force and the centripetal force are equal, so:

(G * m * M) / R^2 = (m * v^2) / r

Simplifying the equation, we get:

v = sqrt[(G * M * r) / R^2]

Given:
- Diameter of the plastic sphere = 3.2 cm, so radius (R) = 1.6 cm = 0.016 m
- Distance from the sphere's surface to the proton's orbit = 2.4 mm = 0.0024 m
- Mass of the proton (m) = 1.67 x 10^-27 kg
- Mass of the plastic sphere (M) = unknown

Before we can calculate the orbital frequency, we need to determine the mass of the plastic sphere. This information is missing from the given data, so we cannot determine the exact orbital frequency.

However, if you have the mass of the plastic sphere, you can substitute it into the equation above to calculate the orbital velocity (v), and then use the relation:

f = v / (2 * pi * r)

to calculate the orbital frequency (f) of the proton.