Final answer:
The charge on the sphere is 111 nC, and the charge on the paint droplet is 20 μC. the correct answer is d) (12.5 mC).
Step-by-step explanation:
The charge on the metal sphere can be calculated using the formula Q=CV, where Q is the charge, C is the capacitance, and V is the potential difference. Here, the diameter of the sphere is given as 0.200 m, so the radius is 0.100 m. The capacitance of a sphere is given by C=4πε0R, where ε0 is the permittivity of free space (8.854×10^-12 F/m) and R is the radius of the sphere.
Plugging in the values, the capacitance comes out to be 4.43×10^-12 F. Now, using the formula Q=CV, the charge on the sphere is given by Q=(4.43×10^-12 F)×(25.0 kV) = 1.11×10^-7 C, which can be written as 111 nC. So, the answer to part (a) is 111 nC.
To find the charge on the paint droplet, we can use the equation qV = 1/2 mv^2, where q is the charge of the droplet, V is the potential difference, m is the mass of the droplet, and v is its speed. Here, the mass of the droplet is given as 0.100 mg, which is equal to 0.100×10^-6 kg.
The speed is given as 10.0 m/s, and the potential difference is the same as before, 25.0 kV. Plugging these values into the equation, we get q = (1/2)(0.100×10^-6 kg)(10.0 m/s) / (25.0 kV) = 2.0×10^-8 C = 20 μC. So, the answer to part (b) is 20 μC.