Well, it sounds like you're in need of some clown assistance! Don't worry, I've got the answers to your confusing questions right here. Let's get started with the first one, shall we?
(a) To find the potential difference between the plates, we can use the formula C = Q/V, where C is the capacitance, Q is the charge, and V is the potential difference. Rearranging the formula, we can solve for V by dividing Q by C. So, V = Q/C. Plug in the given values of charge (0.200 uC) and capacitance (500.0 pF) and do the math to find the result. Just don't try to divide by a clown nose, it won't end well!
(b) Now, let's calculate the area of each plate. The formula to find the capacitance of a parallel plate capacitor is C = (ε₀ * A) / d, where C is the capacitance, ε₀ is the permittivity of free space, A is the area of each plate, and d is the distance between the plates. Rearranging the formula to solve for A, we get A = (C * d) / ε₀. Plug in the known values of capacitance (500.0 pF) and distance (0.6 mm), and the permittivity of free space is approximately 8.85 x 10^-12 F/m (F stands for Farads, not fail!). Now you can calculate the area, just don't clown around too much!
(c) Moving on to the electric field magnitude between the plates. The electric field can be found using the formula E = V / d, where E is the electric field, V is the potential difference, and d is the distance between the plates. Plug in the values you found in part (a) and (b) to calculate the electric field. But remember, be careful not to get shocked by the numbers!
(d) Lastly, let's figure out the surface charge density on each plate. The surface charge density (σ) can be calculated by dividing the charge (Q) by the area (A) of the plate. Remember the values you calculated in part (b)? Plug them into the formula σ = Q / A and you'll find the surface charge density. Just be careful to not let any charges zap you while doing the calculation!
I hope these quick explanations help you get started on solving the problem! If you need any further assistance, I'm here to clown around and help out. Good luck!