A total charge of Q is deposited on a solid metal sphere of radius R. This charge distributes itself uniformly over the surface, yielding an electrostatic field E~ perpendicular =(sigma / e0) n^ just outside the sphere and E~ = 0 everywhere inside the sphere, where  sigma = Q=4 pie R^2 and ^n is the outward pointing unit normal vector. Consider a small patch of surface area dA. We wish to calculate the electrostatic pressure P on this patch due to all the other surface charge. This pressure equals the electrostatic force per unit area, P = (dq)E=dA = sigma E where E is the electrostatic field acting on the patch due to all the other charges. But what value of E do we use? (The total field is zero just inside the surface, but jumps to a nonzero value just outside the surface.) This problem addresses that question. a) Consider the small patch alone. What are the fields ~E ~patch that the patch produces just outside and just inside the sphere? (Write these fields as vectors involving  sigma and ^n.) Hint: If you are infinitesimally close to the patch, even this small patch will look like an infinite sheet of charge. b) According to the superposition principle, the total field just outside the surface must equal a contribution from the patch itself plus a contribution ~E ~other from all the other charge. What is ~E~other just outside the surface? (Again, write the result in terms of  sigma and ^n.) c) In a similar manner, use the superposition principle to calculate the field ~E~other just inside the surface. d) Is ~E~other continuous in both magnitude and direction as you move through the surface? e) What is the electrostatic pressure P on the patch due to all the other charges, in terms of  sigma and e0? f) The largest E-fields that can be produced at the surface of a conductor are approximately 250 X 10^6 N/C. Beyond this value the metal breaks down due to field emission. How much charge would have to be deposited on a sphere of copper 1 cm in radius to produce a field of this magnitude? What would the resulting electrostatic pressure be, in both N/m^2 and atmospheres? (1 atmosphere =101,325 N/m^2.)