When an uncharged conducting sphere of radius a is placed at the origin of an xyz coordinate system that lies in an initially uniform electric fieldE = E, k, the resulting electric potential is V (x,y,z) = Vo for the points inside the sphere and V (x,y,z) = Vo – Egz + Ega³z (x² + y² + z²)3/2 for points outside the sphere, where V is the (constant) electric potential on the conductor. Use this equation to determine the x, y, and z components of the resulting electric field in the following regions. (Use the following as necessary: X, y, z, a, and Eo:)

A thin ring of radius R and uniformly distributed total charge q is located in the xy-plane with its center at the origin. (a) (5 stig) Show that the electric potential on the central axis (the z-axis) of a distance z from the rings center is given by, V = 1 q 4π€о √R² + 2² (b) (5 stig) Using this result, derive an expression for the electric field at any point along the z-axis.

An infinitely long metal cylinder has radius R0 and charge per unit length λ. It is held at potential V0, which you should use as the reference point for this problem. The cylinder is solid (not hollow) and in electrostatic equilibrium. (a) Find the electric potential outside the cylinder, for a distance r > R0 from the center of the cylinder. (b) Find the electric potential inside the cylinder, at a distance r < R0 from the center of the cylinder.

Ra1 +9 -9 Rea Consider two concentric spherical conductors, separated by an isolating material with (absolute) permittivity e. The two conductors have radius R1 and R2, they are put on a potential V and V2, which leads to a charge +q and –q sitting on them, respectively. By the problem's spherical symmetry, we see that the charge on each conductor is distributed uniformly, and that, in spherical coordinates, the electric field between the two conductors is of the form E(r) = -E(r) er. Determine the capacity C using the following steps: 1. Use Gauss's Law in integral form, with N a ball of radius r (R2 < r < R1), to find an expression for E(r) in terms of q. 2. Calculate AV = Vị – V2 using the formula - E•dr Δν and with C the black line segment indicated on the drawing (parallel with e,). 3. The capacity now follows from C = q/AV.