This is a two-dimensional problem. Consider a square on the xy plane with vertices at (0,0), (a, 0), (a, a), (0, a). There is no charge or matter inside the square. The sides perpendicular to the x-axis have a potential equal to zero. The side at y = a has a constant potential φ(0), while the one at y = 0 has a constant potential at -φ. Find φ (x, y) for all points within the square.

You are working at a company that manufactures solenoids for industrial and research use. A client has ordered a solenoid that will be operated by a 1 000-V power supply and must be of length ℓ = 25.0 cm. A cylindrical experimental package of radius rs = 1.00 cm must fit inside the solenoid. The client wants the largest possible magnetic field inside the solenoid. The thinnest copper wires allowed by your company are AWG 36, which corresponds to a wire diameter of dω = 0.127 mm. You determine the maximum magnitude of magnetic field that can be created in the solenoid to report to the client.

P5 This figure depicts a cross-section of a resistor made out of a solid metal sphere of radius r by cutting off its “poles” and its schematic connection to the outside circuit (establishing the direction of the current flow). The resistivity of the metal is ρ. If the voltage drop across this resistor equals to V, what is the magnitude E of the driving electric field right at the center of the resistor?