# Please solve all the problems with detailed solution.

Please solve all the problems with detailed solution.

Homework 1. TOTAL 120 pts 1. A sphere of radius a is located at the back corner of a cube (length of each side L) in air as shown in the figure. The center of the sphere coincides with one corner of the cube at (0,0,0). Assume a total charge of q is uniformly distributed on the sphere, calculate the total flux of through the shaded side. (15 pts) 2. The diagram below shows some of the equipotential lines in a plane perpendicular to two parallel charged metal cylinders. The potential of each line is labeled. (a) If the left cylinder is positively charged, determine the sign of charge of the right cylinder. (2 pts) (b) Sketch the electric field lines produced by the charged cylinders, you have to also include the equipotential lines in your plot. (2 pts) (c) Determine the potential difference VA-VB, between points A and B (3 pts) (d) Calculate the work done by the electric field if a charge of 1 coulomb is moved along a path from point A to point C then to point B and finally to point D. (3 pts) 3. Given a uniform sphere of charge of radius b and volume density  in air; (a) Find the electric field intensity in the regions R>b and Rb and R>a and S>>b). Given uniform line charge density on both conductors to be l , find the capacitance per unit length of the structure. (10 pts) x Q Q -Q -Q 2L 2L x = 0 x Q Q -Q -Q 2L 2L x = 0 x a S x b + - P x a S x b + - P d 7. A capacitor is formed from a segment of two coaxial cylinders as shown in the figure below, neglect the fringing field at the edge of the plates, (a) Using Cylindrical Coordinates, at z = 0 plane, draw the electric field lines and equipotential lines within the dielectric region (use arrows to indicate the proper directions whenever necessary). (4 pts) (b) Express the potential Vo in terms of the electric field inside the region. (6 pts) (c) Applying boundary condition at   plate, express the surface charge density s of the corresponding surface. (6 pts) (d) Integrate the surface charge density you obtained in (c) to find the total charge Q induced on the plate. (8 pts) (e) Calculate the capacitance of the structure. (8 pts) (Cartesian) (Cylindrical) (Spherical) sin x y z r z R V V V V a a a x y z V V V a a a R r z V V V a a a R R R                                    2 2 (Cartesian) 1 ( ) (Cylindrical) 1 1 1 ( ) ( sin ) (Spherical) sin sin yx z z r R AA A A x y z A A rA r r r z A R A A R R R R                                    2 ( ) ( ) ( ) (Cartesian) 1 (Cylindrical) sin 1 (Spherical) sin sin y yx xz z x y z r z r z R R A AA AA A A a a a y z z x x y a a r a r r z A A A a a R a R R R A RA R A                                           