Mid-Term Exam. Electricity and Magnetism, Spring 2013

Mid-term exam. Electricity and Magnetism, Spring 2013.

Instructor: Elbio Dagotto. Feb. 19, 2013. Deadline: Feb. 26, 2013.

(1) Consider a grounded (i.e.  = 0) metallic sphere of radius a in the presence of two point charges located along the z axis (the origin of coordinates is at the center of the sphere). One of the charges is of magnitude q and it is located at position z = 2a and the other is of magnitude q’ and it is at position z= 3a. (a) Using the method of images find the value of q’ such that the whole system charges plus sphere has a net charge equal to zero. You can use results from the book regarding the case of just one charge in the presence of a metallic sphere. (b) Find the magnitude of the dipole moment p created by this distribution of charge when observed at long distances along the z axis. Carry out the calculation for the value of q’ found in (a). (c) Calculate the surface charge density . Again here you can use results from the book. If q is positive, is  positive or negative?

(2) Consider a sphere of radius a with electric potential fixed at the surface with value V()=V0 cos  , where  is the angle away from the z axis in spherical coordinates. (a) Using the method of separation of variables find the electric potential all over space outside the sphere. (b) Using the method of Green functions repeat the calculation. You are allowed to use the Green function suitable for this problem from the book. Leave the result expressed as an integral. (c) Considering the large r limit, find the dominant term in the potential found in (b) and check that it is equal to the result in (a). (d) From the intuition developed in the calculation in item (a), what angular dependence should the potential V() have in order to induce only a quadrupole potential in all space outside the sphere?

(3) An electric point dipole of magnitude p pointing along the z axis is at the center of a sphere of a linear dielectric material (with radius a and dielectric constant . (a) Find the electric potential  inside and outside the sphere. (b) From your results in (a), check that the potential is continuous at the surface of the sphere. (c) Sketch the lines of electric field of the potential outside the sphere. (d) Calculate the polarization P inside the sphere. (e) Calculate pol, namely the surface charge density induced by P, at the positions z=a and z=-a along the z-axis, and from this information qualitatively sketch polat the surface of the sphere. Is the result compatible with the sketch in (c)?