A point charge Q is embedded in a semi-infinite dielectric material, with dielectric constant ₁. It is located a distance d away from a planar interface that separates the first dielectric material from a second semi-infinite dielectric, with dielectric constant €2. Without loss of generality, the boundary between the two dielectric materials can be taken as the z = 0 plane, as shown in the figure. €1 d An adapted "Method of Images" technique can be used to find the electric potential V everywhere: (a) Boundary conditions: What conditions are placed upon the tangential and normal components of Ď (or Ĕ) at the interface between the two dielectric materials? (b) V(s, z>0): Replace the dielectric material €2 with an image charge Q' located a symmetrical distance d from the interface (dielectric material €₁ now fills all space). Using cylindrical coordinates, and giving your answer in terms of Q and Q', what is V(s, z > 0)? (c) V(s,<0): There are no free charges in the region z <0. Assume the potential there is equivalent to a charge Q" located at the position of the actual charge Q (and dielectric material €2 fills all space). Using cylindrical coordinates, and giving your answer in terms of Q", what is V(s, z <0)? (d) Using your boundary conditions for the normal and tangential components of Ẽ (-and-, respectively), show that: = 9-(2)9 €2 Q €2 Q 8-(0)9 Q" (e) Putting it all together, what is V(s, z) everywhere?

A point charge Q is embedded in a semi-infinite dielectric material, with dielectric constant ₁. It is located a distance d away from a planar interface that separates the first dielectric material from a second semi-infinite dielectric, with dielectric constant €2. Without loss of generality, the boundary between the two dielectric materials can be taken as the z = 0 plane, as shown in the figure. €1 d An adapted "Method of Images" technique can be used to find the electric potential V everywhere: (a) Boundary conditions: What conditions are placed upon the tangential and normal components of Ď (or Ĕ) at the interface between the two dielectric materials? (b) V(s, z>0): Replace the dielectric material €2 with an image charge Q' located a symmetrical distance d from the interface (dielectric material €₁ now fills all space). Using cylindrical coordinates, and giving your answer in terms of Q and Q', what is V(s, z > 0)? (c) V(s,<0): There are no free charges in the region z <0. Assume the potential there is equivalent to a charge Q" located at the position of the actual charge Q (and dielectric material €2 fills all space). Using cylindrical coordinates, and giving your answer in terms of Q", what is V(s, z <0)? (d) Using your boundary conditions for the normal and tangential components of Ẽ (-and-, respectively), show that: = 9-(2)9 €2 Q €2 Q 8-(0)9 Q" (e) Putting it all together, what is V(s, z) everywhere?