5. Dipole Moment on PEC Sphere (30 pts): Consider a perfectly conducting spherical ball centered at the origin of a spherical coordinate system. The sphere has radius a, and is

immersed in an external electric field given by Eext =Eext Z. a) Find an expression for the external potential Vext associated with the external electric field using spherical coordinates. (*Hint: First do this in Cartesian coordinates and then convert to spherical coordinates). b) Assume the total potential is the sum of the external and induced potentials V =Vext + Vind. Write down the differential equation that governs this potential and the associate boundary conditions (*Hint: There is no o dependence and the tangential component of the electric field must be 0 on the surface of the sphere). \text { c) Show that if } V_{\text {ind }}=\frac{A \cos (\theta)}{r^{2}}, \text { the total potential satisfies the boundary } conditions and also satisfies Laplace's equation. What is the value of A? d) Find an expression for the total electric field (Eext + Eind). Show that the total field is the sum of the external field plus the fields due to an electric dipole. What is the electric dipole moment p? (*Hint: See expression of electric dipole fields from HW 3). e) Sketch the electric field lines outside of the sphere.

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