A sphere with radius a is centered on the origin and is held at a fixed electric potential 2 V cos * (O) where o is the azimuthal angle of spherical coordinates. Find the potential everywhere external to the sphere starting with the series solution to the Laplace equation.

Consider the six surfaces of a cubic cavity with side length L. One of the surface is maintained at a constant voltage Vo, while the other five surfaces are maintained at a zero voltage. Find the electric potential inside the cavity.

A dielectric sphere in an external field. Consider a simple dielec- tric with permittivity e, in the form of a uniform spherical ball of radius a. It is placed at the origin in an external electrostatic potential (x, y, z) = bxy (where r, y, z are Cartesian coordinates and b is a constant). Find the elec- trostatic potential o and electric field E everywhere. %3D

Consider a circular arc of constant linear charge density A as shown below. What is the potential V, at the origin O due to this arc?

Given an electric potential of   find its corresponding electric field vector. Sol. Using the concept of potential gradient, we have   Since we only have, radial direction (r). Then, the del-operator will be   By substitution, we get   Evaluating the differential, we, get the following  *(q/)

Show Your Solutions In free space, there are 2 concentric spherical surfaces with radii a and b. There is a constant volume charge density between 2 surfaces such that: (0; p=Po; aspsb (0; p>b pca a) BY USING POISSON'S EQUATION find the electric potential everywhere. b) BY USING POISSON'S EQUATION find the electric field vector everywhere

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The potential of a thin spherical shell of radius R is given as V(R, 0) = 3 cos² 0 + cos 0 - 1. Both inside and outside the sphere, there is empty space with no charge density. The questions on this page are based on this system. What is the linear combination of the two Legendre polynomials that will generate this potential (Denote a Legendre Polynomial as P₁, where I is the index of the polynomial starting from 1 = 0)? O a. 2P₁ + P₂ O b. 2P3 + P₁ O c. Po + P₁ O d. None of the listed answers. O e. 2P₂ + P1₁ Of. P3 + P2 What is the radial part of the potential V(r, 0) inside the spherical shell for the Pi with the lowest l (i.e. the pre-factor of Pi)? O a. r O b. None of the listed options. O c. r/R O d. 2 O e. O f. (r/R)² 1/²