Consider an infinite cylinder of radius R as shown below with surface charge den- sity o(6) = a sin 5ộ, where a is a constant. Find the potential inside and outside the cylinder.
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A: All the answers are given in the explanation part.Explanation:Step 1: Step 2: Step 3: Step 4:
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Q: potential at a distance R from its center.
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- 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. %3DGiven 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/). A capacitor consists of two concentric spherical shells (inner radius a and outer radius b). The inner shell is at a potential of Vo and the outer shell is grounded. The dielectric between the two shells has the permittivity of E. (b) Calculate the surface charge density on the inner shell. Vo a O HI
- Please asapThe 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 1/²Some practice with an infinite line charge. (a) What is the electric field on an infinite line charge that has a constant charger per unit length of X? (b) What is the electrostatic potential in this case? Is it possible to make the potential vanish at infinity? (c) Consider an infinite conducting cylinder of radius a that is grounded. Suppose there is an infinite line charge with a constant charge per unity length that is located inside the cylinder, such that it is parallel to the cylinder, but displaced from the central axis by an amount b, with b < a. Determine the electrostatic potential everywhere inside the cylinder. I suggest using the method of images, by placing another line charge outside the cylinder.