Question

SEPARATION OF VARIABLES - SPHERICAL The potential on the surface of a sphere (radius R) is given by V=V0 cos(2). (Assume V(r=[infinity])=0. Also assume there is no charge in or outside, it's ALL on the surface!) a) Find the potential everywhere inside and outside this sphere. (Hint: Can you express cos(2) as a simple linear combination of Legendre polynomials? ) b) Find the charge density () on the surface of the sphere in part a. Sketch or draw it!

Answer 1

To find the potential everywhere inside and outside the sphere, we need to express cos(2θ) as a linear combination of Legendre polynomials. By substituting this expression for cos(2θ) into the given potential function, we can obtain the potential everywhere inside and outside the sphere.

Step-by-step explanation:

To find the potential everywhere inside and outside the sphere, we need to express cos(2θ) as a linear combination of Legendre polynomials. We can do this by using the Legendre polynomial expansion of cos(θ), which is given by:

cos(θ) = P₀(cos(θ)) + 2P₂(cos(θ)) + 2P₄(cos(θ)) + ...

Using this expansion, we can express cos(2θ) as:

cos(2θ) = 2cos²(θ) - 1 = 2[P₀(cos(θ)) + P₂(cos(θ)) + P₄(cos(θ)) + ...]² - 1.

By substituting this expression for cos(2θ) into the given potential function V(r,θ) = V₀cos(2θ), we can obtain the potential everywhere inside and outside the sphere.

Answer 2

a) Vin(r,θ)=(-Vo/3)+(4Vo/3R^2)r^2P2cosθ

b)σ(θ)=((Voεo)/3R)(20P2cosθ-1)

Step-by-step explanation:

Due to the complex variables used to solve this exercise, the solution and the explanation are in the image