Potential Inside and Outside of a Charged Spherical Shell

jkthejetplane · Apr 3, 2021

Typed problem statement: a spherical shell of radius R with surface charge density σ0 on its northern hemisphere and zero charge on its southern hemisphere; find the potential inside and outside, in terms of A, B, r and P_ℓ(cos θ), up to ℓ = 6 (20 points).

So here was my first go around at it:

Handwritten notes titled Hmwk 14, problem 3: a sphere of radius R with the top hemisphere at surface charge sigma naught and the bottom hemisphere at zero, followed by Legendre expansions of the potential inside and outside and the monopole-like integral for V naught.

At first it made sense in my head but don't think my process is correct. Then i noticed the example in the book:

Printed textbook page showing Example 3.9 on finding the potential inside and outside a spherical shell with surface charge density, with the solution's expansions in Legendre polynomials for interior and exterior regions (Eqs. 3.78 and 3.79).

Printed textbook page with equations 3.80 to 3.86 on Legendre polynomial expansion of potential, showing coefficient relations B_l = A_l R^(2l+1) and the Fourier-trick formula for A_l.

Scanned textbook page showing the potential outside a sphere, V(r, θ) = kR³/(3ε₀ r²) cos θ for r ≥ R, equation (3.87), followed by the induced-charge result E₀R³/r² cos θ.

I guess the reasoning isn't 100% there in my head and if i don't have an actual σ, how will i cancel out any legendre polynomials due to orthogonality?

kuruman · Apr 5, 2021

jkthejetplane said:

Homework Statement:: I need help on this question that i thought i understood but i really dont. A semi similar example in the book started to spark some brain juice but i ended up more confused than when i started.
Relevant Equations:: Legendre polynomials

View attachment 280846
So here was my first go around at it:
View attachment 280849
At first it made sense in my head but don't think my process is correct. Then i noticed the example in the book:
View attachment 280850
View attachment 280851
View attachment 280852
I guess the reasoning isn't 100% there in my head and if i don't have an actual σ, how will i cancel out any legendre polynomials due to orthogonality?

You cannot cancel any Legendre polynomials without knowing σ(θ). Take two simple cases that you can do in your head.
1. Uniform density over the sphere implies a 1/r potential and only P₀ is non-zero.
2. σ(θ)=σ₀cosθ implies a 1/r² potential and only P₁ is non-zero.

In short, if you don't know the surface charge distribution and you don't know the potential, you don't know nuttin'.

vanhees71 · Apr 6, 2021

But you know the surface-charge distribution,
$$\sigma(\vartheta)=\sigma_0 \Theta(0 \leq \vartheta \leq \pi/2).$$
Now you can just use the solution for the coefficients posted in #1.

Potential Inside and Outside of a Charged Spherical Shell

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